REPUBLIQUE DU SENEGAL.
UNIVERSITE CHEIKH ANTA DIOP DE DAKAR
THESE DE DOCTORAT D'ETAT DE MATHEMATIQUES
'-----' ""'-----'.~-·=',-_·~·,.-",.,.---·----l
:ONSE!l AF:R!CAIN ET MALGACHE \\
SOUTENUE
'OUR L'ENSEIGNEMENT SUPERiEUR 1
:, i\\. tri:.E. S. -
OUAG,b,DOUGOU !
,{,':"/08 O·5·FfV;·199g·· .. ··· ·JarMonsieur Gane Samb Lô
:i1regi.stré sous n° #·0·2 '3~Q'4 JI
....--
----
en vue d'obtenir le grade de DOCTEUR ES SCIENCES MATHEMATIQUES
SUJET DE LA THESE
CARACTERISATION EMPIRIQUE DES EXTREMES
ET QUESTIONS STATISTIQUES LIEES
Le 6 Decembre 1991
devant la commission composée de:
Professeur Galaye Dia, Président Rapporteur
Professeur Paul Deheuvels, Membre Correspondant de l'Académie
Française, Rapporteur
Professeur Michel Broniatowski, Université de Lille VI, rapporteur
Professeur Hamet Seydi, Université de Dakar

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JE DEDIE CE HUMBLE TRAvAIL A MES PARENTS
ET AMIS, ENSUITE A MA TENDRE ET PATIENTE
EPOUSE, MA COMPLICE EN TOUTE CHOSE.

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REMERCIEMENTS
Je remercie d'abord le Professeur Paul Deheuvels pour l'intêret qu'il a toujours
porté à nos travaux depuis sept ans malgré ses multiples et riches occupations.
Il a.rd'emblée accepté de se déplacer jusqu'à Dakar pour faire partie du jury de
cette soutenance.
Mes remerciements sincères vont aussi au Professeur Michel
Broniatowski pour les ~êmes raisons. Sa courtoisie m'a beaucoup rassuré
quand je fus étudiant en thèse au LSTA de Paris 6.
\\
Que dire de mes anciens professeurs depuis la première année à
l'Université de Dakar, les professeurs Hamet Seydi et Sakhir Thiam.
Le premier mathématicien émérite, a toujours été affable à mon égard.
Il vient encore de le prouver en facilitant l'organisation matérielle de cette
soutenance.
Le second a personnellement organisé mon voyage vers la France après
ma maîtrise. Ce travail n'aurait donc jamais eu lieu sans lui. Qu'ils trouvent
ici, tous les deux, l'expression de ma tès sincère gratitude.
Enfin, "the last but not the least", le professeur Galaye Dia m'a toujours
encouragé, depuis que nous étions ensemble au LSTA, lui préparant une thèse
d'Etat et moi une thèse nouveau régime. Il n'a ménagé aucun effort pour que
cette soutenance soit une réussite. Je lui témoigne ici ma reconnaissance.
Pour terminer, je remercie notre secrétaire de recherches, Mlle Rokhaya SaIT,
pour la compétence avec laquelle elle a tappé une partie de cette thèse.
Gane Samb Lô
Saint-Louis, le 15 Novembre 1991

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SOMMAIRE
TITRE
PAGE
INTRODUCTION GENERALE
1
PREM'IERE PARTIE:
6
CONTRIBUTION A I:lINFERENCE DE L'INDEX D'UNE LOI DE PARETO.
ASYMPTOTIC BEVAVIOR OF HILL'S ESTIMA TE AND APPLICATIONS
6
ON THE ASYMPTOTIC NORMALITY OF SUMS OF EXTREME VALUES
21
\\
.
1
THE WEAK LIMITING BEHA VIOR OF THE DE HAAN-RESNICK
30
ESTIMA TE FOR THE EXPONENT OF A STABLE DISTRIBUTION.
APPENDIX: RESUME DE LA QUESTION DE LA DISTRIBUTION DES
SOMMES DE VALEURS EXTREMES. (Par D. Masan et al.)
44
DEUXIEME PARTIE.
46
CARACTERISATION EMPIRIQUE DES EXTREMES.
46
EMPIRICAL CHARACTERIZATION OF THE EXTREMES:
46
1:
A FA MIL Y OF CHARACTERIZING STATISTICS.
46
Il: THE ASYMPTOTIC NORMALITY OF THE CHARACTERIZING
64
VECTORS
III: THE LAWS OF THE ITERATED LOGARITHM OF THE
CHARACTERIZING STATISTICS.
113
A GAUSSIAN PROCESS LIMIT OF SUM-PRODUCT STATISTICS BASED
ON EXTREME VALUES
147
TROISIEME PARTIE.
168
CONTRIBUTION A L'ETUDE DES ESPACEMENTS.
1- GAUSSIAN APPROXIMATION AND RELATED QUESTIONS FOR THE
SPACINGS PROCESS
168
II - ON THE INCREMENTS OF THE EMPIRICAL K-SPACINGS PROCESS fOR
AFIXED OR MOVING STEP.
186

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QUATRIEME PARTIE.
STATISTIQUES D'ORDRE DANS LES ESPACES LATTICES
MAJORATION. MINORATION ET CONVERGENCE FORTE DES QUANTILES
CENTRALES ET EXTREMES DE VARIABLES ALEATOIRES DANS DES ESPACES
DE TYPE C(S).
203
FIN
INTRODUCTION GENERALE
Cette thése regroupe un ensemble de travaux dontle lien évident est la statistique
d'ordre. Toutes les recherches dont les résultats sont exposés ici nous ont été
proposées ou, ont germé dans notre esprit lorsque nous étions dans le
La.boratoire de Statistique Théorique et Appliquées, (LSTA) de l'Université de
Paris 6 où j'étais étudiant \\en. thèse entre 1983 et 1986. En ce moment,\\toutes
les questions relatives aux statistiques d'ordre, étaient systématiquement
étudiées soit par les professeurs Geffroy et Deheuvels soit par les membres de
leurs équipes. Je voudrais relater ici les circonstanc~s dans les quelles j'ai eu à
m'intéresser à chacun des travaux exposés ici dans ce
document. Ceci nous
permettra aussi de classer les travaux dans un ordre chronologique.
PREMIERE PARTIE.
La première partie de ce document est une contribution à la théorie de
l'inférence sur l'index d'une loi de Pareto. Au début des années 80, la jonction
entre cette théorie d'une part et le problème de la determination de la loi limite
des sommes de valeurs extremes a été
réalisée par l'estimateur de Hill(1975)
de l'index d'une loi stable.
Cela justifie, entre autres raisons, la plus grande
importance donnée à cette estimateur parmi la classe des estimateurs d'un tel
index.
L'inférence sur l'index d'une loi de Pareto a été très vite étendue à toute la
classe des fonctions de répartitions dont le maximum des observations
indépendantes (m.o.i.) est attiré par une loi de Fréchet quand la taille devient
infinie. Par ailleurs, on sait que le m.o.i. est attiré soit par une loi de Fréchet,
une loi de Weibull
ou soit par une loi de Gumbel. Le cas de Weibull est
immédiatement traité travers le cas de Fréchet par une simple transformation
algébrique. Dès lors, l'alternative naturelle des résultats de convergence et de
normalité asymptotique était l'attraction du m.o.i. par une loi de Gumbel. Cette
classe contient les très importants cas
particuliers des lois normale, log-
normale et exponentielle. On comprend dèss lors l'intêret des spécialistes porté
à cette question.

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Le Professeur Deheuvels m'a demandé une après-midi d'octobre 1984
de chercher le comportement de l'estimateur de Hill(1975) lorsque la
fonction de répartition associée était celle de Gauss. Ce fut mon premier
contact avec les valeurs extrêmes. Les résultats obtenus puis généralisés
pour toute la classe L des fonctions de répartitions dont le m.o.Ï. est attiré
par unç loi de Gumbel ont été publiés dans l~ Journal of Applied
Probability(1986). J'ai été très surpris quand le Professeur Deheuvels m'a
demand, de rédiger l'article en Anglais
puis de le soumettre à J.A.P.
Depuis, j'ai pris l'habitude d'écrire mes articles en Anglais.
Mais lorsqutil s'est ,agi de prouver la normalité asymptotique de
l'estimateur de Hill~ la collision aavec le formidable travail de Csorgo-
, Haeusler-Mason s'opéra. Finalement, je trouvais da~s le travail de Csorgo-
,Mason(1986) la manière de résoudre le problème. Il a suffi
d'affiner
suffisamment les, propriétés des éléments de L pour reconduire les
méthodes de Csorgo-Mason. Dans Csorgo-Haeusler-Mason(Ann.Probab.
Vol 19, No2, 783-811), où les lois limites des sommes de valeurs extrêmes
sont entièrement déterminées. Cette contribution
à l'étude de la normalité des
sommes de valeurs extrêmes a été reconnue et signalée (voir annexe, p. ).
Dans la foulée de ces résultats, nous nous sommes intéressés
aucomportement de l'estimateur de DE HAAN-RESNICK(1980) (voir troisième
article) et des estimateurs de Csorgo-Deheuvels-Mason( 1987) (résultats non
exposés ici). Ces deux papiers furent publiés en rapports techniques au LSTA,
Nos 29/1985 et 49/1985)
TROISIEME PARTIE.
Au cours de l'année 1985, Dr. Van Zuijlen a été invité au séminaire
de Deheuvels-Chevalier. Il fit un exposé sur les espacements. Ce domaine
m'enchanta. Le Profeseur Deheuvels à qui je fis part de cela me remit toute une
documentation fraiche et complète non publiée sur le thème.
Je me suis donc intéressé à
l'approximation Gaussienne du processus
empirique basé sur les espacements d'un échantillon issu d'une v.a aléatoire
uniformément répartie sur (0,1). Nous avons montré que
l'approximation

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proposée par Aly-Beirlant-Hprvath(1984) était optimale- dans l'approche que
j'appeleraide Shorack - par la détermination de la limite supérieure. Ce résultat
a été obtenu même avec un pas devenant infini avec la taille de l'échantillon. Ce
résultat indique qu'il faudrait nécessairement changer d'approche pour atteindre
une meilleure vitesse.
Récemment, au colloque "Order Statistics and Nonparametrics: Theory and
Applications" tenu à Alexandrie, 18-20 September 1991, Dr. Aly me confirma
que cette vitesse n'a pas encore été améliorée. Je pense m'intéresser à ce
problème plus tard.
Nous avons étendu certaines propriétés du module de continuité du
processus empirique classique au processus des espacements. Ces travaux
furent publiés en rapports techniques LSTA N 47 et 48/1986 et exposés au
" /"
séminaire Deheuvels-Chevalier(5 Février 1985).
QUATRIEME PARTIE.
Vers la fin de l'an 1986, nous nous sommes intéressé aux
Statistiques
d'ordre dans les espaces de Banach lattices, particulièrement dans C(S), où S est
un espace métrique compact séparable. Cet article est le premier pas. Il fut
d'abord soumis à Journal of
Multivariate Analysis. Les deux rapports des
referees étaient concluants. Je n'ai pas pu, malheureusement, à l'époque (1988)
envoyer une version révisee. Entre autres, l'un des referee confirma l'originalité
de l'étudue après avoir consulté les fichiers MATHFILE.
Nous avons
là pour l'Université de Saint-Louis un beau champ
d'investigations pour nos collègues et étudiants intervenant en Statistique.
DEUXIEME PARTIE.
Cette partie, il est vrai, doit être classée immédiatement après la première.
Mais, chronologiquement, elle constitue notre dernière série de travaux allant
de la période de 1988 à 1991. Nous voudrions la considérer comme notre
nclusion à notre contribution à l'inférence des extrêmes.
Nous caractérisons l'attraction du m.o.i. par une classe de huit statistiques.
Deux statistiques jouent les rôles principaux. La première (notons la comme
dans notre premier article Tn (2,k,l), déjà fort connue, est celle de Hill(1975).
La seconde, notée An(1,k,I), est
dans sa forme, nouvelle. En fait la
discrimination entre tous les cas (Fréchet, Weibull, Gumbel) est déjà réalisée

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par le couple
(Tn(2,k,I), An(1,k,I)). Il faut cependant signaler que cette
r
discrimination a déjà été obtenue par Dekkers-Einmahl-DE HAAN (Ann.
Statist., 1989,17, 1833-1855). Des résultats similaires sont signalés par
Tiago de Oliveira et Arne Fransèn (J. Tiago de Oliveira (ed.), Statistical
1
Extrêmes and Applications, 373-394).
1
1
Nous avions'déjà dit que An(1,k,l) était nouveau en tant qu'estimateur de
l'index d'une lo'i'de Paréto. Lors du Colloque d'Alexandrie, le Professeur DE
HAAN m'a dit qu'il pensait que l'esti mateur Mn de son papier avec Dekkers et
Einmahl avait un lien avet le nôtre. De retour à Saint-Louis, j'ai vérifié que Mn
=2A (1,k,O). Ceci dinimue la portée de notre affirmation. En fait, la forme que
nous avons introduite est dérivée directement d'une intégrale. Elle
a donc
l'avantage d'être plus commode lors de l'étude de la normalité asymptotique.
Une fois la caractérisation obtenue, les questions naturelles relatives aux
techniques asymptotiques s'imposaient d'elles-mêmes.
Elles sont:
1) normalit, asymptotique des statistiques.
2) lois du logarithme itéré.
Les deux articles qui suivent traitent de ces deux questions. En le faisant,
nous avons obtenu des résultats de caractérisation. Ce qui permet d'aller plus
loin plus tard en developpant le calcul. Nous devons signaler le role primordial
joué par les approximations Csorgo-Csorgo-Horvàth-Mason(1986).
Dans le quatrième article, les statistiques Tn (2,k,l) et An (1,k,l)
sont
généralisées à un rang quelconque. Le résultat est un processus
dont l'étude
asymptotique est étudiée. Cette étude nous fait déboucher
sur un processus
apparemment nouveau dont la fonction de covariance
est entièrement
déterminée. Ce processus ainsi mis en évidence
mérite à mon avis, une étude
particulière. C'est un élément de plus pour une future équipe de recherche à
Saint-Louis ou simplement entre Saint-Louis et Dakar.

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CONCLUSION.
D'une manière générale, chacune des parties de notre travail contient
des
directions de recherche. Aucun travail de recherche ne peut ëtre complet et
définitif. Il faut toujours s'arrêter un instant, profiter de ce qui a déjà été fait,
puis répartir; Mon souhait est qu'émerge entre les Université de Dakar et de
Saint-Louis une équipe aussi
célèbré que l'école Hongroise ou l'école
Néerelandaise dans le domaine des probabilités et statistiques. La chose n'est
pas impossible. En tout cas, tout au long de cet exposé, nous avons signalé des
problèmes statistiques dignes de recherèhes de niveau international.
Nous promettons d'y travailler.
\\
Enfin, un mot sur la forme de la thèse. Nous aurions pu tout réecrire en un
seul corps avec beaucoup de rappels. Nous avons préféré avec l'autorisation de
Monsieur Deheuvels, laisser apparaître le chercheur dans ses activités. Les
articles et rapports techniques sont rendus tels qu'ils ont été présentés au public.
Saint-Louis, le 15 Novembre 1991.
Gane Samb Lô

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PREMIERE PARTIE~
CONTRIBUTION A L'INFERENCE DE L'INDEX D'UNE
LOI DE PARETO

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J /\\PPi. jJroh ~3, q=~_ù_~c\\ ( !qt~(~.l
Pr!!l!ed in /srnc!
.-\\SYMPl'Ol'IC nEHA VIOR OF HILL'S ESTJ\\L\\ TE AND APPLIC-\\ l'IONS
.\\ nstrocl
The :,:\\,hlern of eSllnl{llîl~g. the t\\pi.Jnent (lf 3. sta~ie la\\\\' is recè)\\-ing ;::1
inc:::~~5-!;:~ JiilOuni: (If Jilèntinn hèl.';:;~~:': Pi1reto's 12·....· \\~)r Zipf's 13\\\\") d(,;"C,ib:..'~
Ill:!::~, t-:Cl)~;f!Cjl rhL'rH-lr:H~r::i \\"Cl:' Wè;: (5~C c_~ Hill (JÇ·ï:+)). This r;(~h!::'.;~; 1,;:',:-
Grs; .'(lh~J n~' Hill (11.)'7)), \\\\"ho proposed an estJn12[e. 2!id the (()n .... er,::cncé l,f
th2: c,:;li;-',L";(C 10 ~Oin~ rt)Sill .... e e:tnd hnl1c nU.mber was snl)\\';n ta DC fi ch3ra:k.ris-
li,~ \\..',; t;:~,~~ihl~::on ;'lJrlct!nrls bel(ln~in~ {\\..) th~ Fréchtt dom~llll 0; at:,3;"';:()'î
(\\L~~(l~l (1(;0~)
,-\\s a conlribution io 3 cump!cte thCOi)' of inkrtncc for !!l('
u!)~:::r I~)I ,Ji a gcncral distribution functrün, ',.I.e gi\\'c the asymptotic bcha\\':e"
(\\\\e:1k 2:ld S1ron~) of Hil\\"s estimate when the associatcd distribution func<loc
belc)ng, !O th" Gumbd domain Di 3rtraClilln. Exampies, applic3tiuns ""d
Sinlldati(1nS 3rc gi\\'en.
ORDER 5LA TISTlCS, DO~,f..>,lt' OF ATTP, ..>,ITION, FUNITIOSS Sl.OWLY VAR YhG
.>, T
ZERO. ~:OR\\!L~G CO.~STA"TS. CE~-TR'L U'IIT THfèORBI. WEAK LAW OF I..'RGE
1. Introduction 3nd results
For the last 100 ycars, Zipf's (or Pardo's) law, àeflIled by
(1.1 )
C > O.
c > 0,
as x l' +:c
has recei\\cd an increasing amount of attention (cf. Boulenger (1885)). As noted
by Hill (1974). ',','ho gives in Hill (1970), (974), (1975) a useful sUIve)' of this
topie, man: biolagical phê'nomena ,ne weil described by the probabilistic mode!
(1,1 ).
This has been the justification for much work on thc probJem of estimating (.
First, Hill (197.5) proposed. for a fixed integer k, 1 ~ k < n,
T
1
o = k- l
(Xo -._ 1." - Xo-,,-o)
i=l
as the conditional maximum likelihood estimate of c under the assumption that
X" X~,' . ',X" are independent and identical copies of i1 random variable X
Received 7 Fcbruary 19R5; revision received la October 1985 .
• Postal address: SI Résidence d'Athis, 26 Rue de la Plaine Basse, 91200 Athis-Mons, France.
922

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GANE SAMB LO
If (Hl) holds, we have (he following (see e.g. Lemma 3):
(1.2) there exisl sorne constant Co and a positive function s(· ) su ch lhat
f
0(1 - u)= co+ s(u)+
(s(t)/t)dt,
where 5 ( . ) is a function slowly varying at °(SYZ).
Therefore. we can rnake the following assumption:
(H3) (Hl) and (1.2) hold and s(u) is ultirnately non-increasing as u t 0
Our main results are as follows.
Theorem
1.
LeI
(H4)
F(log(' )) E:: D(A) and F(x) is ultirnately continuous as x 1 A
be salisfied Then for anv sequence "n = k satisfying (K), we havc
k
'f
!
PC .. '0 - X,-<.n) ~ 0,
as n ~ + "x.
1=1
Theorem 2
Let (HI) be satisfied. Let (1.2) be reduced to
( 1.2')
0(1 - u) = Co + f' .:UD dt,
u
t
then for any sequence satisfying (K), we have
where
p
Œ
----> 0,
C~ = s(i/n),
i = l,' . " k
o
d
and ~ cfenoles thc convergence in distribution.
Corol/ary
1.
Let the assumptions of Theorem 2 and (H3) be satisfied. If k
sa tisfies
(K 1)
5 (k / n )/ s (1/ n ) ~ l,
as n -> + 00,
we have
1 = Il
-1
"
P
en' k ~ (Xn-i.,o - Xn-k.n)- l, as n ~ + 00.
1 =1
Corol/ary 2.
Let (H2) and (H4) be satisfied. Let k = (n'), 0< 0 < 1. If for
sorne Il, 0 < Il < 0/2, we have
(K2)
n"s(k/n)->+oo,
as n->oo

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AsymplOtic behavior of Hil/'s estimare alld dpplIC<lliolis
then we get
1 =k
(1.4)
C.o· k- I L (Xn-"i
Xn'n)----> l,
almost surely.
(a.s)
0
-
i=l
Coro/lary 3.
Let A> 0 and (H2) be satisfied. Then if (Hl) and (1.2') (or
(H4» are satisfied, the statement (13) (or (1.4» remains true if we replace
i=1.· .. ,k+1
s(u)
by
r(u)--{R(()(I-u»IO(I-u)}
asu10.
C.
bv
rU/n).
and if k satisfies (K) (or (K) and (K2) with s(·) replaced by r(·». Moreover if
log A> O. we may derive sirni\\ar results fer the second logarithm. etc.
Remark.
If A > O. wc haw for large values of n
0<Xo-1+1.o<A,
l~i~ko+l, sinceknln---->O
and
ko--->+oo.
Corollary 3 may be inverted as follows.
Coro/lary 4.
Let R (1)----> 0 as 1 i A and let (H2) be satisfied. Then if (Hl)
and (1.2') (or (H4» are salisfied. the slalemenl (1.3) (or (1.4») remains true if we
replace
Xn"'o
by
exp(Xo',"'o)
s(u)
by
l(u)~{R(O(I-u»)exp(O(I-u»}
asu10
Co
by
{(i/n)
and if k satisfies (K) (or (K) and (K2) with s(· ) replaced by 1(' )). Moreover, if
{(u)----> 0 as u ! 0, we may rcpeat the operation, etc.
Now, we give sorne examples via the expressions of the norming constants
C.o'
Corollary 5 (particular cases).
In each case, (i) (or (ii» will correspond to the
choice of k = (log n) (or k =
o
o
('1'),0 < S < 1).
1. Normal case: X ~ N(O, 1)
(i) (2Iognt2 To':l,
(ii) (2(1- S)log n)T ----> 1, a.s
o
2. Exponential (or gamma case): exp(X) ~ E(1)
.
p
(1) (Iogn)To----> 1,
(ii) «(1- S)log n)To ----> 1, a.s.
3. Logp-normal: X = logp sup(b, Z), Z ~ N(O, 1), logp stands for the pth log
and 10gpb=O: Let Do =(2Iog(knln))n;:~-llogJ(210g(ko/n)Yl2,
Iogo x=l, 'r/x.

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.-tn mploric behavior of Hi/l's esrimare and app/icariolls
(b) Let a = +00. Suppose that there exists a sequence (1;, 1,),1;, ln Î A such
that (I~- tn)/R(tn) is bounded. (ii) would impil that k(t~)~ k(tn) is also
bounded, which is not possible at the same time as a = + x.
(c) Let a = - 00. We use (i) at the place of (ii) in the above case and get that
( l' - 1)/ R (t ) _ - 00.
50, we have proved that
(k(t')-k(I)-a)=> ((l'-I)/R(I)-a)
al 1'.1 i A.
Converse/y, suppose Ihal (l' - 1)/ R (1)_ a.
(0) Let
a be finite. Then for any E > 0, we have for t', 1 near A,
(+ (a - E)R(I)~ l' ~ 1 + (a + E}R(I). Therefore, (13) implies
a - E ~ lim inf k (l') - k (1) ~ lim sup k (r') - k(t) ~ a + E.
t',r t A
(./ i A
(13) Let a = + 00, for any ri> 0, we have for ('. 1 near A : 1';::; 1 + dR (1).
Therefore (B) implies
lim inf k (l') - k (1);::; ri.
,',1 i A
(-y) Let a = - 00; similarly to the preceding case. we gel
limsup k(t')- k(I)~ - d_
,',/ T A
By letting E -
0, and d -
+ 00, we get the other implication of the equivalence
we had to prave.
Lemma 2.
Let FE D(A). If in addition F(x) is continuous for x near A,
then F-'(I - u) = 0(1- u) is slowly varying at °(SYZ).
Proof of Lemma 2.
Let l' = 0(1- u) and 1 = 0(1- uv), with v fixed and
v> O. Beeause of the eontinuity of F(·), we have u = 1 - F(t') = exp( - k(t'))
and uv = 1 - F(t) = exp( - k(I». Hence k(I') - k(l) = 10g(1/v). Lemma 1 im-
plies then (t'-t)/R(t)-log(l/v), which in tum implies that
O(l- u) _
R(O(l- u»
O(l_uv)-I+(Iog(l/v)+o(l»
O(1-u)
But,
by
Lemma
5,
R(t)/I-O
as
ItA,
whenever
FED(A).
Henee
limuloO(l- u)/O(l- uv)= l, which is the announced result.
Proof of Theorem 1.
Since (H5) holds, Lemma 2 implies that the function
H (l - u) assoeiated with F(log(·» is SYZ.
Note that O(l - u) = log H(1 - u) as u 10. Recall also the well-known
representations

---

-13-
Asymproric behavior of Hill' s esrimare and applicarions
Pro of of Lemma 3.
The proof füllows from Theorem 1.4.1 and Theorem
2.4.1 of De Haan (1970).
Lemma 4.
Let the assertion (ii) of Lcmma 3 be satisfied. then FE D(A). If
in addition F is continuous as x i A, we get
.
s ( u ) _
~IPÀ R (00 - u» - 1.
Praof of Lemma 4.
The proof is given in Lo (1986), Lemma 4.
Lemma 5.
Let (HI) and (H2) be satisfied, then R(I)/1-1-0 as 1 i A and
R(O(I- u» is SVZ.
Praef of Lemma 5.
First, we remark that Lemma 4 implies that R (00 - u»
is SVZ and by Lemma 3 of Lü (1986), R (1)/1 -1- 0 as 1 i A
Proof of Theorem 2 (continued).
Since (HI) is satisfied, suppose that (1.2) is
reduced to
0(1- u)= c + f
ù
(s(I)/t)dt.
Thus, for each i, 1 ~ i ~ k, k satisfying (K), one has
(2.5)
X"-i+I."-X
~
d n
OO-Vi.n)-O(I-V'+ln)=fu··'"(S(U)/U)dU.
U'"
Now, since s(u) is SVZ, it admits the Karamata representation:
(f
s (u ) = z ( u )exp
(w ( v )/ v )d v) ,
(2.6)
Z(u)-1-Z,
O<z<+:o,
w(u)-1-0,
asulO.
So, for each i, 1 ~ i ~ k, for any ai•n • V" ~ a,n ~ V i +'"' we have for large n,
(2.7)
{s(a,")/s(i/n)} = {z(a,n)/z(i/n)}exp ( - fn'" (W(v)/v)dv).
Also
1fa,. ~ dvl ~ sup 1w(v)/· suI? max (' log~ v," l, IIOg~ u'+I.n 1)
1/'1
V
vEl..
I:;,;:;;n
l
l
(2.8)
=: R~' R~
where In = (O,max(k/n, Uk+I")) is a random interval.
However, it is not difficult to see that the sequence R~ is bounded in
probability (see e.g. Lemma 13 of Csorgo and Mason (1984)). Furthermore, since
w (u) -1- 0 as u t 0, it foHows that sup{ W (v), v E 1.} -: 0 independently of a," and

---

-15-
Asymploric behavior Dj Hill" s eslimare and applicarions
R (0(1- u))/s(u)~ 1, as u! O. Now, since s(u) is SVZ and (n/k)Uk." ~ 1, a.s.,
we get by Lemma 5
R(O(l- U
))/s(k/n)-'.>1,
kn
a.s.
which completes the proof.
Proof of Corol/aries 3 and 4.
These two corollaries will follow from the
lemmas below. Define by L the set of distribution functions F satisfying (Hl)
and (H2).
Lemma 7.
Let
A > O. Then if X
has a
distribution
function
FEL,
log sup(O, X) has a distribution function 0 ELand
S(O-'(I- u))~ {R(O(l- u))/O(1- u)},
as 1< L0,
where
_J'''F' (1 - 0 (v))
S(I)-,
(l-O(I))dv,
-
00 < 1 < log A.
Conversely, we have the following.
Lemma 8.
Let R (I)~ 0 as 1 i A. Then if X has a distribution function
FEL, then exp(X) has a distribution function Z ELand
R(O(I-u))-{T(Z'(I-u))/Z-'(1-u)}
as u LO.
where
_J'AI-Z(V)
T(t) - ,
1- Z(I) dv,
Proof of Lemmas 7 and 8.
These lemmas are proved in Lo (1986), via
Lemmas 9 and 10. Note that (H2) implies that log sup(O, X) exists almost surely if
A >0.
Proof of Corol/aries 3 and 4.
By Lemma 7, we see that if for instance (Hl) is
satisfied, the same pro pert y is also true for log sup(O, X). So, we can write (1.3)
with s (. ) replaced by r('), where r(u) is derived from De Haan's representation
for O-I(1-U) as in (1.2). But, we see also that 0(1- u)= co+J~(s(t)/I)dt =>
s(u)= uQ'(l- u). Since
0'(1- u) = log 0(1- au),
a = P(X > 0),
r(u) = ug~~l_-u~) = u (0-'(1 - u)')
is SVZ by Lemma 2 and (1.2). Hence,
0-'(1 - u) = - 0-'(1- YI) + f !ljJ dl,
for u ~ YI ~ 1.

---

-17-
AsympiOcic behavior of Hill' s estimate and applications
3. Applications and simulations
As remarked above, Hill (1974) desçribed sorne basic models which follow
(1.1) or (Ac). We note that ail these models are c10sely related to problems based
on extreme values. As already noticed, the works of several authors (Csorg6 and
Mason (1985), Csorg6 et al. (1985), Hall (1982), Hill (1970), (1974), (1975), Mason
(1982» have entirely settled the properties of T" under the assumption (Ac). It
fallows from their results that if X
X
are the observations of X, we can
io
2 , •• "
X o
verify if (Ac) holds. In that case, we proceed as follows:
3.1. Identification of the upper tail of a distribution.
(i)
Chaose 0,0< 0 < 1,
(ii) Choose k. = (n 6 ),
(iii) Calculate T. = k ~1 L::~ (X.-'+l .• - X"-".) for large values of Il.
(iv) If T. is very near c, 0 < c < + 00, Ihen by Theorem 1 of Mason (1982),
(Ac) holds.
1
d
(v) But (Ac)? FE D(A)? c- (X,," - Q(l-l/n»~A,
and therefore, we can use (v) for predictions about a critical value of X,,".
However, il is not always certain that T. converges 10 a finile slrictly positive
number. For example, if we want to know whether F salisfies (Ac) or if F(log(· »
is the distribulion function of the standard Gaussian random variable, how could
we proceed?
3.2
Comparison between a regular rail and a gaussian tai/.
We want ta know
if
(Ll) F(log(x» = x -Ile (1 + DO (x -b», c > 0, and b = 1/2e
or
(Ü) XI = log sup(O, Z), Z ~ N(O, 1)
where F(·) is Ihe unknown distribution function associated wilh Ihe observa-
tions XI, X 2, •• " X. of X.
(i)
Choose k = (n 112), then
(ii) If (LI) holds, we have
(a) (Mason (1982»
T" ~ c, a.s.
(b) (Hall (1982»
nI/4(T"-c)~N(0,1).
(iii) If (Ü) holds, we have
(a) (Corollary 5, Part 3, p = 1), log nT. ~ 1, a.s.
(b) (Lo (1986»
n 1I4(D"T" - 1)~ N(O, 1), D" = log n(l + 0(1».
Thus, we see that we are now able to test (Ll) against (LI). If we choose
D = {( C - t:) ~ T" ~ (c + t:)} as the accepted region of our test, (i) and (ii) give
the characteristics of that test. To test (LI) against (LI), One can choose
15 = {1 - t: ~ T" . log n ~ 1 + d as the accepted region wilh a small value of t:.
3.3. Comparison
between
an
exponential
and
a
gaussian
tail.
Let

---

-19-
Asymptotic behavior of Hill' s esrimate and applications
Remarks.
1. One might be surprised to find that our simulations are not sufficiently
good, considering the large size of the sam pie spa ce (n = 4(00). However, only
the k extreme observations (k = 62 or 63) are used for the calculation of Tn •
Taking that into account, the theoretical part of this paper is relatively weil
illustrated by the simulations. SpecificaIly:
2. Column 4 illustra tes the almost sure convergence of HiIl's estima te for the
Pareto law: T
converges almost surely to 1.
nJ
3. The identity of columns 2 and 3 is a consequence of the choice of Xi. It is
c1ear that if Xi = ( - 21og(1 - u; »If2, we get Tn 2 = ~ Tn 1. However, this choice is not
arbitrary. Indeed, if Tnz is the true value obtained from the use of the true
quantile function, we have Tn*z = Bn(li)' where (see e.g. statement (2.\\3)).
U
1
=1
{(-'21
(1-
.»1/2(1
[Og(-IOg(I-
i)-47T+O(l»)}
og l,
og
og
u,
+
4 log(l - u )
i
= ~ log 2 + ~ log Yi + 0 ((lOg log 1 ~ uJ / 410g 1 ~ uJ .
With our data, 3937 ~ i ~ 4000, k = 62 or 63, we have
log li = 410g 2 + ~ log y, :!: 0.07,
therefore
4. We now test (U): (Xi) are the order statistics of a N(O, 1) random variable,
against (L4): (Xi) are the order statistics of the standard exponential law. If we
choose R. = {l- a ~ log nT ~
nl
1 + a} as our accepted region and choose a =
0.29 as the significance level of our test, the power of the test will be f3 ~ 0.74,
and Rn will be Rn = {0.75 ~ Tnl 'Iog n ~ 1.2533}. Here, we accept (U) since the
table gives the value 0.7554 for (log n . LI) for n = 4000.
5. Column 5 gives the ten first values of the order statistics of the uniform
random variable. One may work with the highest or the lowest values since
{l- u;, 1 ~ i ~ 4000} ~ {U4ŒX)-'+I' 1 ~ i ~ 4000}.
Conclusion.
De Haan and Resnick (1980) and Csorg6 et al. (1985) have also
given estimates of c under the assumption (Ac). In future papers, we shaH
describe their asymptotic behavior using the assumptions of this paper.
5. Remarks and furtber generalizations
Remark 1.
Deheuvels et al. (1986) have recently shown that the De Haan
representation (1.2) holds whenever F(' ) belongs to D (A).

---

- 2~ -
î l -
Journal of Sialislicai Planning and Inference 22 (1989) 127-136
Nonh·Holland
A NOTE ON THE ASYMPTOTIC NORMALITY OF SUMS Of
EXTREME VALUES
Gane Samb Lü
Universüé Paris 6, L.S. T.A. T.45·55, E.3., 4 Place Jussieu, 75130 Paris 05, Fronce
Recei\\'td 15 January 1988; revised manuscripl received 16 March 1988
Recommended by P. Deheuvels
Absrracr: Lei X I,X2• ••• , be a sequence of independenl random variables wilh common distribu·
lion funnion Fin Ihe domain of allraction of a Gumbel exneme value distribulion and for each
imeger n 2: 1. let X I• " ••• "X",n denole the order statislics based on lhe first n of these random
ft
variables. Along wilh related results il is shown that for any sequence of positive imegers k. - 00
and k./n-O as n-oo. lhe sum of lhe upper k n extreme values Xn-k.'l.n+'··+X•. n' when
properly cenlered and normalized, converges in disl ribulion 10 a slandard normal random variable
N(O.I). These results conslilule an extension of resuhs by S. Csorgo and D.M. Mason (1985).
AMS Subjecr Classification: 62E20, 62G30. 6OF05.
Key words: Order stalistics; extreme values; Gumbel law; asymplotic normality.
1. Introduction
Let X"X2• •••• be a sequence of independenr random variables with common
distribution function F and for each integer n ~ 1. Jet Xl,n ~ '" ~Xn.n denote the
order statislics based on the first n of these random variables. Csorgô and Mason
(1985. 1986) have recently shown among other results that if
I-F(x)=L*(x)x-a
as x-oo,
(1)
where L· is a slowly varying function at infinily and a~ 2, or if F has exponential·
like upper tails, meaning
l'~ (1 - FlY» dyl(l -F(x» --+ C as x --+ 00.
(2)
•. :r
where 0 < c< 00, then for any sequence of integers satisfying
(K)
there exist sequences An> 0 of normalizing constants and Cn of centering constants
i
such that
f!
as n-cx>,
(3)
0378·3758/89/$3.50 © 1989. Elsevier Science Publi,hers n. V. (Norrh.Holland)

---

-22-
..
"'1'
C.s. Lo / Asymplolic norma/ily of ulremes
The case (1) is contained in the lheorem of Cs6rgo and Mason (1986) and the case
(2) is Theorem 1.5 of Cs6rgo and Mason (1985).
An application of Theorem 2.4.1 of de Haan (1970) (Lemma 1 below) combined
with Fact 1.4 of Cs6rgo and Mason (1985) shows that (2) implies the existence of
sequences of normalizing constants On and centering constants b such that
n
(4)
where G is a Gumbel random variable with distribution function
P(G::5x)=e.\\p(-e- X )
for -oo<x<oo.
Whenever such sequences of constants can be chosen 50 that (4) holds, we say that
Fis in the domain of altraction of a Gumbellaw, wrilten FeD(A).
One of the purposes of this note is to show that (3) holds more generally than
under condilion (2), thal is, Fe D(A) is sufficient for (3) to hold. This will be a con-
sequence of our main resulls stated in the next section. We shaH also obtain sorne
funher extensions of the resuhs of Cs6rgo and Mason. The proofs are given in Sec-
tion 3.
2. Siaiemeni of main resulls
First we introduce sorne notations. LeI
Q(s) = inf{x: F(x)~s},
for O<s::5 1,
with Q(O) = Q(O+), denote the in'verse or quantile funclion of F. WrÏle
'1
'\\
02(S) = \\
\\
(min(u, v) - uv) dQ(u) dQ(v),
for O::5S::5 1.
• 1 - 5 • 1 - 5
For any O<P< 00, set
'1
C(S,P>=s-p\\
(I-u)PdQ(u),
forO::5s::51.
,,1 - 5
For convenience, when P= l, we set c(s) =c(s, 1). (Refer 10 the nexl section for our
inlegral convention.)
Let D*(A) denote the subclass of D(A) consisting of ail distribution funclions F
whose quantile function Q satisfies
'1
Q(I -s)=a+ \\ u-1r(u)du,
, s
for ail s~ 0 su fficielllly small, ois a fixed constant and ris a striclly positive function
slowly varying at zero. The fact thal D*(A) is a subclass of D(A) follows from
Theorem 2.4.1 of de Haan (1970).

---

-23-
- 29 -
G.S. Lo / Asymp/o/ir no,malilY olex/,emes
whenever these integrals make sense as Lebesgue-Stieltjes integrals. In this case, the
usual integration by parts formula
r r
f dg +
g df =g(b)f(b) - g(a)f(a)
is valid.
The proof of our theorem will follow c\\osely the proofs of the results of Csorgo
and Mason (1985), substituting their technical lemmas concerning properties of the
.quantile functions of distribution functions salisfying (2), by those describing
properties of lhe quantile functions of FED(II). We therefore begin with these
technical lemmas.
Lemma I. FE D(II) if and only if for each choice of 0 s x, y, w, z < 00 fixed, y * w,
Q(I -sx)- Q(I -sz)
log x-log z
as sLo.
(8)
Q(I -sy) - Q(I -sw)
logy-Iog w
(This is Theorem 2.4.1 of de Haan (1970).)
Lemma 2. Whenever FED(il), c(s,{J) is slowly varying ar zero for each choice of
0<{J<00.
Proof. We have to show that for each 0<).<00 and O<{J<oo,
c(ls, {J)/c(s. (J) --> 1
as s Lo.
(9)
Choose any 0<).<00 and 0<0< 1. Then for ail s>O small enough we have
tl-.ls (1 - u)pdQ(u) =JJ:~::::., (1 - u)pdQ(u)
..
sI: ).PsPOiP{Q(I-lsOi+I)_Q(I-lsOi)}.
(10)
;=0
Applying Lemma 1 gives
Q(I - )'Ou) - Q(I - ).u) --> 1
as ulo.
(II)
Q(I- Ou)- Q(I - u)
Select any 0<1:<00. From (Il) we have that, for ail s>O sufficiently small, expres-
sion (10) is
..
s (1 + 1:) I: ). PSPOiP{Q(I - SOi+ 1) - Q( 1- SOi)}
i=O
(1 + 1: »). P ..
~ 1- sO" 1
s
pI: sPO(i+ I)P
dQ(II)
8
,=0
• J -sO'
(1 +I:»).P JI
s
P
(l-u)dQ(u).
o
I-s

---

-24-
- 30 -
ri
C.S. La / AsymplOIic narmaliry uf extreme.<
Thus for ail 5>0 sufficienlly small,
(1 + e)
c(As, p):5 -----rJïi c(s, Pl.
(12)
Observing that for ail 5>0 small enough,
• 1
00
\\
(1 - u)1l d Q(II) ~I ).t'sIlO(i + \\lB {Q(I - ).sOi + 1) - Q(I - AsB i )},
,·1 - ..h
1=0
we see that by an argument very much like the one jusl given, we have for ail 5 >0
suflïciently small,
(13)
Assertion (9) now follows from inequalities (12) and (13) by the fact that B can be
chosen arbitrarily close to one and e arbitrarily close 10 zero. This completes the
proof of Lemma 2.
The following lemma is related 10 Theorem IA.3.d of de Haan (1970) and its
pmof is based on a modification of the techniques used 10 prove this theorem. For
derails sec Deheuvcls et al. (J 986).
Lemma 3. Whenever FE D(1), there exists a constant - 00 < b < 00 sllch that Jor 011
O<S:5t.
·'1
., .,)
Q(I-s)=b-c(s)+ \\ u-1c(u)du.
(14)
, 5
./
Lemma 4. Whellever FE D(/1), for eacll O<x<oo,
.
Q(I -xs) - Q(J -5)
hm
=-logx.
( 15)
5JO
c(s)
Proor. Applying Lemma 3, we have for any O<x<oo and for ail 5 sufficiently small
Q(I-xs)-Q(l-s)
c(s)-c(xs)
1 \\'5 c(u)
- " - - - - - = - - =
+ -
-
du.
c(s)
c(s)
c(s) . X5
U
Silice C is slowly varying at zero, both
inf{c(u)lc(s): u E I(s)} -
J
and
supt c(u)Ic(s): U E I(s)} - 1
a~ 510, where 1(5) is the closed intcrval formed by xs and s. From these two facts
Ihe proof of Lemma 4 follows immedialely.
Lcmma 5. Whenever FE D(1), Jor any 0 < P < 00,
c(s,P)/c(s)--+I/P
ass10.
( 16)

---

-25-
- 31 -
c.s. Lo 1 Asymptotic normolity of extremes
Proof. Let Q(I - s) =0 Q(l - Slip). Since by Lemma 1 for any choice of 0 <x, y < 00,
y*l,
Q( 1 - xu) - 12(1 - u)
Q( 1 - X \\IPU IIp) - Q(l - u \\Ip)
Q(I - yu) - 12(1 - u) =0 Q(I - y I/PU IIP) - Q(I - U I/P)
converges as u10 to logx/logy, we conclude that QED(/I).
Let
.1
ê(S)=OS-1
(1 -1I)dQ(u),
for o<s< I.
\\
• 1 - J
A change of variables shows [hal ê(sP) =0 c(s, p> for 0 < s< 1. Thus
c(s, p>
ë(sf!)
- - - - -
c(s)
c(s)
Q(l - 2s) - Q(I - s)
Q(l- (2s)P) - Q(I - sP)
=0
X --=---------
c(s)
Q(l - 2s) - Q(l -s)
which by Lemmas 1and 4 converges to 1/pas s 10, completing the proof of Lemma 5.
Lemma 6. Whenever FE D(/I),
a2(s)/(2sc2(s))~1 as s10.
(17)
Proof. The proof is based on Lemma 5 and fol1ows almost exactly as the proof of
Lemma 3.3 of Csorgô and Mason (1985). Therefore, the details are omitted.
The proof of the following lemma is an easy consequence of the Karamata
representation for a slowly varying function.
Lemma 7. Let an be any sequence of positive constants such that an ~ 0 and
non ..... 00. Also let L be any slowly varying funetion at zero. Then for any O<P< 00,
(18)
We now describe the probability space on which the assenions of the theorem are
assumed· [0 hold. M. Csorgô, S. Csorgô, Horvath and Mason (1986) have con-
structed a probability space (D, V, P) carrying a sequence VI' V 2, ••• , of indepen-
dent random variables uniformly distributed on (0, 1) and a sequence BI' B2, ••• , of
Brownian bridges such that for the empirical process
Gn(S)=onI/2{Gn(S)-s},
O~s~ l,

---

- 32 -
-26-
r'
)
C.S. Lo / Asymp/o/Îc normali/y of ex/remes
and the quantile process
where
and, with UI.n:s. ... :s. U n.n denoting the order slatistics corresponding to UI' ... , Un'
_[Uk.n
if (k-I)ln<s:s.kln, k=I •...• n.
Un(s)-
U
'f-O
I.n
1
s- .
wc have
sup n VOla,,(s) - 8,,(s)l/(I - s) - o', + 1/2 = 0/,( 1).
( 19)
Osssi
with Bn(s) =Bn(s) for Iln:s.s:s. 1-lin and zero elsewhere and
sup
nV'IPn(s)-Bn(s)/I(I-s)-v,+ 1/2=Op(I).
(20)
OS''''1 -I/n
where VI and V2 are any fixed number such that O:s. VI < 1 and O:s. v2:s. t. The state-
ment in (19) follows from Theorem 2.1, while the slalemenl in (20) is easily inferred
from Corollaries 2.1 and 4.2.2 of the above paper.
Thoughout the remainder of the proof of our theorem we assume that we are
o
Da the probability space of Csôrgo et al. (1986). Since the sequence of random
variables XI' X 2, ... , is equal in distribution to Q( U I ), Q( U2 ), ... , we can and do
assume thal the first sequence is equal to the second.
First assume FE D(A). We shall establish (5). Applying integration by parts we
sec that the lefl side of (5) equals
'1
'I-k/n
dQ(s)
-(nlkn)I/2c(kn ln)-1
a n(s)dQ(s)+nk;112 \\
•
(I-Gn(s)-knln)---
\\
• 1 - *.In
. u. .•..•
c(kn ln)
:= Ll I•n + Ll 2•n •
We shall firsl show lhat
wilh Rn = ope 1).
From (19) we have for any O<v<t.
su p lan(s) - Bn(s)l/(I - s) - v... 112 = °pen - V).
(21)
Os.sl
Notice (hal for any such V,
la (s)-B (s)\\ (n )1/1 \\'1
IRnl:s. sup
n
-V:I/1 -
(I_s)-v, 'l2 dQ(s)/c(kn ln)
0";5"; 1 (1 -s)
k n
. 1_ *.In
.. :

---

- 33 -
-27-
"1
1!i1
C.S. Lo 1 Asympfofic normalif." of exfremes
From (21), we oblain
c(k /n 1_ v)
R
=0 (n-" n"
"
,)
k' V
l.n
P
)
c(kn/n)
n '
which by Lemma 5 equals op(I).
Aiso
which by Lemma 6 is
-2a~(I/Il)/a~(k,,/Il)
as n~oo.
From Lemmas 2 and 6 we infer lhal a 2(s) is regularly varying of exponenl one al
zero. Hence, by Lemma 7,
a2(I/Il)/a2(kn/n)~O as n~oo,
which yields R2•n = ope 1).
Thus we have proved R" =op(l).
Nexl we show lhal Ll ~.n = ope 1). Choose any 1< A < 00 and set
T,,(A) = IIk,; 1I2C(k" ln) -III - Gn(I - k nln) - kIl /n\\ { Q(r,; (A» - Q(r,;(A»},
where
Ak
r;(A) = 1 - _n
and
n
NOlice lhal since for ail 5 in the closed interval formed by U n - k •• n and I-k,,/n,
Il - G,,(s) - k,,/nl ~ Il - Gn(I -kn/n)-kn/nl,
we have for any ) < A< 00,
Since (K) implies (cf. Balkema and de Haan (I975» thal
p
n(l- Un-k•. ,,)/kn--+ 1 as n-+oo,
(22)
lhe lower bound in lhe above inequalily equals one.
Hence for each 1< A< 00,
(23)
Observe for each 1 <). < 00,
\\
(
kn)
(
ETnO)~lQ l-i;; -Q
Ak")1
1--;;- j/c(k,,/II).

---

-28-
-
34 -
C.S. Lo / Asyrnp/o/ic normoli/y 01 ex/rernes
Applying Lemma 4 we see that this last expression converges to 2 log A, which yields
lim lim sup ETn(A)=O.
(24)
À 11
n ..... 00
The facl lhal .1 2.'1 = op(l) now follows by an elementary argument based on (23)
and (24). This compleles lhe proof of (5).
Nexl consider (6). NOlice lhal since FED*(A),
' )
C(S)=S-1
r(l-u)du .
\\.1-5
Thus since r> 0 and slowly varying al zero Theorem 1.2.1 of de Haan (1970) gives
r(s)/c(s) --+ 1
as s 10.
(25)
Th~ !eft sidc of (6) equals
k~f2 \\'1- u•.•..• r(u)
1/2 r(knln)
- - -
-
du = -kn - - {Iog(l- U n_k ln) -Iog(knln)}.
c(knln) .. k.ln
U
c(knln)
•
k~f2 J.I-U......
du
*
*
- - - -
(r(u)-r(knln»-:=.1ln+.12n'
c(k" ln)
k.ln
u '
.
The same argumenl based on (20) as given in Csôrgo and Mason (1985) shows lhal
_k~i2{log(l - Un _k•. n) -Iog(kn ln)} = Yn+ op(l).
Therefore by (25) and lhe facl thal Yn=op(l) we have
.1tn= Yn+op(l).
Sincc r is slowly varying al zero we get for each 1< A< 00 as n --+ 00,
sup Ir(s) - r(knln)llc(knln): -
:ss:s -
--+ O.
(26)
[
~
A~]
An
n
The facl lhal .1rn = op(l) now follows easily from Y = Op (1 ), (22) and (26) com-
n
pleling lhe proof of (6).
Since FE D*(/J) we have
• 1
JJn(kn)-knQ(I-knln)=n
r(l-s)ds .
\\
... I-k,,/n
Asserlion (7) is now a direcl consequence of (5) and (6).
finally we prove lhe convergence in dislribulion of Zn' Yn and Zn - Yn 10 N(O, 2),
N(O, 1) and N(O,I), respeclively, as n --+ 00. NOlice lhal lhe Zn random variable in
(5) is normal Wilh mean zero and second moment (J2(knln)/(knc 2(kn)/n), which by
Lcmma 6 converges 10 1 as n --+ 00. The Y random variable in (6) is normal Wilh
n
mean zero and second momenl 1 - k nln --+ 1 as n --+ 00.
The Zn - Yn random variable in (7) is normal wilh mean zero. Applying Lemmas
5 and 6 il is easy 10 verify lhal E(Zn - Yni --+ 1 and n:-' 00.
This compleles lhe proof of the lheorem.

---

-29-
- 35 -
G.S. Lo / Asymptotic normality of extremes
Acknowledgemenl
The aUlhor is eXlremely gralefui 10 David Mason, Erich Haeusler and Paul
Deheuvels, for helping him 10 complele Ihis work.
References
Balkema. A. and L. de Haan (1975). Limit laws for order slalislics. In: P. Révész, Ed., Limit Theorems
of Probabilit)' Theory, Coll. Malh. Soc. Bolyai, Vol. Il, 17-22.
Csorgo, M., S. Csorgo, L. Horvàlh and D.M. Mason (1986). Weighled empirical and quanlile processes,
Ann. Prob. Il,31-85.
Csorgo, S. and D.M. Mason (1985). Central limil Iheorems for sums of extreme values, Math. Proc.
Cambridge Philos. Soc. 98, 547-558.
Csorgo, S. and D.M. Mason (1986). The asymplotic distribution of su ms of extreme values from a
rcgularly varying distribution, Ann. Probab. 14,974-983.
Deheuvels, P., E. Haeusler and D.M. Mason (1988). Almosl sure convergence of Ihe Hill eslimator.
Math. Proc. Cambridge Philos. Soc. \\04,371-381.
Deheuvels, P., E. Haeusler and D.M. Mason (1989). Laws of the iteraled logrithm for su ms of extreme
values in rhe domain of allraclion of a Gumbcl law. /Juil. Sci. Math., to appear.
de Haan, L. (1970). On Regular Variation and its Application to the Weak Convergence of Sample
E~tremes. Malhemalical CenirC Traci No. )2 (Amsterdam).
Hill, B.M. (1975). A simple general approach 10 inference about che tail of a dislribution. Ann. Starist.
J, 1163-1174.
Lo, G.S. (1986). AsymplOlic behavior of Hill's eSlimale and applications. J. Appl. Probab. 23,922-936.
Mason, D.M. (1982). Laws of large numbers for su ms of e."reme values. Ann. Probab. \\0,756-764.

---

-30-
THE l,vEAK Llî'IITH:G B':::AVIOR OF THE DE HAAN /RESNICK ' S
ESTIMATS Of THS EXPO~ENT OF A STABLE DISTRIBUTION.
by
LO Gane Samb, Université Paris 6.
-L.S.T ..~_.
Abstract~
The problem of estimating the exponent of a stable la\\" receives a conside-
rable attention in the recent literature. Here, we deal with an estimate of
such an exponent introduced by De Haan and Resnick when the corresponding dis-
tribution function belongs to the Gumbel's domain of attraction. This study
permits to construct new statistical tests for the inference about the upper
tail of a distribution and may beapplied to many biological phenomena. ~xamples
and simulations are given. The limiting law are shown to be the Gumbel's law
::
and particular cases are ,given
with norming constants expressed with itera-
ted logarithms and exponentials.
. .. '
Address: LO Gane Samb, Université Paris 6. L.S.T.A. Tour 45-55, E.3.,
A, Place Jussieu. 75230, Paris Cedex 05.
Keys words and phrases: Regulary and slowly varying functions, Domain of attrac-
tion, norming constants, Order Statistics, Limiting law.

---

1
1
-31-
1
1iTRODUCTION AND RESULTS:
Many biological phenomena seem to fit the Zipf's form:
l - G(x) = C x-l/c,
c>Q and C>O.
r .1)
r
instance, we can cite the plot against r of the population of the r-th lar-
st city ( see e.g. Hill(l974) ). This motivated considerable works on the pro-
em of estimating c. More generally, if X ,X
I
2, .... , X are independent and
n
ientical copies of a random variable ( rv ) X such that F(x)::: P( X~ x) satisfies
Lim 1 - F(Log(tx»
::: t- l / c
\\c)
Jf t>O,
l - F(Log(x»
t
x-t=
~veral estimates ( Hill(1975), S.CsorgB-Deheuvels-Mason(1983) ) have been pro-
osed. De Haan anG ~esnick (1980) introduced
T
= (X
- X
) /Lôg k
n
n,n
n-k,n
here Xl
,X
, ... ,X
are the order statistics of X 'X " " , X and k is
,n
2 ,n
n,n
I
2
n
sequence of integers satisfying
K)
O<k<n,
k ::: ken)
-r + <Xl,
k/n -r Q
as
n -r + co
De Haan and Resnick(1980) have proved that (Ac) implies under sorne conditions
P
~hat :
(i) T
-r
C,
in proba bili ty
( î )
n
Log k
ct
Jnd
(ii)
( T -c) + fi
, in distribution ( -+ )
C
fi
-x
·,.;here
I\\(x) = exp( e
) is the Gumbel's law.
In order to contribute to a complete asymptotic theory for the inference
about the uPger tail of a distribution ( as specified in LO(1985a), Section 3),
Ive deal here with the asymptotic behavior of T
in the case where (Ac) fails.
n
Notice that (Ac) means that F(Log(.»
belongs to the Fréchet's domain of attrac-
tion. Here, we restrict ourselves to the case where F(Log(x»
belongs to the
Domain of attraction of the Gumbel's law D(fI). The results we have obtained
yield statistic:al tests for many situations. These results are stated in this
section, proved in'SectionII and illustrated by simulations in Section III

---

-32-
true if we replace
X
,X
k
by
Log X
, Log X k
n,n
n- ,n
n,n
n- ,n
Q(l-s)
by
Log Q(l-as), a = P(X>O)
r(s)
by
s(u) = R(Q(l-u»fQ(l-u)
and if k satisfies Ks(À) at the place of Kr( À). Moreover, if Log A>O, we may
repeat the operation.
Conversely
Corollary 2: Let r(u)+O as ufO. then if (Hl) and (H2) are satisfied,
(1.3)
and (1.4) hold if we replace
X
, X
by
n,n
_
exp( X
), exp(X
k
)
n k n
,
n,n
n- ,n
Q(l-u)
by
exp( Q(l-u) )
r(u)
by
t(u) = exp(Q(l-u»
R(Q(l-u»
and if k satisfies Kt(À) at the place of Kr(À).
Corollary 3 ( particu1ar cases)
Here, we restrict ourselves to the case where
k=(.Log n).Q.). ( )denotes the
integer part, and .Q. is any positive number.
1) Normal case: X'VN(O,l)
.!.
(i)
.Q.(2Log n)2(LogLog n ). T
.(1+0(1»
- .Q.(LogLog n)(l+o(l»
~
A
n
(H)
(2Log n)~ T -- ~
1
n
2) exponential case : exp(X) 'V E(l). ( or general gamma case, see the proof )
(i) (1+0(1» . .Q.( "Log n) (1ogLog n) T
- t(LogLog n)(1+o(l»
- j
A
n
1
(ii)
(Log n) T
1
n
\\
3) Sup'pose that X a,;.s-L08p Sùp(ep_151), 2), Z'VN(O,l),p~l, Lo-g (resp. e
)
p
p
1
denotes the p-th~logarithm -( resp: exponential-) with by ~onvention LogO x =1,~
1
L
C
(
TI h=p-1
!
et
n 'V 2Log n)
h=l
Logh(2Log n)
1
(i) .Q.(LogLog n) C
T - .Q.(LogLog n)(l+o(l»
j
A
n
n
1 (H) C T
~
1.
n
n
1
1

---

-33-
Before the statements of the results, we need sorne notations.
Define
A = inf h,
F(x)
1 }
B = Sup h, F(x)
o }
R(t)
(l-F(t))-l fA (l-F(v)) dv,
B~t<A
t
Q(u)
-1
{
F
(u) = inf X, F(x)~u} is the quantile function of X.
We shall assume, when appropriste, that
(Hl) F(Log(x))~D(A)
(H2) F(x) is ultimately strictly ·increasing and continuous
\\ve will prove that i f (Hl) anè (H2) are séitisfied, then FE.:D(/\\) ( see Lemma 2)
So, we can use th~ De Haan's representation for the quantile function associated
with a distribution function F such that F€D(/\\)
and F(x) is ultimately strictly
increasing as xtA ( see De Haan(1970), theorems 1.4.1 and 2.4.2):
(1. 2)
Q(l-u) = c
+
rÇu) + fl (r(t)/t) dt, as u +0
o
u
where c
is sorne constant and r(u) is a positive function slowly varying at
o
zero ( S. V. Z ).
Finally, define this assumption on k
(Kr(À))
k satisfies (K) and
r(l/n)/r(k/n)
+
À
Our main results are
Theorern:
Let (Hl) and (H2) be satisfiecl, then
"
(i) for any sequence k sati~fying (K), we have
( 1.3)
(X
n - X k
)/Leg-k
a
n ,
n- ,n
(ii) for any sequence k satisfying (Kr(À)), we have
x
- X
Log k
{
n, n
n-k,n } _
Q(l-l/n) - Q(l-k/n)
(1.4 )
À./\\
( -:'7 )
r
k;n
r(k/n)
Log k
a
r(k/n)
and h
= {Q(l-l/n)-Q(l-k/n~ /r(k/n)
n
n
Corollary 1: Let A>O. Then, if (Hl) and (H2) hold,
(1.3) and (1.4) remain

---

1
1
-34-
1 Remark ( important ): Mason(198Z) has proved that the Hill's estimate
H
= k...,l Ii:xk ( X
- X
)
n
i=l
n-i+1,n
n-k,n
1 is characteristic of distributions satisfying (Ac) in the following sense: sup-
pose that A= +00, then for any real number c, O<c<+oo, one has
(
T
c, for any sequence k satisfying (K)
n
1
if and only if F satisfies (Ac).
In order ta compare H
and T , we remark that this property is not obtained for
n
n
T . Indeed, the Mason's distribution defined by
n
1
F-1(1_Z-m)
m,
m=0,1,2, ....
1
p-l(l_u) = m +(2-m_u) 2m+1 , if 2-m- 1<u<2-m
satisfies the fol10wing pro pert y
F- 1(1_s) - F-1(1-bs)
-1
(Log 2)
as
b -++ ro and bs + 0 .
Log b
.
-1
-1
Thus, by lett1ng b= U
.U
, s=U ·
, we get T
!
(Log 2)
k , n
1 ,n
1 ,n
n
1
where we have used the wellknown representation X
. 1
~ F- (U .
n-1+ ,n
n-1+1
and U
< U
<
<U
are the arder statistics of a sequence of independent
1,n=
2,n=
=
n,n
rv'sunifor~ly distributed on (0,1). However, 1-F(Log(x»
does
not vary regulary
-1
at infinity, in other
words, does not satisfy
(Ac), c=(Log 2)
.
( see e.g.
Mason(1982), Appendix ).
11- Proofs of the results:
First, we show how to derive corollaries 1 and 2 from the theorem. To begin
with, we need 4 lemmas and we define L as the set of distribution
funetions F
satisfying (Hl) and (H2)
r(u)
Lemma 1: If FE. L, then (1.2) holds and
1. It follows that
Lim R(Q(l-u»
u+a
R(Q(l-u»
is S.V.Z.
Lemma 2: Let A>O. then if X has a distribution function FE-L, LogSup(O,X) i5 de-
fined a.s. and has a distribution function GEL and

---

-35-
Log A
S(G-l(l-u)) '\\,R(Q(l-u»
, as u+O, where
S(t)= S
l~G(v) dv, -oo<t<Log A
Q(l-u)
I-G(t)
t
Lernma 3: Let R(t)-tO as ttA. Then if X has a distribution function FEL, exp(X)
has a.distribution function H such that
HE.L and
-1
A
R(Q(l-u)) '\\, T(H
(l-u)) as u+O, where T(t)
B
A
=
(l-H(t))-l Se (l-H(v)) dv, e <t<e
l
H- (1-u)
t
Lemma 4: If FE:.L, then Q(l-u) is S.V.Z.
Proofs of 1emmas l, 2, 3,4
Lemmas l, 2 and 3 are proved in LO(1985b) via 1emmas 3.2, 3.3 and 3.4.
Lemma 4 is proved in LO(1985a) via 1emma 2.
Proof of coro11aries 1 and 2:
Let G be the distribution function of LogSup(O,X). Lemma 2 says that (Hl)
and (H2) imp1y that FED(A). But G(Log x) = FI (x), where FI is the distribution
function of Sup(O,X). And it is obvious that FlE:D(A) if A>O. It fo110ws that
(Hl) and (H2) are true for G. So, we may write (1.3) and (1.4) for G. Further-
-1
more, we have G
(l-u) = Log Q(l-as), a=P(X>O) and 1emmas 1,2 say that we may
-1
R(Q(l-u))
replace r(u)'\\,R(Q(l-u)) by S(G
(1-u))'\\,60_u)
Finally, r-emark that i f A>O
k satisfies (K), we have for large
values of n, Sup(O,X
k
) = X k
and
1
n- ,n
n- ,n
,
Sup(O,X
) = X
,a.s .. With the above remarks, we can see that the corol1ary
n,n
n,n
1 is proved.
Coro11ary 2 is proved in a simi1ar way with 1emma 3.
Proof of the part Ci) of the theorem:
-1
Let GCx)
F(Log x). Since G~L __ , 1emma 4 implies that G
(1-u) is S.V.Z.
At
this step, we need theKaramata'srepresentation for functions S.V.Z:
-1
1
(2.1)
G
(l-u)
eCu) exp (S
(E(S)/S) ds ), c(s)+c, O<c<+=, e::(s)+O as s+O.
u
Sa,
(2.2)
Q(l-u)
LogG-l(l-u) = Logc(u)
+ fl (E(S)/S) ds,
u

---

1
1
-36-
We recall tha t
1
d
( 2 . 3) {X.
, l.~iSn} ~ {Q (U . .), 1S,.sn }, U.
~ 1-U
where
l, n
- -
l , n
- -
l,n
n-i+1,n'
r
o=u
.s ....
o .s. U
-S U
oS U
.s U 1 =1 are the order statistics of a sequen-
,n- 1 n- 2 ,n-
-
n,n -
n+ ,n
,
ce of independent rv's uniformly distributed on (0,1). Therefore,
(2.3) implies
(2.4)
(Log k) T
:::: X
- X
k
}
*
~ Q(l-U
- Q(l-Uk+l,n) =(Log k) T
n
n, n
n- ,n
1 , n
n
Let's apply (2.3). We ob tain that
ILogC(Ul,n)/c(Uk+l,n)i+iLOg(nuk+l,n)/
(2.5)
o < *
T
<
SUPO<S<U
IE(S) 1
=
n =
Log k
Log k
==k+l,n
<
Obviously, we have
1;
~
(2.6)
Anl
0, since Ul,n
0 and Uk+1,n ~ 0 if k satisfies (K).
By (2.1), ~e have also that
(2.7)
SUPO<s<U
IE(s)1
:::: 0
(l)
p
==k+l.n
Therefore, ~e can see that (2.5), (2.6) and (2.7) will imply the part (i) of
theorem 2 if we prove that (LognU
1
)/Log k = 0 (1). But ( see e. g. De
k + ,n
p
Haan and Balkema(1974»
1
k
(2.8)
n k-ï
(U
)
i
kn-;-
N(O,l), which implies
,
-1
.
p
(2.9)
n (~+l)
U
1
.
-+-
L
k+ ,n
We deduce from (2.9) that
-1
(2.10)( n k
U 1
)
1. So,
k+ ,n
';
:
(2.l0b)
(Log nU
1
)/Log k = 0 (1)
k+ ,n
p
which completes the proof of the part (i) of the theorem.
Proof of the part (ii) of the theorem:
Let (Hl) and (H2) hold. Then, (1.2) holds.
(1.2)
Q(l-u) = c
+ r(u) + f1 ( r(s)/s ) ds, as u+O
o
li
We have
*
T
={Q(1-U ,n)-Q(1-1/n)}/Log k
+ {Q(l-k/n)-Q(l-U +
n
1
k l ,n)}/Log k
+ a . b /Log k, .
n
n
(2.11)

---

-37-
w~ere a
and b
are defined in the statement of the theorem. First, we prove that
n
n
(2.12)
A 4/a
o
n
n
U
By (1.2), we have A
= r(k/n)-r(U + ,n) + Jk~~l,n ( r(s)/s ) ds,
n4
k l
Remark that since r(u) is slowly varying at 0, we have on account of (2.10) that
r(k/n)!r(U
1 ' )
k
+
l, in probability
+ ,n
F'.lr thermore ,
U
(2.14)
a~l Ifk~~l,n ( r(s)/s ) dsl ~ ~Og(nk-luk+l,n)ls~ilr(~/~)1
n
where
l
=(Min(~, U 1 ), Max(~ ,Uk 1 )) is a random interval
n
n
k+ ,n
n
+ ,n
At thiS step, we need a lemma
Lemma 5: Let r(u) be a positive function S.V.Z: Let (u ) te a sequence of rv's
n
1 -,\\
~p-d (d ) be a sequence of real
numbers such that
n
-1
u
J 0,
(d .u ) = 0 (1), and (d .u )
o (1) an n+t=
n
n
n
p
n
n
p
then
r(s)
r(s)
1+0 (1)
<
Sup
1+0 (1), as n + +00
P
InJf r(1/ d )
r(l/d )
p
.,
SE
n
sEJ
n
n
..
n
,.
(
:~!; 'j:';
where
un)' Max (~ ,un))
,:. ;~;~
n
.(
.;
Proof of lemma 5: The proof is the same'as that of lemma 3.5 in LO(1985b)
Proof of the part (ii) ofthe.theorem ( continued )
By (2. lOb) ,
~ U
~ 1. Sa, we may apply lemma 5 ta (2.14) and get
n
k+l,n
U
(2.15)
-1 If k+l n
( /
1
a
k / '
(r s)
s ) ds.'S-o·(1). (1+0 (1))
n
n
-
p
P
Combining (2.13) and (2.15), we get (2.12)
Now; we concentrate on A
and show that
n3
(2.16)
A 3/a
À.A
, i f k satisfies
(Kr(À))
n
n
We have
A 3 = Q(l-U
)- Q(l-l/n)
n
l ,n

---

-38-
(2.17)
r(Ul,n)-r(l/n) - I0/ n (r(s)/s) ds,
l,n
Recal1 that
-x
(2.18)
P( n U n~ x)
+
e
as n ++ co
l ,
-1
This means that nUl
= ° (1) and
(nUl,n)
= 0p(l) . Thus, we may apply lemma
,n
p
5 and get
(2.19)
r(l/n)!r(U
)
X lasn++co
l ,n
Then, if k satisfies (K(r(À)), ~e have
P
(2.20)
+
0, as n t + .xl
Now, let
n
B
= I l /
( r(s)/s ) ds
n
Ul,n
We have
ds
B /a
= {r(l/n)/r(k/n)}. I l / n
{ r(s)
n
n
U
r(l/n)
s
l,n
Then, it fol1ows
from lemma 5 and the fact that r(u) is positive that i f k satis-
fies (Kr(À)), we have
(2.21)
{B / a} + À Log nUl
Log nUl
. 0 (1)
n
n
,n
,n
p
By (2.18), we see that
-x
-e
Lim P ( - Log nUl
< x
) = e
= fi (x)
,n
n t+ ex>
We get finally that if k satisfies (Kr(À)), one has
d
(2.30)
B /a
+
À./\\
n
n
(2.16) and (2.12) together imply the theorem.
Proof of the corollary 3:
Previously in the occasion of our study of the same particular cases for the
Hill's estimate ( see LO(1985a), lemma 5 and corollary 5 ) we have proved that
(3.1)
Lim
R(Q(l-u))/p(u)
+ l ,
where peu) = u Q'(1-u),
Q'(u) = dl(u) for
u-}O
values of u
near 1.
With
(3.1), we may handle the different points of corollary 3. Here, we
concentrate on the case where k =(Log n)~) , ~>O

---

-39-
J,-
2/2
Jx
- 2 - t
I-Normal case: F(x) =
(2n)
e
dt
_00
Remark that F(Log x) is the distribution function of the log-normal
law.
It follows that F(Log(.))~D(A). (H2) is obviously true. On the other hand, it
is well known that
l
L08L08(1/s) + 4n+o(1)
(3.2)
Q(1-s)
(2Log(1/s)) 2
+
l
1
ê.S
si-O
2 (2Lo~(1/s))2
l
(3.3)
p(s) == s Q~ (1-s)
(2Log(1/s))-2 (1+0(1)), as s~O.
Notice that we might have used ( see Galambos(lS7G), p.66 )
-1
-3
R(t) == t
(1+o(t
))
and P(s)""R(Q(l-s))
1
1
l
Then
a-
== P- (k/n) (1+0(1)) == (2Log n)2 (1+0(1)) and
n
L08L08 n »)
a
b
Q(l-l/n)-Q(l-k/n)
t~08L08 ~1 ( 1 + O(
n
n
(2Log n)2
Log n
Therefore,
b
=={tLogLog n } (1+0(1)
n
-ax
2-Exponential case: F(Log x) = 1-e
,
a > 0
More generally, since the tail of the quantile function associated with a ge-
neral gamma law y(r,a), r>O, a> 0, admits the expansion
-1
1
(3.4)
H
(l-u) =(Log u)
(1+0(1))
the behavior of T
is the same for all gamma laws because (3.4) doesn't depend
n
on r or o.. Tha t' s why we only consider
x
F(Log x) = 1 - e-
Therefore
-1
Q(l-u) = LogLog(l/s) ,
peu) == (Log(l/s))
then
a
== (Log n)
(1+0(1)), b
=(tLogLog n ) (1+0(1))
n
n
At this step, we apply the theorem.
3-1n this case, we have,
T
= {Log
2
- Log
2
k
}/ Log k
n
p
n,n
p
n- ,n
for large values of n, where 2
,2 ,n, ... ,2
are the order statistics of a
1 ,n
2
n,n
sequence of independent and
standard Gaussian rv's

---

1
-40-
1
1
We have also
2
_t /2
(3.6)
e
dt ), where G is the distri-
r
butiori functionassociated to Sup( e
10), Z), Z'VN(O,l) and m=P(Z > e
10 ))
p-
p-
1
Since X= Log Sup(e
1(1), Z), one has
p-
Q(l-u) = Log
G- 1Gl-mu)
Log {(2Log(1/ms)1+ LogLog(l/s~)+Cil)}
p
p
2 (2Log(1/ms)2
It follows from (3.3) and (3.6) that
p -l(s) = (2Log(1/s)) TI;:r- l Log (2Log(1/s))1
j
Then
'-
1
a
'V 2Log n n~-Pl-
Log, n
n
J=
J
and from (3.6) we deduce after sorne calculêtions
b
= ~LogLog n (1+0(1))
n
Remark that the part 3 of the corollary might have been derived from th~ part 1
of the same corollary after p applications of corollary 1. We have given the
normal case as example but such an operation is possible whenever Z, has a dis-
1
tribution function F such that F(Log(. ))E. D(A) and Log
1 A > O. Even F(Log('))
p-
E: D(~), where ~(x)=e-l/x is the Fréchet's law, we can have the part 3 since
F(Log(·))E.D(~) impliesthat F(·)E.D(A).
III- Simulations
Her~, we will illustrate the behavior of T in the three cases
n
Ci) exp(X)'VE(l),
(ii) exp(X) = Sup(O,Z), Z'VN(O,l) , (iii) F(Log x) = l-l/x
For making our simulations, we have generated an ordered sample u ' 1~i~4000
i
from a uniform rv. Therefore, we have constructed
Ci) an order sample of the standard exponential law
y. = - Log(l-u.)
1
1
.1
and defined T
= ( Log Yn - Log Yn-k )/Log k, k=(n 2
n1
)
Tni = C ( Yi )
n

---

1
1
-41-
1 (ii) an ordered samp1e of the standard Normal law for U.t 1
l
1-
x. =(-2Log(1-u.) )2 ( see e.g. 3.2)
l
l
J
and defined T 2 = C
(x.)
n
n
l
,
(iii) an ordered sample of the Pareto'S law
-1
z. = (l-u.)
and defined T 3 = C
(z.)
l
l
n
n
l
Before we proceed any farther, we remark that with x_= (-2log(1-u.)), we get
l
l
1
*
1
Tn2= "2 T · In fact, we have T
2 T
n1
n2
n1
LogLog(l/(l-u
k
)
+ 0 (
n- •n
)
4 Log k LogO/O-un_k,n)
where T~ is the exact value of T
if we use the true quantile function. With
n2
*
*
l
our data, we get T
= ! T;l ± 0.0179 and
(Log n) T
=(2 Log n)T
± 0,14
n2
n2
nl
The simulations
are given as follows:
n
(~Log n) T
(Log n)T
n1
n2
U4000-i+1
3991
0.3294
0.3294
0.3302
0.002435
".
3992
0.3391
0.3391
0.4042
0.001631
;ifi~~:
f;~~~,i;~r{
3993
0.3625
0.3625
0.4334
0.001620
3994
0.3550
0.3550
0.4324
0.001337
3995
0.4598
0.4598
0.5954
0.000988
3996
0.4693
0.4593
0.6124
0.000437
.'~""~
3997
0.4689
0.4689
0.6130
0.000418
3998
0.4977
0.4977
0.6625
0.000308
3999
0.5116
0.5116
0.6963
0.000297
4000
0.6942
0.9942
1.0332
0.000095
1
2
3
4
5

---

-42-
Comments :
1°) The right co1umn gives the first values of (u ). With the symmetry of the
i
uniform law, we have {l-u , 1~i~4000} ~ {u4000-i+1 ' 1~i~4000 l,
i
1
2°) if k~(n2), simi1ar calculations as in the proof of the part 2 of corollary
3 show tha t a ' " Log n and
a . b '" Log 2. Therefore, if 1
0.p.notes the De Haan/
n
n
n
nI
Resnick estic8te for exp(X) '" E( 1) ,~ we get
(4.1 )
(~Log n) T
Log 2
n1
the same considerations from
the part 3 of coro11ary 3 ( p=l) yield
(4.2)
(Log n) 1
Log 2
n2
where T
denotes the De Haan/Resnick estimate for X = Log Sup(O,Z), Z"'N(O,l).
n2
Notice that (4.1) and (4.2) are weIl illustrated by our simulations since
Log 2 '" 0.69314
. ~'.
3°) The column 4 illustrates the results of De Haan/resnick (1980)
T--
î 1
iï.3
IV- Conclusion
R,
We have proved that for a nice choice of k ( for instance k"'{Log n}
),
we can find the norming constants d
such that
n
d
. T
î
1
n
n
In addition, we havegiven the limit law as the Gumbel
distribution. The same
work has been already done by De Haan and Resnick(1980) under the hypothesis
(Ac). So, for a wide range of distributions be10nging in D(A), we can der ive
from that statistical tests. For instance, we may obtain a recognition code
between a Gaussian sample and an Exponential sample based on (4.1) and (4.2).
Similar tests are specified in LO(1985a)

---

-43-
REFERENCES:
- Balkema., A.A and De Haan, L.(1974): Limitlaws for order statistics. In:
Colloquia. Math. Soc. Boylai. Limit theorems of Probabibility, Keszthely,
17-20.
_ CsorgH, S., Deheuvels, P and Masan, D.M.(1983): Kernel estimates for the tail
index of a distribution. Technical report, L.S.T.A., University Paris 6.
Ta appear in Annals of Statistics.
- CsorgH, S. and Masan, D.M.(1984): Central limit theorems for sums of extreme
values. Proc. Car;;br. P;:il. ~~ath. Soc.
- Davis, R. and Resnick, S.I.(1984): Tail estimates motivated by extreme value
theory. Ann. Satist., Vol 12, N°4, 1467-1487
":1
l, .,~.
De Haan, L.(1970): On regular variation and its application ta the weak con-
. -~.: :
.'~ '.•
c.
vergence of sarnple extreme. Mathematical Centre, Amsterdam.
lilil~~k
- De Haan, L.and Resnick, S.1. (1980): A simple asymptotic estimate for the tail index
.
of a stable distribution. J. ~oy. Statist. Soc. D 42, 83-37
- Galambos, J. (1978): The asymptotic theory of extreme arder Statistics. Hiley,
?'Î
tfew-York.
;/~".
. :«t
Hill, B.M.(l974): The ran~,:-frequency
form of Zipf's law. Journ. Amer. Assac.
.(
Dec., Vol 69, nO 348. Theory and Methods Section.
- Hill, B.M.(1975): A simrle
general approach to the inference about the tail
index of a distribution. Ann. Statist. 3. ·1163-1174
LO, G.S.(1985a): Asymptotic behavior of Hill's estimate. Tech. Report, n028,
L.S.T.A., University Paris 6. Submitted
- LO, G.S.(1985b): On the CLT for sums of extreme values. Tech. Report. n030,
L.S.T.A., University Paris 6. Submitted.
- Mason, D.M.(19~2) Laws of large
numbers for sums of extreme values. Ann.
Probabb. Vol 10, n03, pp.754-764.
- Teugels, J.L.(1981); Limit theorems on order Statistics. Ann. Probab.9, 868-880.
~i
,]
,~~I

---

,
"
:
1
1
)
",
)
''',
)
''',
DEUXIEME PARTIE.
CARACTERISATION EMPIRIQUE DES EXTREMES.

---

-44-
The Annals of Probahil ity
1991. Vol. 19. No. 2. 783-811
THE ASYMPTOTIC DISTRmUTION OF EXTREME SUMS
By SANDOR CSORG6,1 ERICH fIAEUSLER AND DAVID M. MAsON 2
University of Szeged, University ofMunich and
University of Delaware
Let X I • n -'> •.. -'> Xn • n be the order statistics of n independent ran-
dom variables with a common distribution function F and let k
be
n
positive integers such that k n ..... co and kn/n ..... a as n ..... co, where
o -'> a < 1. We find necessary and sufficient conditions for the existence of
normalizing and centering constants An > 0 and C
such that the se-
n
quence
converges in distribution along subsequences of the integers ln} to nonde-
generate limits and completely describe the possible subsequential limiting
distributions. We also give a necessary and sufficient condition for the
existence of An and Cn such that En he asymptotically normal along a
given subsequence, and with suitable An and en determine the limiting
distributions of En along the whole sequence ln} when Fis in the domain
of attraction of an extreme value distribution.
1. Introduction and statements of results.
Let X, Xl' X 2 .•. be a
sequence of independent nondegenerate randorn variables with a cornmon
distribution function F(x) = P{X s x}, X ER, and for each integer n z 1, let
Xl n S
... S X n n denote the order statistics based on the sample Xl"'" X n ·
Th'roughout the paper k n will be a sequence of integers such that
k
-> 00
and
as n -> oc;
Tl
(1.1 )
or
k
=
[na]
with 0 < a < 1,
n
\\1
•
where [.] denotes integer part. (We shaH refer to the first case as the case
a = 0.) The study of the asymptotic distribution of the (properly normalized
and centered) surns of extrerne values
~ ,
k n
n
( 1.2)
L X
L
X
Tl + 1 - i . n
=
i • Tl
1
i~l
i=n-k + 1
n
was initiated in [6) for the case when a = 0 in (1.1) and under the restrictive
1
Reeeived April 1988; revised December 1989.
.
Ipartially supported by the Hungarian National Foundation for Scienti~c Research, Grants
1808/86 and 457/88.
1
2Partially supported by the Alexander von Humboldt Foundation, NSF Grant DMS-88-03209
and il Fulbright grant.
, .
AMS 1980 subject classification. Primary 60F05.
Key words and phrases. Sums of extreme values, asyrnptotic distribution.
1
783
1
,,,'
\\1

---

-46-
..
1
, ' 1
1
J
' ' \\
J
' ' \\
EMPIRICAL CHARACTERIZATION OF THE EXTREMES:
A FAMILY OF CHARACTERIZING STATISTICS.
Gane Samb Lô
Université Saint-Louis - Université Paris VI
(LSTA).
Abstract. Given 'only a sequence of i.i.d.
random variables with common
unknown distribution function F, we provide a class of four statistics
i
which character~zes the limit law of the largest or ~he smallest
1
observation. No icondition is required on F.
i
AMS 1990 subject classifièation.
Primary 62G,
Secondàry 62F.
Key words and phrases.
Ext~eme value theory,
order statistics,
empirical distribution function,
domain of attraction of the maximum,
characterizatio~.
Mailing and permanent address. UER de Mathématiques Appliquées et
d'informatique, Université de Saint-Louis, bp 234. Sénégal.
Research affiliation. LSTA,
Université Paris VI,
T44/45,
3E,
4 Place Jussieu.
Paris Cédex 05.
France.
1

---

,",
,
-:-47-
, ' 1
1
) '4,
) '4,
I- Introduction and statement of results.
a)
Statement of the problem.
Let Xl'
X , · · · be aisequence of independent copies
(s.i.e)
of a real
2
rahdom variable
(r.v.)
X with p(x~x)=F(x), xem. A major step of the
extreme value theory consisted in determining necessary and sufficient
1
1
conditions on F for/the convergence in distribution to ai non-degenerate
:
1
l
'
distribution M of the maximun X
=max(x , ... ,X ),
when l t is
;
n,n
1
n
1
;
appropriately centered and normalized by two sequences of real nurnbers
(a >0)
1 and
(b)
l'
that is
n
n~
n n~
(1.1)
IyIxem,
lim
p (X ,', ~a x+b ) =M (x) .
n~+oo
n,n
n
n
,.
~If
(1.1)
holds,
i t is said that F is attracted to M or F belongs'to the
'domain of attractior of M, written FeD(M) .
\\
Several authors such as Fréchet (1927) , Fisher and Tippet(1928),
Gnedenko(1936)
led to a complete solution of the probabilistic problem
of finding conditions under which
F belongs to D(M)
along with the
determination of the sequences
(a ) and
(b ).
Further developments and
n
n
other
references
may
be
found
in
de
Haan (1970)
and
Galambos (1978) .
Pickands(1975)
also gave necessary and sufficient conditions on F in
order that FeD(M).
We summerize here the classical characterization of
the extreme in
Theorem A.
If FeD(M),
where M is not degenerate,
then M is
necessarily one of these three types of distribution
x
A(x)=exp(-e-
), xe~,
(Gumbel's type);
-0
~
(x)=exp(-x
),
xe~ , 0>0,
(Fréchet's type of parameter 0);
o
+
~ (x)=exp«-x)O), xe~_,o>O, (Weibull's type of parameter 0).
o
Specifically,
(i)
FeD(A)
iff,
IyIte~, lim
(l-F(x+tR(x,x ,F))/(l-F(x))=e- t ,
x~x
0
o
2

---

.,,,;
.,";
"
1
1
J
' ' \\
J
' ' \\
-48-
-1
JX
where R(X'XO,F)=(l-F(X))
0
l-F(t)
dt,
-oo<x<x '
o
x
x =x
(F)=sup{x,
xe~, F(x)<l}i
o
0
l-F(Àx)
(ii)
FeD(~ ),
~>O, iff x
(F)=+oo and for all
-~
À>O,
lim
À
i
~
0
l-F(x)
x~+lXl
(iii)
FeD(~ ), ~>O, iff x
(F)<+m and F(x -1/·)
e D(~ ).
~
0
0
~
1
i
Recènt characterizations of D(A),
D(~)=U
0 D(~ ) and D(~)=U
0 D(~ ),
~>
~
~>
~
:
i
usihg a,unified approach are available in de Haan(1970). Our aim is to
.~ Igive statistical, i.e.,
empirical characterizations of the class
1
~tr=D(A)U D(~)U D(~) of distributions attracted to sorne non degenerate
~:distribution. Our first motivation cornes from Resnick(1987):
.;.
Il Suppose 'that
by stretching the imagination,
one believes the data: to
\\
\\
i . • be
modeled by iid assumption.
Usually,
that
is the
Gumbel,
so
called
double exponential,
which is chosen ...
the underlying d.f.
may not be
extreme value,
but we robustly hope it is at least
in a domain of
attraction . .. "
This raises the problem of choosing the right extremal type ln
modeling sample extremes.
ls the systematic use of the double
exponential law justified? Thus,
the non-parametric statistical
problem is the following:
Given only
the
independent
observations
Xl'
X ... vith
common
unknown
2
distribution function F,
is it possible to ansver these three questions
(ql) Does not F lie in r7
(q2) Does F lie in r7
(q3) i f Fer,
to vhich of D(A),
D(~) or D(~) F is attracted7
We shall answer any of these questions.
3

---

, ' 1
1
, ' J
t
}
''\\
}'\\
- 4 9-
On an other hand,
many theorem limits for statistics under hypotheses
like FeD'(L\\),
FeD (q,)
or FeD (t/J) ,have been continuing ta be established;
1/
particularly when FeD(q,),
(see Mason(1982),
Csorgo,
S,-Mason(1984),
1/
Lô(1986-89),
Csorgo,
S.-Deheuvels-Mason(198S),
Smith(1987),
Davis and
Resnick(1984)
etc ... ).
It is clear that a correct an?wer of questions
,
i
(ql),
(q2)
and ~q3) will yield a general rule for the application of
1
i
1
such results. This was our second motivation.
1
1
1
1
1
Mason(1982)
partly answered the problem.
He proved
a
Theorem B. Let k=[n ], O<a<l, t=O,
then
-1
-1
i=k
.
T
(2, k, t)
= k
L'
Il (y
,
1
- Y .
)
n
l= 1 +
n -l + , n
n -l, n
converges in prbbability to 1/7f,
O<7f<+co,
iff FeD(q, )\\, where Y.
7f
l,n
10gX.
,
i=l, .. :. , n,
Xl
:S ... :sX
are the order statistics of Xl' ... ,
l,n
,n
n,n
X
and log(.)
stands for the natural logarithm.
n
b- The characterizing statistics.
Troug'hout Xl'
X , '"
will denote a
s.i.c of a
r.v.
with P(X:sx)=F(x),
2
F(l)=O. However, we rather work with the the sequence Y.=logX.,
i=1,2 ..
l
l
and P(Yl:Sx). We introduced
(
k D)
k- 1"i=k
j=i
. (
/
A l ,
, c- =
L .
IL .
1 J
1- <5 •.
2) ( Y .
- y
.
) (y
.
'
- Y .
)
n
l=l+
J=l+
lJ
n-l+l,n
n-l,n
n-J+l,n
n-J,n
and combine i t with Hill's statistic to set the following characterizing
statistics:
-1/2
T
(l,k,t)=T
(2,k,t)A
; T (2,k,t)
is defined above;
n
n
n
n
T (3,k,t)=(Y
k()
-y
k(
»/T (2,k(m),t(m»; T (4)=Y
;
n
n-
p , p
n-
m
n
n
n, n
T (S)=T
(2,t,1); T (6)=(Y
-y
k
)-l T
(2,t,1);
n
n
n
n-l, n
n-
, n
n
T
(7)=(Y _
n-Yn-k
n
n l
)-2A
(l,t,l); T (8)=n- V (Y
-y
k
)-1;
,
,n
n
n
n-l,n
n-
,n
4

---

:
1
"
1
}
.,,"
}
.,,"
-50-
}
.,,"
T(9)
=(y-Y
)/(y-Y
k
)
n
0
n-l,n
0
n-
,n
·(only defined when y
= log(sup{X,F(x)<l})<+oo),
o
L
L
where v is a positive real number,
p=n+[n ], m=n- [n ], L=2-a-o,
with
1
!
1<o+(3<o+a<2,
o,.
is Kronecker' s
symbol and,
finally,
k and t are non
1J
negative integers such that
O~=t(n)<k=k(n)<n, (k(n)/n, t(n)/n)~(OrO) as n~+oo.
1
•
1
1
~e should notice that Dekker, De Haan!
aand Einmahl(1989)
introduced
~*=k-1~~=k1'(Y . 1 -y k )2 and obt~ined the discrimination of the
1
n
LJ=
n-1+ ,n
n-
,n
:
*
2
three extremal types with the couple
(T
(2,k,t),
A /T
(2,k,t)
). This
n
n
n
will be obtained again here since direct calculations show that
*
Ais.:
n
the double of An'
and let rr
,p<n, denote the projection of ~n on ~p.
p,n
c- The results.
a
Theorem 1: Let FEI,
then for all sequences k=[n ] , t=[n(3],
0.5<(3<a<l,
for any v>O,
i)
rr ,7(Yn)
converges almost-surely to sorne rr4,7(~)' ~Ee, as n~;
4
ii)
y
converges in probability to sorne ~Ee. Specifically,
n
1)
If FED(A) ,
then ~=(l,O,O,yo'O,O,O); YO=+oo or YO<oo.
2)
If FED(_ ),
then ~=(l,l/7,O,+oo,l/7,O,O); 7>0.
7
{
-1} -1/2
3)
If
FED (!/J 7)'
then ~= ( 1- (2 +7)
, 0 , 0 , y 0 ' 0 , 0, 0) ;
y 0 <00 and 7> 0 .
iii)
in addition,
4)
If FED(A)U D(-7)'
then Tn(8)~O, a.s., as n~.
5)
If FE D(!/J7) ,
then Tn(9)~O, a.s.,
as n~.
This theorem can be inverted as follows.
5

---

" ,
" ,
1
, ' 1
1
..
1
1
"
1
-51-
J
''',
J. ''',
CX
[3
Theorem 2.
Let k= [n J, t= [n J,
0<0<0. 5<[3<cx<1,
1<[3+0 <CX+O <2,
T=2-CX-0,
1
2v=min(l-cx,
0+[3-1).
a)' I:5 Y ~
lA as n~+oo with YO=+oo.
If further T
(8,[3/2)~
0 as n~, then
1
n p
n
p
1)' FeD (A)
whenever c=l and d=f=O i
1
2)
FeD(~ ) whenever c=l and d=f=l/~, ~>O.
~
i
b)
Let Y
~p lA as n~ with YO<oo.
!
n
1
t
i
4)'
If c=l,
d=f=O and T
(8,[3/2)~ 0 as n4+oo, then FeD (A) i
1
n
p
i
2
c
5)1
If 1~c<v2, d=f=O and T (9)~ 0, thenlFeD(~ ), where ~=-2+ --2--'
1
n
p
~
c
-1
r
These two theorems show that the family of statistics
(quoted below
as a,Family of Characterizing Statistics for the extremes,
FACSEXT)
do
r
char,cterize the extremes.
r
In the next section,
we shall recall or introduce sorne technical
lemmas.
Sorne of them are known or may be easily derived from results
r of De Haan(1970). However Lemmas 7 and 8 will be completely proved
because of their roles in the proofs of the theorems.
r
r
II- Technical lemmas.
Set
z
R(X,Z,F)=(l-F(X))-lI
(l-F(t))
dt,
X<Z~X (F) with R(x,x ,F)~R(x,F) i
0 0 ·
x
"
r
W(x,z,F)=(l-F(x))-l IZ I~l-F(t)) dt dy, x<Z~Xo(F) with W(x,Xo,F)~W(x,F)i
x
y
x
1
1
1
G(x)=F(e ) i
F-
(u)=inf{x,
1
F(x)~u}, F- (O)=F- (O+)
Lemma 1.
For any ~>O,
(i)
FeD(A)
iff GeD(A)
and R(x,G)
as x~x (G)=y i
o
0
(ii)
FeD(~ ) iff GeD(A)
and R(x,G)~l/~ as x~y i
~
0
6

---

-52-
"
,
, ' ,
)
J
' ' \\
J
' ' \\
(iii)
FeD(W ) iff FeD(W ); and then R(x,G)~O as x~Yo;
a
a
(iv)
FED(W ) iff x
(F)<oo,and F(x -l/o)ED(W ).
a
0
0
a
Lemma 2.
(De Haan(1970)) ~
,
W(x,F)
(i)
FED(A)
iff W(x,F)
and R(x,F)
are finite for x~x
and l lm "
2 1.
o
i
x~x R(x,F)
01
1
W (x, F)
(1-11 (2+'1) ) .
x
(F)<+~ and lim
1
,
o
i
x~x
R(x,F)2
i
,
o
1
1
1
j
Funçtions s(u),
O<u<l,
such that for all À>O,
lim
0 s(Àu)/SCu)=l, are
u~
called Slowly Varying functions at Zero
(SVZ)
and are greatly involved
in-our proofs. Here are sorne of their properties.
Lemma 3. Let s(u),
O>u>l, be SVZ. Then,
(i)
it admits Karamata's representation
(KARARE)
(2.1)
s(u)= c(u)
exp(J~ t-lb(t) dt), O<u<l,
where
(b(u) ,c(u))~(O,c) as u~O, O<c<oo.
-0
(ii)
for any 0>0, U
s(au)~+oo as au~O, a~+oo.
Lemma 4.
(i)
FED(~ ) iff ulla F-l(l-U) is svz.
a
(ii)
FED(W)
iff u-l/a
(x (F)-F-l(l-u))
is SVZ.
a
, 0
(iii)
FED(A)
iff there exists a SVZ function s(u),
O<u<l, and a
constant b such that
1
(2.2)
F-l(l-U)=b-s(u)+Ju t-ls(t)
dt,
O<u<l.
Proof.Cf.
Theorem
2.4.1
of
de
Haan(1970)
for
(i).
(ii)
is
easily
derive from
(i)
and Theorem A. The representation
(2.3)
is originally
7

---

•
1
,
-53-
J
' ' \\
due to de Haan(1970).
It is now generalized by Deheuvels-Hauesler-Mason
(1988)
so that i l will be qoted as the DDHM representation.
i
The following two lemmaswill be used to overcome discontinuity
problems.
1
Lemma 5.
Let G be any distribution function.
Then
1
-~
1
i
(i)
for
any u,
O<u<l,
G(G
i(l-u))=u
or
lies
on
a
constancy
interval
-1
-1
1
ofG
i
(ii)
for
any
-oo<x<too,
G
(G(x) )=x
or
lies
on
a
constancy
interval of G.
Proof.
Omitted.
Thi$
lemma combined with Lemma 1
in Lô (1986b)
and Lemma 4
in Lô (1989)
\\
.
\\
along with the Balkema-de Haan representation(1972)
(Cfi
Smlth(1987))
yields
Lemma 6.
Let GED(A),
then
-1
(i)
(l-G (G
(l-u)) /u -7 1 as U-70.
1
(ii)
R(G-
(1-U))
is SVZ.
The following two lemmas will be useful for our characterizations.
Lemma 7.
Let F be any distribution function satisfying
(i)
R(x)
and W(x)
are finite for x<x
(F) i
o
(ii)
(z-x)/R(z)-7+oo as X-7X ,
Z-7X ,
X<Zi
o
0
2
(iii)
(z-x)
/W(x)-7+oo as X-7X ,
Z-7X ,
X<Zi
o
0
then,
R(x,z)/R(x)-71 and W(x,z)/W(x)-71 -7+00 as X-7X '
Z-7X ,
x<z.
o
o
8

---

, '
, ' ,
1
J
'4,
-54-
Proof. Direct computations give
(2.4)
R(x,z)=R(x) (1-E1) and 0~E1~(1+(Z-X)/R(z))-1
and
2
-1
(2.5) W(x,z)=W(x) (1-E2-E3)j
0~E2~(1+(z-x) /W(z))
j 0~E3~R(z)/(z-x).
These two remarks combined with Points
(i)
and
(ii)
prove Lemma 7.
\\
i
Lemma 8. Let !Fer, then for u=l-G(x) ,
v=l-G(z), v/J~o as x~y , z~Yo'
!
i
0
(i) (z-x)/R(z)G)~+ooj
(ii)
(z-x)2/W(x,G)~+00, as x~Yo' z~Yo·
Proof. Lemma 1 in Lô(1986)
implies Part
(i)
for GeD(A).
If FeD(w ),
~
KARARE combined:"with·Formula
(2.4.6)
of de Haan(1970) yields Part
(i),.
Part
(ii)
follows from Part
(i)
and Lemma 2 above.
\\
\\
To finish with .this section we recall properties of empirical distribu-
tion functions
(EDF). The EDF associated with Y ,.· .'Y
is defined by
1
n
(2.6)
G (x)= #{i,
l~i~n, y.~x}/n, xeR.
n
l
Let U (s),
O~s~l, be the EDF associated with the n
first
observations
n
of a s.i.c. of a uniform r.v. on (0,1),
U '
U ' . . . . .
1
2
These basic facts will be constantly used.
Fact 1. We may WLOG and do assume that
{l-G (x), xeR, n~l}={U (l-G(x)), xeR, n~l}.
n
n
Fact 2.
For any~, 0<~<0.5, lim sup
n~ suPO
11U (s)-sl<oo.
n~
~s~
n ·
cx
Fact 3.
Let k=[n ],
0.5<cx<1.
Then kU
/n~l, a.s., as n~+oo.
k ,n
Fact 4.
9

---

-55-
"
1
•
, ) 4 ,
) '4,
Un(s)
Un(s)
------<suPU
< <1
<À)=l.
s
-s-
s
l,n
Fact 5.
For n fixed,
there exists a
sequence
(t
) >1 such that t
Y
P p-
p~ n,n
as p~oo and for all p~l,
U
(s)
U
(l-G (x) )
U
(s)
n
.
f
n ,
f
ln
<
ln
~
sup
n
<su
U
1-G (x)
1-G(x)
-
Pu
~s~l
s
1
~s~l
s
0< <t
n
-x-
O~x~t
,
p
l,n
P
II
Fact 2 is derived from Theorem 4.5.2 and Formula 1.5.3 both in Cs6rgo
and Révèsz(1981).
Fact 3 is a simple consequence of Fact 2.
Fact 4 is
obtained from direct computation.
Fact 5 is implied by Lemma 5 with
1
t
=Y
when G(G-
(l-U
))=1-U
, t =Y
-l/p otherwise.
p
n, n
1 , n 1
, n
p
n, n
111-
Proofs of the theoremJ.
x
From now on R(x,.)
and W(x,.)
are used only for G(x)=F(e )
Furthermore,
Fact 1 means
(3.0)
{y,
,
J, n
1/ Proof of Theorem 1.
Ci.
u =l-G(Y
k
),
v =l-G(Y
) ,
k= [n J,
0.5<(3<Ci.<1.
n
n-
,n
n
n-l,n
First we have to prove that
(3.1)
O~v ~u ~O and v /u ~O, a.s.,
as n~.
n
n
n
n
By Facts 1,2 and 3,
we have
o
1-G
(y
)+o(n-
)
U
1
+o(n- O)
(3.2)
v /u =
n
n-l,n
__l_+__,~n
~
n
n
-0
-0
1-G
(y
k
)+O(n)
U
1
+O(n
)
n
n-
,n
k + ,n
a.s.
as n~ whenever 0<0<0.5, (3+0-1>0.
The proof of Theorem 1 will follow from the partial proofs of State-
ments
(Sl),
(S2),
etc . . .
(Sl)
T
(2,k,l) =R(x ) (1+0(1)),
a.s.,
as n~, with x =Y
k
'
z =Y
n
n
n
n-
,n
n
n-l,n
10

---

·,
, ' ,
1
: '
j
'~
j
.~
-56-
It is easy to check that
,(2,k,l)=nk-lJZn
(3.3a)
T
(l"':G
(t))
dt,
n~l.
n
n
;
x n
By Fact 2,
for any~, 0<~<0.5,
(3.3b)
T
(2,k,l)=R(x
,Z
){1+0(n-a-~+1(z -x )/R(x ,Z ))}, a.s., as n-7ro.
n
n
n
n
n n
n
,
Choose ~ such that ~=~+~-1>0. By Lemmas 7 and 8 and Statements (3.0)
and
(3.1) ~ we have
!
(3.4) R(xl,z )/R(x )~1, a.s., as n-7ro.
n
n
n
We get
(3.5) T,(2,k,l)=R(x ) (1+0(n-~(z -x )/R(x )), a.s. as n-7ro.
n'
n
n
n
n
Either GeD(~~), ~>O and we apply Formula 2.4.6 of de Haan(1970):
R(x )/(y -x )~q=(l-K)/K as n-7ro with K=(1-1/(~+2)) and 0.5<q<l, to get
n o n
J~
{ -1
}\\_~
-~
(3.6)
O~n
(z -x )/R(x )~ 2q
(z -x )/(y -x ) n
~2n
/q~O as n-7ro.
.
n
n
n
n
n o n
1
Or FeD(A)U D(~ ). By Lemma, u1/~ F- (1-U) is SVZ when FeD(~ ). It may
~
~
1
also be shown from
(2.3)
that F-
(1-u)
is SVZ. Now using (3.0)
and
KARARE,
we get in both cases for c,
O<c<~,
1
1
1
1
(3.7a) -2 (u /v )À-c~F-1(X
)/F-
(X
k
)=F-
(1-U
1
)/F-
(1-U
1
)
n
n
n-1 n
n-
,n
1+ ,n
k + ,n
À+E:
~2 (u /v)
,
n
n
a.s. as n-7ro,
where À=l/~ when FeD(~ ) or À=O when FeD(A). Since
~
1
1
G-
(1-u)=logF-
(1-U)
and since u /v - na-~, a.s. as n-7ro, it follows
n
n
(3.7b)
z -x =O(log n)
a.s. as n-7ro.
n
n
Thus,
(3.8)
O~(z -x )/(n~R(x ))=O( l07~ x
; ;
),
a.s.,
as n-7ro.
n
n
n
n~
n~
R(x)
n
-1
By Lemma 6,
R(G
(l-u))
lS
SVZ,
and since u -(k/n),
a.s. as n-7ro,
n
1
R(x )_R(G-
(1-k/n)),
a.s.,
as n-7ro.
Hence,
by Lemma 4,
n
11

---

" ,
"
1
) 4 ,
) '4,
-57-
(3.9)
n~/2R(X )_n~/2R(G-1(1-k/n))~a.s., as n~.
n
(3.3b),
(3.5),
(3.6),
(3.8)
and
(3.9)
together prove
(Sl).
2
2
(S2)
A (l,k,l)=(T
(2,k,n/T (l,k,l))
=W
(x ) (1+0(1) )=KR(x)
(1+0(1)),
n
n
n
n
n
n
a.s.,
as n~, wh~re K=l if FeD(A)U D(~)
and K=1-1/(7+2)
if FeD(~ ).
7
1
1-G (t)
dt dy.
n
By Fact 2,
(3.11)
A
(l,k,l)=W(x ,Z ) (1+0(n-/l(z -x )2/W(x ,Z )), a.s. as n-7+00
n
n
n
n
n
n n
•
By Lemmas ~ and 8 and Statements
(3.0)
and
(3.1), we have
(3.12)
W(x ,Z )/W\\ (x ) -7 l, a.s.,
as n-7+oo.
n
n
n
Hence,
Lemma 2 yields
(3.13)
W(X )/R(x )2 -7 l,
a.s.
as n-7+oo.
n
n
It follows from
(3 .11),
(3.12)
and
(3.13)
that
(3.14)
A
(l,k,l)=W(x)(1+0(n-/l(z -x )2/ R (x)2)),
a.s.
asn-7+oo.
n
n
n
n
n
But
the
calculations that
led to
(3.6)
and
(3.8)
showed that
for
all
p>O,
O«~ç,
for any Fel,
(3.15)
(z - x ) ( n- P R(x )-ç -7 0,
a . s .,
as n-7+oo.
n
n
n
Thus
(3.13)
and
(3.14)
ensure
(S2).
-1/2
(S3)
T
(l,k,l)
-7 K
, a . s . as n-7+oo.
n
(Sl)
and
(S2)
prove
(S3).
(S4)
T
(3,k,l)
-7 0, a.s. as n-7+oo.
n
The proof is direct using KARARE and DDHM's representation and SVZ
functions properties. We therefore omit it.
(S5)
T
(4)~ Y , a.s. as n-7+oo.
n
0
12

---

: ,
, ' \\
1
..
1
j
'4,
This is obvious.
-58-
1/0 if FED(cP
)
(S6)
T (2,t,l)
~
0
{
n
p
0 if FED(W)U D(A) .
See Mason(1982)
for FED(cP),
Lô(1986a)
for FED(A)
and Dekkers and al.
(1989)
for the remainder case.
(S7)
T
(6)
~
0,
as n~+oo.
, n
p
We use the device of Fact 5 in
(3.3a)
by consedering the integral
as an improper one with respect to the upper bound. Remarking that
1
(t/n)-(l-G(Y
)),
and putting Zn(l)=sup{IU
(s)/sl,
U
~s~l}, we get
n-l,n
n
l,n
(3.17)
T
(6)~2 Z (1) R(z )/(z -x).
n
n
n
n
n
This together with Fact 4 and Lemma 8 prove
(87).
(S8)
T (7)~
0,
as n~+oo.
\\
n
p
As for T
(6), we have
n
(3.18)
T (7)~2 Z (1) wez )/(z -x )2 ~ 0 as n~+ro.
n
n
n
n
n
(S9)
If FED(A)U D(cP),
then T (8)~O,
a.s.
as n~+oo.
n
v
Recall that T
(8)=n- /(z
-x ). Now by DDHM's representation
(cf.
Lemma 4)
n
n
n
and by
(3.0), we have for aIl À>l,
ÀC
1
1
(3.19)
z -x ~G-1(1-c )-G- (1-ÀE )=S(ÀE ) -S(E )-J
n t- s(t)dt,
n
n
n
n
n
n
E n
for large n,
where c =U
1
. Now,
the properties of SVZ functions easily
n
1+, n
yield for any fixed c,
0<c<0.5;
-E+(l-E)log À ~1,
(3.19)
z -x ~(l-c)s(t/n){-E+(l-c)log À} ~ 0.5 s(t/n), a.s. as n~+oo.
n
n
Thus, by Lemma 4,
V
(3.20)
nV(z -x )~0.5 n
s(t/n)~, a.s. as n~+oo,
n
n
which proves
(S9).
Finally
(S10)
If FED(W),
then Tn(9)~O, a.s.
as n~+oo.
is a simple consequence of Lemma 4.
13

---

·,,,;
"
1
1
J
' ' \\
J
' ' \\
-59-
8tatements
(81)-(810)
together prove Theorem 1.
2- Proof of Theorem 2.
First us~ Fact 2 ih (3.3a) ang get for any 0<~<0.5,
(3.21)
T
(2,k,t)=R(x,z)
{1+0 (n-~(z -x )/R(x,z ))}=:R(x,z )(1+0 (B )).
n
i n
n
P
n
n
n
n
, n
n
p
n
[
'
The identity
(3.2~) belowand (3.2)
together yield
1
- (~+o-a)
i
.
[
O~B ~2n
, al. s. as n~. Chooslng ~ close to 0.5 irrplies
n
1
,
(3.22)
R(x,z )=Ti(2,k,t) (1+0 (1)), as n-Hoo.
n
n
n
p
8econdly,
(3.23)
A (l,k,t)=W(x ,z ) {1+0 (n-~(z -x )2/W(x ,z ))}, as n~+oo.
;
n
n
n
p
n
n
n
n
Using the same techniques,
we get, under the assumptions of Theorem 2,
2
for c =l/K,
(3.24)
W(x,z )=K\\T
(2,k,t)2
(1+0
(1)),
as n~+oo,
n
n
n
p
which together with (3.22)
yield
·(3.25)
W(x ,z )/ R (x ,z )2 ~
K,
as n~+oo.
n
n
n
n
n
p
We now want to drop z
in
(3.25).
It suffices to check whether the
n
conditions of Lemma 7 are satisfied. We will do that only for R(x ,z )
n
n
and the method will extend to W(x ,z ) word for ward. Apply Fact 5 ta
n
n
T (2,t,l)
and get for Z (2) =inf{ lU (s) I/s, U
~s~l},
n
n
n
1 ,n
y
(3.26)
T (2,t,l)~Z (2){R(z )/Z (1) -(nit) f 0
1-G(t) dt}
n
n
n
n
Yn,n
1
8ince T
(2,l,l)
is bounded in. probability, u- 1J1
1
(1-s)dG-
(s)=0(1)
n
-u
as u~O, by Propositions 3 and 4 and 8tatement
(20) all in Mason(1982).
This with this simple idendity for any distribution function G,
.
1
-1
Yo
(3.27)
Vu,
O<u<l, J
(l-s)dG
(s)=J_
(l-G(x))dx
1 -u
1
G
(l-u)
together imply that
14

---

, ' 1
1
-60-
:
1
, ' .
1
J
' ' \\
J
' ' \\
Yo
Yo
-1
1
-1
nJ
1-G (x)
dx=nJ
1-G(x) x
(nUl
) (U
J
(l-s) dG
(s) =0 (1).
Yn,n
,n
1 ,n
1-U
P
G- 1 (l-U
)
l,n
1 ,n
Thus "
(3 . 28)
T ( 6 ) 2:Z
(2) {R (z ) / Z ( 1) (z - x
- T (8, #3) 0
( 1) } .
i l
n
n
n
n
n
n
p
Thus,
the convergence of T
(8,~/2) to zero, Fact 4 and (2.28) imply
n
(3.29)
R(z )/(z -x )~ 0, as n~+oo,
n
n
n
p
whicn is also immediately implied by T (9)~ 0 as n~+oo since
n
i
p
1
R(z )/(z -x ):s(y -z )/(z -x )=l/(-l+l/T
(9)).
n
n n
0
n
n
n
n
It follows that R(x ,z )/R(x )~ 1 as n~+oo. By the same methods
(replace
,
n
n
n
p
T (6) par T (7)): and remark that lim sup W(x):s(lim sup R(x))2)
, we also
n
n '
x~yo
x~yo
have W(x ,z )/W(x )~ 1 as n~+oo and thus
n
n
n
p
(3.3D)
lim
W(x )/R(x )2=K,
in probabi1ity.
1
n~+oo
n
n
1
In view of Lemma,s 1 and 2,
Theorem 2 will be proved i f we show that
for Y~Yo' thereexists a subsequence xn(m) with Y~Yo iff n(m)~ and
(3.31)
lim
W(x)/W(x ( )) (resp R(x)/R(x ( )))=1 in probability.
y~yo
n m
n m
The
proof
of
this
is
quite
direct
and
technical
aand
requires
the
convergence of T (3,k,f)
to zero. We state in the
n
15

---

.. '
J
'C\\,
-61-
APPENDIX
Recall basic facts
n-Psn- P + n-1s(k+1)sn-P + 2n- 1 for k=[na ] ,
p=l-a.
By fact 2,
-P
-0
- p - 1
n
-cn
SU '
sn
+2cn
, a . s . ,
as n~+oo, 0<0<1/2,
k +l,n
O::s:0+a-1<1,
o<c< min
(0,01,p/(1+p),p/(2
+ p)).
Set a
P
=n-
, n~l. The sequence a =1> ... > a.>a. 1>" .makes a
n
1
J
J+
'
partition of
[0,1].
For x~y , U=l-G(x)~O,
there existsÎat each
o
step of this limit an integer n such that a
susa
::s:a.
Let
n+2
n+1
n
~
~
m=(n)=n-[n ],
~=2-a-o, p(n)=n+[n ], ~=2-a-o.
Remark that O. 5 s p<1.
One has a
-a =- [n~] f' (ç ) with f (x) = x- P
m
n
n
and mSÇ sn.
Thus,
for large values of n,
n
\\
0
a
-a ~p(l-c)m
and a
-a ~p(l-C)D-o.
m
n
n+2
p
..
Since m (n) -p (n) -Tl, we have for o<c<min (p/ 11+p) ) , (p/ (2-'p) ) ,
-0
'
usa sa -p(l-c)sa -cm
~J~( ) 1
'
m
m
m
m + ,m
d .
-0
-0
usa
2~a +p(l-c)p
~a +2cp
SU
( )
_
n+
p p
k
p +1., P
and
u-Uk(m)
+ 1,m- Uk(p)
+ 1,p a.s.,
as «~o.
yo
JYo JYo
Put M(x)
=
1-G(t)
dt,
m(x)
=
I-G(t)
dt dy.
J x
x
y
-1
Using G
(G(x))s x for all x for all d.f.
Gand noticing that both
M( .)
and m ( .)
are nondecreasing, we obtain
O::s:M (8-1 (1 -
1
U
) - M(G -
( 1 - u) ) sM (x ) - M(x) s JXp 1-G(t)dt.
k(n)+l,m
m
x n
Using
(3.2),
(3.29),
(3.24b),
(3.24c),
we get
16

---

, ' 1
1
)
''\\
-62-
x
-x
0::$1+0
(1) - (1+0
(1))
R(x)
«1+0
(1))
m
p «1+0
(l))T
(3,k,t).
P
p
R (x )
P
R (x )
p
n
m
m
and
2
( )
(
)
1[
T
(3,k,l)
0::$1+0
(1) -(1+0
(1))
W x
<
1+0
(1)
K-
n
__
]
p
p
p
W(x )
R (x
)
m
, m
These twoi last formulas yield
W(x )
n
W(x)
Lim
= ç$=> lim
ç
n--7 +
i
R (x ) 2
R(x)2
lXli
x1'y o
n
Important remark.
De Haan and Resnick(1980) proposed
C =(Y
-y
k
)/log k
as an estimator of the index of a stable law. '
n
n~n
n- ~n
It is shown ini16(1986) that this estimator does not characterize D(I)
\\
as Hill~s estimator does.
REFERENCES.
Il
[1]
Csorgo, M.
and Révèsz,
P. (1981).
Strong Approximations in
Probability and Statistics.
Academic Press,
New-York.
Il
[2]
-
Csorgo,
S~ Deheuvels,
P.
and Mason,
D.M. (1985); Kernel estimates
of the tail index of a distribution. Ann.
Statist. 13,
1050-1077.
Il
[3]
-
Csorgo, Sand Mason,
D.M. (1985).
Central limit theorem for sums
of extreme values. Math.
Proc.
Cambridge Philos.
98,
547-558.
[4]
-
Dekkers, A.L.M.,
Einmahl,
J.H.J.
and ne Haan,
L. (1989). A Moment
estimator for the index of an extreme-value distribution. Ann.
statist.,
17,
',1833-1855.
[5]
-
ne Haan,
L. (1970) .On Regular Variation and its Applications to
the Weak Convergence of Sample Extreme.
Mathematical Centre Tracts,
32, Amsterdam.
[6]
ne Haan,
L.
and Resnick,
S.I. (1980). A simple asymptotic estimate
for the index of a stable law.
J.
Roy.
Statist. Soc.
Ser.B,
83-87.
17

---

.. ":
-63-
J
'~
'~
[7J
Deheuvels,
P., Hauesler,
E and Mason,
D.M.
(1990).
Laws of
iterated logarithm for sums of extreme values in the domain of
è
attraction of a Gumbel law.
Bull.
Sc. Math.,
2
série,
114,
61-95.
[8]
Fisher, R.A.
and Tippett,
L.H.C, (1928).
Limiting forms of the
frequency distribution of the largest or smallest member of a
sample.
Proc; Cambridge Philos; Soc. XXIV,
Part II,
180-190.
[9]
Fréchet, M. (1927).
Sur la loi de probabilité de l'écart maximum.
!
i
Ann.
Soc.
Polonaise de Math.,
6,
93-116.
l
'
[10 ]
- palambos, J. (1978). The Asymptotic Th~ory of extreme Order
1
:
Statistics. Wiley, New-York.
1
1
1
[11 ]
bnedenko, B.V. (1943).
Sur la distribution ::',~,mite du terme maximum
d'une série aléatoire. Annals Math.
44.
[12]
- Leadbetter,
M.R. and R6otzen,
H. (1988).
Extremal Theory for
stochastic processi~s. Ann.
Probab.,
16,
431 .. 478.
[13]Lô,
G.S.(1986a). T:hèse de doctorat,
Univers:i.té Paris t;
[14]
- Lô,
G.S. (1986b) .Asyr:1ptotic behavior of Hill,' s
estimau' and
\\
0
\\
applications.,
J. Appl.
Probab.'~ 23, 322 - 93 ':. "
[15]
- Lô, ,G.S. (1989). A note on the asymptotic nc".~mality of sums of
extreme values. J.
Statist.
Plann.
Inference,
22,
127-136.
[16]
- Mason,
D.M. (1982).
Law of large numbers for extreme values.
Ann.
Probab,
10,
754-764.
[17]
Pickands,
J. (1975).
Statistical inference using extreme value
theory. Ann.
Statist. 119-131.
[18J
- Smith,
R.L. (1987).
Estimating tails of probability distribution.
Ann.
Statist. 15,
1174-1207.
[19]
- Resnick,
S.I. (1987).
Extreme Values,
Regular Variation and
Point Processes.
Springer Verlag,
New-York.

---

;
1
\\
:
1
)
.",
)
",
-64-
EMPIRICAL CHARACTERIZATION OF THE EXTREMES Il
THE ASYMPTOTIC NORMALITY OF THE CHARACTERIZING VECTORS.
Gane Samb La.
Université Saint-Louis & Université Paris VI-LSTA.
Abstract.
Let Xl' X ,
.. , be a sequence of independent random variables
2
with cornrnon distribution function F such that F(l)=O and for each
n~l,
let Xl
~ .. ~ ... ~X
denote
the order statistics based on the n first
,n
n,n
of these random variabes.
Lô
(1990)
introduced a class of four statistics
-including this new estimator of the square of the index of a stable law,
1/{-k1~ i=k ~j=i j (1-0 . . /2) (logX
. 1
-logX
.
) (logX
. 1
-logX
,
)},
L . 0 IL.
0
1
lJ
n - l + ,n
n - l , n
n - ] + ,n
n - J , n
l={.+
J={.+
. 2
where
(k, e)
is a couple of integers such that k-Hoo,
k/n~O ,e /k~O, as
n~+oo, log stands of ,the Natural logarithm and o .. is Kronecker's symbol-
lJ
8
from which he set 1R -vectors that characteribe the wholedomain of
attraction of the sample extreme and each particular domain of attraction
(the Gumbel,
Fréchet and Weibull one) .The limiting laws of these vectors
are completely determined in this paper.
It is shown that the single
elements
of this family of Characterizing Statistics of the Extremes
(FACSEXT)
and their ratios are asymtotically normal as e~+oo. But sorne
ratios become asymtotically extremal whenever e is bounded. The use of a
unified approach enabled to obtain the multivariate asymptotic normality.
AMS 1990 subject classifications:Primary 62E20,
62GI0
i
secondary 60F05.
Key words and phrases. arder statisticsi
extreme value theorYi
domain of
attraction of the sample extremesi extremal and Gaussian lawsi
representation of distribution functionsi
invariance principesi
multivariate normalitYi
characterization.
Mailing address.UER Math & Informatique,
BP 234, USL,
Saint-Louis,
Sénégal
Research address. LSTA,
Université Paris VI,
T.45-55,
E.3,
Place Jussieu.
75230 Paris Cédex 05.
France.
1

---

1
" ,
'"
)
'\\
, '
)
'\\
-65-
1 l - INTRODUCTION
Lô
(1990)
characterized the class of distribution functions
(d.f,)
F
1 attracted to some nondegenerate d.f. M (written Fe D(M)) by four statistic
while no condition was required on F. This empirical and unified approach
r
includes detection procedures of the extremal law of a sample and
1
statistical tests.
In both cases,
one has to determine the limiting laws
of this Familiy of characterizing Statistics of the Extremes
(FACSEXT)
which is our aim.
The reader isreferred to Lô
(1990)
as a general introduction to this
paper and to Leadbetter and Rootzén
(1982)
and Resnick
(1~87) for detailed
references on extreme value theory.
However,
we recall that if F E D(M) 1
where M islnot degenerate,
then M is necessarily the Gumbel type of d.f.,
x
1
A(x)=exp(-e-
),
for xER,
or the Fréchet type of d.f.
of parameter 0>0,
•
(x)=exp(-x-O)X[O
[(X) 1
for xER,
o
,+~
or the Weibull type of d.f.
of parameter 0>0,
o
I/Jo(x)=exp(-(-x)
) X]_oo,
0] (X)+(l-X]_oo,
0] (x)),
xER,
where X
denotes the indicator function of the set A.
A
Several analytic characterizations of D(.)= U
0 D(. ),
D(I/J)=U
OD(
0>
0
0>
D(A)
and ï
= D(A)
U D(.) U D(I/J) exist. We quote here only those of them
involved in our present work.
Theorem A.
1)
Karamata's representation
(KARARE).
a)
FeD(. ),
0>0,
iff
o
1
1
1
(1.1)
F-l(l-U)
=
c(l+f(u))
u-
/ o exp (J b(t)t- ) ,O<u<l,
u
where sup
(If(u) l,
Ib(u) I)~O as u~O and c is a fini te positive constant
-1
' {
}
F
(u)
= lnf x,
F(x)~u , O<u~l,
is the generelized inverse of F with
2

---

, . ,
1
"
1
J
'4,
J
'4,
J
'4,
-66-
-1
F
(0+).
b)
FED(~ ), r>O, iff x (F) = sup{x, F(x)<l}<+oo and
r
0
-1
l/r
J1
-1
(1.2)
xo(F) -F
(l-u)= c(l+f(u))
u
exp(
b(t)t
dt),
O<u<l,
u
where c,
f(.)
and b(.)
are as in
(1.1).
2)
Representation of de Haan, Deheuv~ls, Haeusler and Mason
(DDHM).
,FED(A)
iff
1
(1.3a)
-1
J - 1
F
(l-u)=d-s(u)+
s(t)t
dt,
O<u<l,
u
where d is a constant and s(.)
admits KARARE
1
1
(1.3b)
s(u)
= c(l+f(u))
eXP(J
b(t)t- dt),
O<u<l,
u
C,
f(.)
and b(.)
being defined as in
(1.1).
11
\\
1
Now,
let Xl' X ,
... be a sequence of independent copies
(s.i.e.)
of a
2
real random variable
(r.v.) X with d.f.
~(X~x)=F(x). Being only concerned
the upper tail of F,
we assume WLOG that X~l and define a s.i.c.
of the
r.
v.
Y=logX denoted Y1 'Y2'
with G(x)=~(Y~x)=F(ex ) ,x~O.
Finally
Y
=logX
~ ... ~Y
=logX
and Xl
~ ... ~
are the respective arder
1 ,n
1 n
n,n
n,n
,n
n,n
statistics of Y ,·.· Y
anè Xl'
... X
for each fixed n~l.
1
n
n
The characterizing statistics extracted from the FACSEXT are
-1/2
T (l,k,f)
= T
(2,k,f)A
(l,k,f)
n
n
n
i=k
j =i
1
A (l,k,f)=k-
l
l
j (1-0 .. /2) (y
.
1
-y
.
) ( Y .
- Y . )
n
i=f+1
j=f+1
lJ
n-l+,n
n-l,n
n-J+1,n
n-J,n
T (2,k,f)
i(Y
- y ) .
n
n - i +f, n
n - i , n '
- v .
-1
n
(y
n
-y
k
)T (2,k,f)
i
n-{.,n
n-
,n
n
3

---

:
1
1
J
.",
J
",
-67-
T (2,f,1)
T
(6,)
=
(y
_y-I
)
T
(2,f,1)
n
n
n-f, n
n-k, n
n
-2
-u
-1
T
(7)=(y
n
-y
k
)
A (l,f,l)
T
(8, u) :::n
(y
n
n-l:-,n
n-
,n
n
0
-y
k
)
i
n
n-l:-, n
n- ,n
T
(9)
(only defined when x
(G)
y < +(0)
n
o
o
where k and f are integers such that l~t<k<n, v is any positive real number
and Q ..
is Kronecker's symbol.
lJ
We recall here the second theorem corresponding to the "necessity part
of the characterization of Lô
(1990).
a)
If lf \\(1) = (T
(1),
. . . . , T (7),T
(8,(3/2))
~ (I,d,O,y ,d,O,O,O,), then
n
n
n
n
p
0
i)
FED(A)
whenever d=O and
ii)
FED(~l/d) whenever O<d<+oo and necessarily Yo
+ 00,
where ~ stands for the convergence in probability.
p
b)
If T
(4) l' yo<+oo and lf (2)=(T
(l),
... ,
T
(7)) ,TA9))
p(c,O,O'Yo,O,O,O)
n
n
n
n
2
with
l<c<~, then FED(!/J ), o=-2+c/(c -l).
o
This motivates a systematic investigation of the limit laws of the
FACSEXT.
We do not include T
(4)=Y
in our study for that its case is
n
n,n
classical in extreme value theory.
Namely,
we must find
non radom
sequences Π(i,k,t)
and «
(i,k,f)
such that.
n
n
d
Œ
(i,k,f)
(STAT.-«
(i,
k,f))
~ NDD,
n
l
n
where STAT.
is an element of the FACSEXT,
a ratio or a vector of its
l
d
elements,
~ denotes the convergence in distribution and NDD is sorne
nondegenerate distribution.
It will be se en farther that NDD is,
in most
cases,
a Gaussian r.v. or an extremal one in some others.
4

---

: '
-:68-,-
-
J
' \\
'\\
Let us now classify the elements of r in our convenience. From (1.1),
(1.2)
(1.3),
it is clear that for each of the three domains,
the couple
(f,b)
represents a subset having elements only distinguishable by constant
We then write for this class
(f,b)
(cf. Lemma 1 in Lô
(1990)),
F=(f,b)
E
D(A)
iff
Il
-1
-li
(1.4)
G
(l-u)=d-s(u)+
s(t)t
dt,
O<u<l
u
where s(.)
verifies
(1.3b)
F=(f,b)
E
D(~ ) iff
r
1
1
(2.5)
G-
(1-u)
= -(logu)lr+log c+log(l+f(u)) Il b(t)t-
dt,
O<u<l
u
F=(f,b)
ED(W ) iff Y =x (G)=sup{x, G(x)<l}<+oo and
r
0
0
1
'
li
Il
1
( 1. 6 )
Y0 - G-
(1 - u )
c (1 +f (u))
u r e xp (
b (t ) t - dt),
O<u<l.
u
Our results below will show that one has asymptotic normality with
no condition on f ( .)
nor on b ( .)
but wi th a
"random noise ",
that is
CT
(i,
k,
i)
(STAT.
- «
(i
,K, i) ~ NDD,
n
l
n
r::
k- 1
with K
~ 1. But,
when we attempt to have a non ramdon centring sequenc
p
«
(i,
k,
i),
only f ( .)
makes problems.
In earlier studies of T
(2, k, i)
fo
n
n
"
instance, Hall
(1982),
Csorgo and Mason
(1985),
Lô
(1989)
imposed f=O.
In
"
*
*
particular Csorgo and Mason (1985),
Lô
(1989)
denote
D (~)
and D (A)
respectively the obtained classes.
*
This condition is not restrictive at all for that,
for example,
D(A)
includes any element of D(A)
having an ultimate positive derivative,
that
is the most important cases: normal,
log-normal,
exponential etc
...
Nevertheless,
we characterize here the possible limits and then, obtain
simple results under general conditions like
(1.7)
f' (u)
exists and uf' (u)~O as~O,
or
5

---

-69-
" ,
)
''\\
)
'\\
) . ''\\
(1.8k
(1),
... , k
(p))
k
(i)1/2 sup(f(k
(i)/n),
f(U
(')
1
))~O, i=l, ... ,·
n
n
n
n
k n l + ,n
p
or a combinat ion of
(1.7)
and
(1.8k
(1),
... ,.kn(p)),
n
(1.9k
(1),
. . . . ,kn(p))
f(u)
f
(u) (1+f
(u))
where f
satisfies
(1.7),
n
1
2
1
f
(u)
~ 0 as u ~ 0 and k (i)1/2
(i)/n)sup(f
(k
(i)/n) ,f
(U
(i)+l,n))
2
n
f1 (k n
2
n
2
k n
pO,
i
= l, ... , p,
where p is a positive integer,
k
(i)
is a
sequence of integers such that f
n
th
each fixed i,
l~i~p, k
(i)/n~O and U
is the k
minimum among n
n
k ,n
independent r.v.~s uniformly distributed on (0,1)
(see
(2.2)
below)
Let 1(0), l(k
(1),
... ,k
(p))
and l(O,k
(l), ... ,.k
(p))
be the class of al
n
n
n
n
.d.f.'s F=(f,b)
satisfying
(1.7),
(1.8k
(1),
... ,kn(p))
or
(1.9k
(1), ... k
(};:
n
n
n
\\respectivel y . Each of these three confitions is quoted as a
regularity condition.
A related problem consists ln replacing the centring sequence by 1/ 0 for
FED(~ ).
It has been studied by Haeusler and Teugels
(1985) .They obtained
o
general analytic condition for the asymptotic normality.
This problem will
not be considered here for space reasons.
Each single statistic is systematically treated apart in Sections III
IV and V while the ratios are studied in Section VI. Section VII is devote
to multivariate limit laws as best achievements. AlI the results are given
into unified invariance principles based on the same Brownian bridge. We
therefore begin to describe the structure of the limiting laws.
II
- Description of the limiting laws.
Il
II
Csorgo,
Csorgo,
Horvàth and Mason
(1986)
have constructed a probabili
space carrying a sequence U , U ,
... of independent and uniform r.v.'s on
1
2
(0,1)
and a
sequence of Brownian bridges
{B
(s),
O~s~l}, n = 1,2, ... such
n
that for aIl v,
0<V<1/4,
6

---

: '
-70-
J
' \\
J \\
1/2
1/2-v
-v
(2.
1)
sUP (l/n):SS:Sl-l/n 1n
(Un(S)-S)-Bn(S) 1/(s(l-s))
a
(n
),
p
and
1/2
1
1/2-v
(2.
2)
sUP (l/n):ss:Sl-l/n
a (n- v )
1n
(S-Vn(S)-Bn(S)
/(s(l-s))
p
,
where U (s)
=
j/n for U.
:Ss<U.
1
is the uniform empirical d.f.
and
n
J ,n
J+
,n
v (s)=U.
for
(j-1)/n <s:Sj/n,
l:Sj:sn,
and V (0)
= U
, i s the uniform
n
J,n
n
1 ,n
quantile function and,
finally,
U
:S ... :sU
are the order statistics
1,n
n,n
of U '
. . . U
with by convention U
=O=l-U
.
1
n
a,n
n+1,n
From now on,
all the
invariance principles
are
assumed to hold on
this
probability' space and we therefore may use the general representation for
\\
'
the empirical d.f.
G
based on YI'
... ,
Y
and for the o~der statistics
n
n
YI
:S . . . :sY
by their uniform counterparts
:
,n
n,n
(2.3a)
{l-G
(x),
O:sx<+lXl,
n~ a }= {u (1- G (x) ),
O:sx < +lXl ,
n~ a} .
n
n
-1
. }
(2.3b)
{y .
,1:sj:sn, n:S1 } = {G
(l-U
. 1
),
l:SJ:sn,
n~l .
J ,n
n-J+
,n
We introduce these notations
z
RI (x, z G)
(l-G(x) )-lI (l-G(t)) dt, x<z:sy 0'
X
IZ Z
z
R
(x, z,G)
=
(l-G(x))-
J. ... l
1
(l-G(t))
dt dY1"
.dy
_
p
x
P 1 '
x<Z:sy ,p>l
o
YI
yp-1
R (x,y ,G)
- R (x,G),
p~l
p o p
«(k,t)
nk -lI z n
1-G(t)
dt
'[(k,t)
x n
with «(k,l)
-
«(k)
and '[(k,l)~'[(k).
7

---

: ,
-71-
: '
' J
'"
J
' "
If the d.f.
is not specified in R (')'
i t is assumed that R ( )-R
(
G)
p
p ' - p " "
x
for G (x)
F(e ).
Finally, put
N
(O,k,i)
{(n/k)1/2sZn B (l-G(t))
dt } dt}/R (x)
i
n
x
n
1
n
n
1
1(2
N
(2,.)
= - (n1 .) : B
(. 1n),
n~1 i
n
i
n
~ (i)=nU
1
li, n~l.
n
1 + ,n
We prove in )the next· sections that each T (i)
is asymptotically one of
n
these r.v.'s or la linear combination of them.
Their abymptotic laws are
described in
Theorem C.
1°)
For each n,
the vector
(W (k),
W (i)),
with W (k)=
n
n
(N
(O,k,i),
N (3,k,i),
N (2,k))
and W (e)
=(N
(O,i,l),
N (3,i,l),
N (2,i))
n
n
n
n
n
n
n
is Gaussian
.
If
further i~+oo,
k~+oo, k/n~O, ilk~O n~+oo, and FEl,
then
6
Wn(k), Wn(i))
converges in distribution to an R -Gaussian r.v.
(W(l), W(2))
where W(l)
and W(2)
are independent vectors with the same
covariance matrix :
2(/y+1)/(O+2)
6(/y+1) (0+2)
J
3(/Y+1)/(/Y+3)
FED (l/J ),
if
/Y
[
(/y+3) (/Y+4)
-1
-1
and
8

---

, ' 1
1
: ,
J
.,,"
-72-
6
,
if FED(A)
UD(~),
-1
where the symmetric matrices are given only in one side.
2°)
Let e be fixed,
then for all XE~, P(E
(e)~x) converges to
n
- ex , j = e ( . 1 ) - 1 (n ) j
1
-
e
L j = 1 J.
< . X ,
x~ 0 .
P (E (t) ~x) =
{
o elsewhere.
We do remark that Var
(W(l))
lS
obtained for FED(A)UD(~) by putting
o = + 00 in Var(W(l))
for FED(~ ). This fact occurs almost always in this
o
paper.Then,
t~roughtout, for any
expression depenqing on 0
with 0<0< + 00,
it is meant by 0<0< + 00 resp.
0
=
00)
that FED(~ )
----....:..-..-----'----------=-------"-------=----------'=---------------'--'--0
(resp.
FED(A)
U D(~)). Also, any limit or inequality with presence Jf n is
assumed to hold as n~+oo.
Proof of Theorem C.
Part 2 is obvious since a uniform r.v.is attracLed by
weibull's d.f. Cf. Hall
(1978)
to get the precise expansion.
As to Part 1, Nn(O,.,.)
are Riemann integrals sa that any finite linear
combination of N (0, .n.),
N (3,.,.)
and N (2,.)
is limit everywhere of a
n
n
n
sequence of r.v.'s of type
(Darboux'
sums)
j =p
j =q
,
,
'"C •• (p,q)
B (t .. (p,
q)),
L. 1 L. 1
lJ
n
lJ
J=
l=
as p~+oo, q~+oo. But a Brownian bridge has Gaussian finite-dimensional
distributions. Thus,
each finite linear combination of the coordinates
(W (k),
W (e))
is a normal r.v.
so that
(W
(k),
W (e))
lS
a Gaussian
n
n
n
n
vector.
That W (k)
and W (e)
are asymptotically independent follows that
n
n
lim
Cov(W (e),W (k))
is the 3 x 3 null matrix by Formula 7.2 below.
n +00
n
n
Formulas
(3.34),
(3.43),
(4.31),
(4.36)
and
(6.10),
Lemmas 4.1 and 4.3
9

---

.. ,
..
1
1
, ' 1
1
J
.,,"
J
.,,"
-73-
below yield Var(i)),
i
== 1,2.
III -. Limit laws for T (2,k,t)
and T
(5).
- - - " - - - - - - - - - - n
n - -
Theorem 3'.1
: Let Fer let
(k,t)
be a couple of integers satisfying
(2.4a)
l<k==k(n) <n,
k~+oo and K/n~O
<
<
/
/
1/2-1)
(2.4b)
1-t<k,
31),
0-1)<1 2, t k
~O,
(with 1) == 0 if FeD(~) U D(~), 1»0 if FeB(A)), then
1 2
«(k,t)k /
{T
(2,k,t)
-
«(K,t)}
N
(O,k,t)+o
(1)
Q N(O,O'2(1)),
n
n
p
0
2
where 0'0(1)
== 2(1+1)/(1+2),
0<1~+00 and «(k,t) is obtained by replacing
x
by
x in« (k, t) .
n
n
Remark 3.1.
(2.4b)
holds whenever t is fixed and t~+oo.
Corollary 3.1. Under the ,same assumptions and notations used ln Theorem
2.1, we have
1 2
<<(k,t)k /
(T
(2,k,t)
-«(1<) )=N
(O,k,t)+o
(1)
Q N(O'O'o2(1)) f
n
n
p
y
where 0<1~+00 and «(K
-lJ 0
nk
x n
Corollary 3.2.
Let FeD(~1)' 0<1<+00.
If
(k,t)
satisfies
(2.4)
with 1)=0,
or
less strongly,
there exists
1
0«<1/ 0 such that t -(+1/ 0 / k 1/2-ç+1/ 0 ~O,then
a)
(y -x ) -l k 1/2 (T (2,k,t) -<<(K)) Q N(O,
2/(0+1) (0+ 2 ))
i
o
n
n
-1
-1
b)
and one can replace G,
G
, Y.
by F,
F
,X.
respectively in Part a.
J,n
J,n
"
Remark 3.2. Although the invariance principles hold on Csorgo and al. (1986
probability space,
the convergences in distribution themselves are true
wathever is the probability space.
10

---

-74-
.. ,
. ,
)
4,
"
Theorem 3.1 was proved by Csorgo and Mason (1985)
and Lô
(1989)
when
1
1
1
FED(A) U D'(t!». But their proofs used u- I
(1-s)dG-
(s)=r(u)
instead of·
1-u
R (x,G) .Remark that R (x)=r(u)
for u=l-G(x)
if G is ultimately continuous
1
1
and increasing. We do not require at aIl such assumptions and since the
elements of the proof of Theorem 3.1 are great(y used in aIl the remainder
of the paper, we should reprove i t rigorously ;ln a simultaneous treatment
of aIl cases FED(A),
FED(~) and FED(~) .
We now characterize the asymptotic normality of T
(2)
when attempting
n
to replace «(R)
by «(k).
For this,
put
D.Oa)
0
(o)=f(U
1
)-f(o/n)
n
0+
, n
*
-1 1/2
T
(2, k, t)
= R (x)
k
(T
( 2 , k, t) - <<( k) )
n
1
n
n
(3.0c)
NA(l,k,t)=N
(l,k,t)+e
(0)N
(2,k),
e1(0)~(0+1)/o,e2(0)=0+l,e2(+00)=1~
n
1
n
Theorem 3.2.
Let F=(f,b)
Eland let
(k,t)
satisfy
(2.4). We have
1 2
T * (2,k,t)
N (l,k,t)+e
(0)k
/
0
(k) (1+0
(1) )+0
(1),
n
n
2
n
p
p
2
2
3
2
2
with Nn(l,k,t)-N(O'~l (0)), Π(0)=(0 +0 +2)/0 (0+2),
O<o~+oo. Then,
1
2 2 2
lim
~1(0)+~ (-00)+2e (0)CoV(N (l,k,t)N (-00) )=Œ
n~+oo
2
n
n
c
c)
T* (2,k,t)~+00 (resp.-oo)
iff N (-00)
~+oo
(resp.-oo).
n
p
n
p
This characterization is very simple and general since one has
2
v
n(U
1
-k/n)k- 1 /
=N
(2,k)+O
(n-vk-
) by
(2.2)
and hence nU
1
/k~ l, as
k + ,n
n
p
k +,n
p
11

---

, ' 1
1
"
,
J
':>,
J
':>,
-75-
k/n-70.
Examples such as ln Hauesler and Teugels
(1985)
may be treated ln
!
very simple ways since in all their models f (u) =:f
(u) (l+f,2 (u))
with
l
f~ (u)-70 as U-70 [for instance f (u)=a.up,p>o, f (u)=a. (log(l/U))-P,
l
l
P>O,
f
(u)=a exp(-b/u)]
and f
(u)-70].
For all
these models,
we have
l
2
Corollary 3.3.Let F=(f,b)
E rand
(k,e)
satisfies
(2.4) 'If further
(1.7)
or
(1.8k)
or
(1.9k~ holds, then
1
* ,
d,
2
T
(2,k,e)=:v (2) (T
(2,k,e)-c
(2))=N
(l,k,e)+o
(1)
-7' N(O,(J'l(~))'
for
n
n
n
n
n
p
O<~:S+CXl.
One proves this corollary from Theorem 3.2 by using (2.2).
Let
(1.7)
be satisfied,
thus
,
kl/2~ (k) = nk-l/~(Uk 1 -k/n) (u f' (u )) (1 +0 )) =N (2, ik) +0 (1) + (u f (u )
n
.
,+
,n
n
n
p
n
r-
p
n
n
X(l+o: (1))
-7 0,
where nu /k -7 1. Now,
if
(1.8k)
holds,
thus
p
p
n
p
If
(1.8k)
holds,
thus
-70
by
what is above.
p
We should remark that lim uf' (u)=O is a
fairly general condition since i t
U-70
holds whenever the limit exists. Also limiting laws of T (5)=T
(2,é,l)
n
n
are particular cases of results stated above.
12

---

-)6-
~'"
)
.'"
..'
' )
PROOF8.
They largely use
technical results
in Lô
(1990).
Use
(2.1)
and
(2.3)
ta get
(3.1 )
R (x )-l k 1/2(T (2,k,e)-«(K,e))=N
(O,k,e)+Z 1(k)+(e/k)1/2
1
n
n
n
n
-
Z
(~)1/2 Z
-lJ
=
R (X )
n{a:
(l-G(t)) -B
(l-G(t)) }dt,
n
n3
1
n
-
n
n
x n
1 2
where a:
(s)
(resp.
(3
(s))
= n /
(U
(s) -s)
(resp.
n~/2(s-V (s))
n
n
n
n
1
G-
(1-U
1
)
Z
(0)
±{ (n/ 0) 1/2 J
0 +
,n
B
(l-G(t))
dt}'/R
(x
),
n1
1
n
1
n
G-
(1- 0 /n)
and
1
\\
nk 1 / 2
G- 1 (1-U
)
Zn2(0)=±{
o+l,n B
(l-G(t))
o
J -1
n
G
(l-o/n)
We shall treat each error term into statements denoted
(81.3), (2.3) ,etc ...
(81.3)
Zn3 ~ 0,
for e~+oo.
First,
we have by
(2.2)
1 2
d
(3.2)
ne-
/
(Un
1
-e/n) =N
(2,e) +0
(1)
~ N(O,l).
l:-+
,n
n
p
sa that nU n 1
/e ~ 1 and hence
l:-+
,n
p
(3.3)
V À>l,
lim
1P(e!Àn):5Un 1
:5:(Àe)/n)
= 1.
n~+oo
~+
,n
For convenience,
if
(3.3)
holds,
we say "for all À>l,
one has e/ (ll.n):5Ue+1,
:5(Àe)/n with Probability as Near One as Whisheè
(PNOW)
for
large values
of n".
Hence,
(3.4)
G(t):5:1-1/n,
uniformly for x :5:t:5:Z ,
n
n
with PNOW as n is large.
8econdly,
13

---

: '
: ,
-77-
J
.~
).~
Lemma 3.1. (Cf.
Fact 5 in Lô(1990).
Let h(o)
be a bounded function on
(a,l)
a>O and G any d.f.
If the integrals below make sense as improper ones,
the
G- 1 (l-a)
G- 1 (l-a)
1J-00
h ( (l-G (t) p (t) dt 1:ssuPa:ss~:l1 h (s) 1J
1p (t) 1dt.
-00
Combtning
(2.1),
(3.4)
and Lemma 1 yields for sorne v,
O<V<1/4,
-
z
(3 .5)
{(n/k)1/2-VJ_n
(l_G(t))1/2-V dt}/R
(x ).
,
1
n
x n
We need three lemmas at this step.
-1
Lemma 3.2.
If FEI,
then
(l-G(G
(l-u) ))/u~O.
Proo~ of Lemma 3.2.It follows from Lemma ~ in Lô (1990)
that either
1
G(G-
(l-U))
~l-u or 1-u lies on sorne constancy interval of
-1
1
G
,
say ]G(x-) ,G(x)],
so thatG-
(l-u)=x and hence l~(l-G(G - Î-(l-u)))
/u)~((l-G(x))/(l-G(x-))). Now, by Lemma 1 in Lô (1990),
either GED(~ ),
O<r<+oo,
and consequently KARARE holds
r
(3.6)
1-G(x)
c (x)
x
(G)
o
where c(x)~c, O<c< + 00 as x ~y and p(t) ~O as t~+oo ;
o
or GED(A)
and by de Haan-Balkema's representation
(cf.
Smith(1987)),
(3.7)
1 - G(x)
= c(x)
eXp(JxL(t) -l dt ),
- 00< X <yo'
-00
where c(x)~l as x~y
and L admits a derivative L' (x)
such that L' (x)~O as
o
x~yo' In both cases,
(l-G(x) )/(l-G(x-))=c(x)/c(x-)~las x~y . This
o
completes
the proof.
Lemma 3.3. Let G (X)=l-(l-G(X)r,
r>O,
then
r
(i)
GED(~ ) ~ G ED(~
) with x
(G)=x
(G )=y
and
r
r
rr
0
0
r
0
14

---

..
1
1
, ' 1
1
J
' ' \\
-78-
1-G(z)
z~y
,
o
1-G(X)~0.
(ii)
GED(A)
~ GrED(A) with x (G)=x (G )=y
and
o
0
r
0
-1
~ r
as x~y ,
z~y
,
(l-G (z) ) / (l-G (x) ) ~O.
o
0
Proof of Lemma 3Î.3. a)
Let GED(I/1"1)'
Thus
(3.6)
holds for Gr by putting
r
c
(x)
c(x),
~
="1r and p
(t)
= rp(t).
Hence G ED(I/1
) and x
(G)
=
r
!r
r
r " 1 r
0
x
(G ).
Further, by Formula 2.5.4 of de Haan
(1970)
or Lemma 4.1 below,
o
r
-1
-1
(3.8)
RI (x,G)/(yo-x) ~("1+1)
, RI (x,Gr)/(Yo-x)~("1r+1)
, as x~yo'
By Lemmas 7 and 8 of Lô
(1990),
1-G(z)
(3.9)
RI (x, z,G) /R (x,G)~
(x,G)~l
1
l, R (x, z,G) /R
as x~yo' 1-G(x) ~O
2
2
whenever GED (1/1) lor GED(A)
(hence whenever FEr).
(3.8) \\ and
(3.9)
together
prove Part i).
b)
Let GED(A). By (3.7),
GrED(A)
with xo(G)
x
(G
)
y
and thus
x
r
o
(cf.
Lemma A in Lô
(1990),
for any tER,
(3.10) (l-G
(x+tR (x,G ))/(l-G
(x))
~e-t, as x~Yo'
r
1
r
r
t
(3.11)
(l-G(x+tR(x,G))/(l-G(x))
~ e- , as x~y .
o
Combining (3.10)
and
(3.11)
implies
-tir
(3.12)
(l-G(X+tR1(x,Gr))/(l-G(x))~e
, as x~Yo'
If for a sequence x ~y , one has R (X ,G)/R
(X ,G )~v, O<v<r-c,
as n~+oo,
n o
1
n
1
n
r
thus,
by
(3.12),
lim inf
(l-G(x +tR
(X ,G) )/(1 - G(x ))
n~+oo
n
1
n
n
~exp(-t(r-c)/t) »e-t,t>o, which is absurd because of (3.10).
Whence,
lim inf
RI (x,G)/R
(x,G )~r. similarly, one gets lim sup
1
x~Yo
r
x~Yo
RI (x,G)/R
(x,Gr)~r. Finally RI (x,G)/R (x,Gr)~r, as x~Yo' which combined
1
1
with
(3.9)
proves Part ii).
15

---

" ,
)
''\\
" , '-7~-
-1
Lemma 3.4.
i)
If eD(A)
U D(cf» , then R (G
(l-u))
lS 810wly Varying
1
at Zero
(8VZ).
1
ii)
If FeD(~ ), then R(G- (l-U)) lS Regularly Varying at
'a'
-1
Zero
with exponent 'a'
-RVZ).
Proof of Lemma 3.4. a)
Let FeD(A)
U d(cf», thus GeD(A) by Lemma 1 in Lô
(1990).
Next,
Lemma 3.2 above: and Lemma in Lô
(1986a)
yield for i\\>0,
i\\~1,
1
-1
-1
i
1
(3.13)
{G
(l-i\\u) -G
(l-u) }/R(G (l-u))~ -log i\\, as u~o,
and Lemma 4 in Lô
(1989)
implies
( (3 d 4 )
{G- 1 (1 - i\\ u) - G- 1 (1 - u) }/ s (u) ~ - log i\\, as u~°,
wheie s(.)
is 8VZ and defined as in
(1.3).
Hence
1
(3.15)
R(G-
(1-U))_S(u)
as u10'
1
so- that R(G-
(l-U))
is 8VZ and Part i
is proved.
b)
Part ii)
is easily derived by Formula 2 _5.4 of de Haan
(1970)
(cf.
Lemma 4.1 below)
and Formula
(1.6), above.
We return back to our proofs of Theorems. By
(3.5)
and Lemmas 3.2,
3.
U
and 3.4,
(3.16) IZn31:S0p(k-U)
R (xn,G1/2_U)/R1 (xn,G)
= 0p(k-
)
1
=
0p(l),
where we have taken
(3.8)
into account.
This proves
(81.3).
(82.3)
Zn3 p 0, when t is fixed.
We have
(3.17)
1 Zn3 1 =Zn3 (1)
+ Zn3 (2) ,
with
1
G-
(1-t/n)
(3.18)
Zn3(1)
= (n/k)1/2{J_
ICX
(l-G(t))-B
(l-G(t)) \\dt}/R
(X
) ,
n
n
1
n
x n
u
which is 0
(k-
) by same arguments used in
(3.5)
(one also has G(t):Sl-l/n)
p
and
16

---

, ,
:
1
:
1
}
''\\
-80-
(3.19)
±Zn3(2)
{(n/k)1/2JZ~1
la (l-G(t))-B (l-G(t))1 dt}/R (X ).
1
n
G
(l-t/n
n
n
1
By'Theorem C,
(3.20)
lim
lim
lP(t/Àn)
::sUn 1
::S(Àt)/n)=l,
.
À~+oo
n~+oo
<.+
, n
w~ich is quoted as
1
«(t/ (Àn) ::SUt 1
::s (Àt) /n holds with PNOW for large values of n and À».
i
+
,n
T~us, with PNOW for larce values of n and À,
1
-
i
t U / À )
(3.21)
Zn3 (2)::s {(n/k) 1/2 J n
la (l-G(t)) -B (l-G(t)) Idt}/R (x )
1
t
(Àt)
n
n
n
n
1
where t
(o)=G-
(1- o/n)
and
\\
n
*
1/~ Jt U/À)
(3 . 22)
Z 3 (2) ) ::s4 {n/k)
L
n
n
t
(Àe)
n
- G /
(t)
dt
1 2
Yo
yo
J
- J
1 - G /
(tldt/
1 - G /
(t)
dt}.
1 2
1 2
t
(t/À)
t
(k)
n
n
By Formula
(2.10),
Lemmas 7 and 8,
all in Lô
(1990),
and Lemma 3.3 above,
*
*
both terms in brakets tend to zero and
(Z 3(2))~0 and hence Z 3(2)~ 0 by
n
n
p
*
Markov's inequality sinee Zn3(2)
is nonnegative.
Consequently,
Zn3(2)pO,
which,
in turn,
together with
(3.17)
and
(3.18)
proves
(32.3).
(33.3)
Z l(k)
~ O.
n
p
We have,
as in
(3.2),
for any À>l,
17

---

: ,
.. '
-81-
J
' \\
J
' \\
with PNOW as n is large.
Furhermore,
' <
1/2J t n(k/À)
(3.24)
IZn1 (k)I-3{(n/k)
t
(Àk)
1-G /
(t)
dt }/R
)
1 2
(X
1
n
i
n
R
(x ,G /
)
1
n
1
2
:53x
X{R
(t
(Àk) ,G /
) (1+0(1»
-R
(t
(k/À) ,G /
) (1+0(1»}
1
n
1 2
1
n
1 2
R
(x
,G)
1
n
,
/R
(t
(k),
G /
).
1
n
1
2
The term in:brackets tends to zero for FED(A)
U D:(~) and to À1 /À_À- 1 /À for
FED(~o)' alr by Lemmas 3.3 and 3.4. 8ince À is arbitrary >1, [(Z:l (k»~O
in all cases and finally
(83~3) holds by Markov's inequality and
(3.~3).
(84 .3)
(t/k) 1/2 Z 1 (t)
~ O.
n
p
\\
\\
This is proved exactly as
(83.3)
when t~+oo. When t
is fixed,
one uses
(3.20)
instead of
(3.2)
and the proof of
(83.3)
is valid again.
1 2
(85.3)
tk- / Z 2(t) ~ O.
n
p
a)
Let FED(~ ),
thus GED(~ ).
By Lemma 4.1 below,
Lemmas 3.2,
3.3,
3.4
o
0
above and Theorem C,
(3.25)
-1/2{
Z 2 (t) =tk
( 1 +0
(1) (y - z ) / y
- x ) - 0
(1 )
n
p
0
n o n
p
(y -z )/y -x )}~ 0,
o
n o n
p
1 2
whenever lim sup
t/k /
<00.
In this
special
case,
one
can choose t
n~+oo
.
satisfying
(3.26)
3~,O<1/o, t1-~+1/o~ 0,
since for all ~,O<~<l/o, sup(IYo-zn l , Iyo - znl)/(yo-x )
n
:52 sup([
(t)l/ o ,l)
(t/k)-E+1/ o .
n
b)
Let FED(~a)' thus R (x,G)~l/a
(cf.
for instance Lemma 1 in Lô
(1990».
1
Using Lemma 3.2 and Theorem C,
one gets
18

---

1
,
",
J
' ' \\
: '
'J,\\
-82-:'"
J
' ' \\
1
1 2
(3.27)
Itk- /
Z 2 (t) l=cx(l+o(l)) IR (z ) (1+0
(1)) -0
(l)R
(z ) 1 (tk- 1 / 2 )
n
1
n
p
p l
n
'
where the 0
(1)
is 1+0
(1)
as t~+oo. Whence,
p
p
r
2
(3 . 2 8)
tk - 1/
Z 2 (t)
~ 0 l'f f l'
nk - 1 / 2
n
p
lm sup n~+oo ~
<00.
,
c)
Let FeD(A).
Thus GeD(A)
and by Lemma 3.2,
1 2
1 2
(3.29) Itk- /
Z 2(t) l::::tk- /
{R
(Z
) (1+0
(1))+0
(l)R-(z )}/R
(x
)
n
1
n
p
p
1
n
1
n '
But, by
(3.3)
and Lemma 3.4,
where c is small enough. Whence
1
-
2c
.with PNOW as n is large.
Putting T/= (1- 1-c
) /2 completes the proof
of
(85.3)
whenever
(2.4b)
holds.
:It remains the normal term N
(O,k,t).
We have
n
2
Lemma 3.5.
Let Fer,
If t/k~O,
then N (O,k,t)-N(O,s
(0)),
where
n
n
2 (1+1)
--~2-' 0<1::::+00
1+
Proof of Lemma 3.5.By Theorem C,
N (O,k, ,t)
is a normal random variable.
n
rts variance is
(3.32)
S~(O)=((n/k) JZn JZn
)
{min ((l-G(t), 1-G(s)) - (l-G(t)) (l-G(s)) }dsdt
x
x
n
n
Conseridering the cases {s<t} and {s::::t}
for the function in brakets yielë
z
z
2
2
(3.33)
s~(O)={(2n/k)Jn Jn(l-G(t) G(s) ds dt }/R (X )-2R (x ,Z )/R (X ).
1
n
2
n
n
1
n
x
s
n
By Lemmas 7 and 8 in Lô
(1990)
(see Formula
(3.9)
above),
Theorems 2.5.6
19

---

r
..
1
•
1
}
'~
,
)
.~
-83-
and 2.6.1 of de Haan
(1970)
(see Lemma 4.1 below)
and Lemma 3.2, above,
1
2,
2
(3.34)
s
(O)-2R
(x )/Rl(x )-2K(Q), where K(Q)=(Q+l)/(Q+2),
O<1~+oo.
n
2
n
n
1
THEOREM 3.1 is thus proved by (3.1),
(81.3),
... ,
(85.3)
and Lemma 3.5.
1
COROLLARY :,3.1 is obtained from Theorem 3.1 by showing that
:
y
l
2
(86 .3)
{n~-1/2)l 0 1-G(t) dt}/R (X ) - (ek- / ) (R (Zn) /R (Xn)~O'
1
n
l
l
:
zn
But,
returning back to
(85.3), we see that
(86.3)
holds for FED(~) iff
lk-1/2~, for FED(!/J ) whenever (3.26) is satisfied and.for FED(A) whenever
Q
(2.4b)
holds.
COROLLARY\\ 3.2 follows from Corollary 3.1 Formula 2.6.3 of de Haan (1970)
1
(8ee Lemma 4.1 below)
and Formula
(3.26)
above.
It remains to prove Theorem 3.2.
For this,
we need
Lemma 3.6. Let F=(f,b)
Eland k satisfies
(2.4a),
then,
1
+0
(1)=e
(1)N
(2,k)+e
(o)k / 2Q
(k)(l+o
(1))+0
(1).
p
1
n
2
n
p
p
PROOF.
a)
Let FED(A).
(1.4)
and (3.15)
yield
U
-lI
(3.35)
112
1/2
k+l,n
-1
ç (k)=k
(1+0
(ll){s(U
)/s(k/n)-1}+k
(s(k/n)
s ( l l t
dl.
n
p
k+l,n
k/n
By
(1.3b)
and
(3.2)
sup (resp.
inf)
where Tn
is the closed interval formed by k/n and U
n
•
Thus
k +("n
U
-lI
1/2
k+1, n
-1
-1/2
(3.36)
k
s(k/n)
s(t)t
dt=(l+o
(1) )+0 (nk
(U
1
-k/n).
k
k/n
p
p
+ , n
20

---

1
:
1
.. ,.;
J.",
1
-84-
Also,
1
1
1 2
(3 . 3 7 ) k / 2 {s (U
1
) / s (k / n) - 1 }== k /
l'
( k) (1 +
k
0
( 1) ) +0
(nk - 1/ 2 (Uk
- k / n) ) .
+ ,n
n
p
p
+l,n
1 (See (3.38) - (3.40) or more details).
b)
let FED(~}. This case is exactly the preceding since GEID(A}
and
(1.4)
1
holds.
r
c)
Let FED(~ }. Use R: (X }-(y -x )/(1'+1) and get
l'
1
' n
0
n
x({y -x )/(y -x ) }-1}.
o
n o n
Now,
by
(1. 6) ,
(3.39)
where E:
sup{b(t),
tsmax(U
1
'
k/n)} ~ O. Since nU
1
/k ~ l,
n
k + ,n
p
k +,n
p
1 2
1 2
(3.40)
ç (k)=(1'+1)k / 1' (k) (1+0 (1)}+o-1(1'+I)nk- /
(U
1
-k/n) (1+0
(1)).
n
n
p
k + ,n
p
Now
(2.2), (3.36), (3.40)
and Point b)
just below complete the proof of Lem!
3.6.
We return to the proof of Theorem 3.2.By
(3.1),
(S1.3),
...
(86.3), (3.41)
x
1 2
v
(2) {T
(2,k,i)-c
(2) }=N (O,k,i)+k /
{k
QI-n l-G(t) dt}/R (X }+o (1).
n
n
n
n
x
1
n
p
n
Using Lemmas 3.2 and 3.6 gives
1 2
=N
(O,k,i)+e
(1'}N
(2,k)+e
(1')k /
1'
(k)(1+0
(1)}+0
(1),
n
1
n
2
n
p
p
from which the characterization is obvious.
To compute [(N (O,k,i)N
(2,~
n
n
21

---

" ,
.'"
" ,
)
-85-
Also,
1 2
1 2
1 2
(3.37)
k /
{S(Uk 1
)/s(k/n)-1}=k /
0
(k) (1+0
(1))+0
(nk- /
(Uk
-k/n)).
+ ,n
n
p
p
+1, n
-(See
(3.38)
-
(3.40)
or more details).
b)
let FED(~). This case is exactly the preceding since GED(A)
and
(1.4)
holds.
1
c)
Let FED(I/J ). Use R (X ) .. (y -x )/(0+1)
and get
' o
1
n
0
n
.
1
(3 . 38)
ç (k) - k / 1;{ (y -x )- (y - x )} / {(y - x ) / (0 +1) } _ (0 +1) k 1/2
n
o n
0
n o n
N ow,
by
(1. 6) ,
(3.39)
l~(l+l'n(k)) (~k U
)C n +l/l'_l,
k+l,n
- x n
where c
sup{b(t),
t~max(Uk 1
,k/n)} ~ O. Since nU
n
k 1
/k ~ l,
+
,n
p
+ ,n
p
1 2
1 2
(3.40)
ç (k)=(l'+1)k / l' (k) (1+0 (1) ) +0'-1 (l'+1)nk- /
(U
1
-k/n) (1+0
(1)).
n
n
p
k + ,n
p
Now
(2.2), (3.36), (3.40)
and Point b)
just below complete the proof of
3.6.
We return to the proof of Theorem 3.2.By (3.1),
(S1.3),
.. ,
(S6.3), (3.41)
x n
1 2
v
(2){T
-J-
(2,k,t)-c
(2)}=N
(O,k,t)+k /
{n
I-G(t)
dt}/R
(x
)+0
(1).
n
n
n
n
k x
1
n
p
n
Using Lemmas 3.2 and 3.6 gives
(3.42)
T * (2,k,t)
n
from which the characterization is obvious.
To compute ~(Nn(O,k,e)Nn(2,k)
21

---

: '
: ,
J
.",
J",
-86-
we recall that Œ(B(s)B(t)=min(s,t) -st if {B(t),
O~t~l} is a Brownian
bridge,
and an easy calculation yields
(3.43)
Œ(N (O,k,l)N )) ~-l.
n
n
This and Lemma 3.5 suffi~e to calculate ~l (0). All the proofs are now
complete.
1
1
IV - Limit laws for A (l,k,l).
- - - - - - - - - - - - n , - ' - ' - '---'--
We, need sorne generalized forms of Lemmas due to de Haan (1970)
or to Lô( 1990) .
Lemma 4.1. Let Fel,
then for any integer p~l,
. - .
-1
(i)
R (x,F)-(x -x)P{n~-81(o+j)
}, as x~x
whenever FeD(w ),
p
0
J=
0
0
and
Proof of Lemma 4.1.
(i)
is obtained by routine calculations from
(1.2).
ii)
is easily proved from Lemma 2.5.1 and Theorem
2.5.2b of de Haan
(1990)
showing that
y
G1=G e D(~)~G2(o)=1-J 0 1-G (t)dt eD(~) and R (X,G )-R (x,G ) as x~xo(G).
1
1
1
1
2
o
Applying this p times gives
(ii).
Lemma 4.2. Let Fel. Then for any p~l,
( i )
( z - x) p / R
(z , G )
~ +00,
p
(ii)
R (x,z,G)/R
(x,G)~l,
P
p
as x~x (G),
z~x (G),
(l-G(z))/(l-G(x))~O.
o
0
22

---

: ,
)
'~
" ' ...}p-
Proof of Lemma 4.2.
(i)
is easily derived from Part ii)
of Lemma 4.1 above and Lemma 8 of Lô
(1990). To prove
(ii), put
z
z
z
Yo
Yo
(4.1)
m(a,b,x)
J
J
. .. J
J ... J 1-G(t) dt d~a,b'
x
xl
x a - 1 YI
Yb-1
where d~a,b= dx 1 ··· dXa_1xd~1" . dYb_1= d~axd~b and
Yo
m(O,b,x)
=Jx
and
m(a,O,x)
- G(t)
dt d~ .
a
Straightforward manipulations yield for any p~2, for z<x<y ,
\\
0
j =p
(z -x)p-j
m(O , J' , z)
)
(4.2)
Rp(x,Z)=Rp(X)
((m(o,p,z)/m(o,p,x)-Lj=l
(p_j)!
X
m(O,p,x)
As in (2.10)
in Lô
(1990),
P
-1
(4.3), 05m(O,p,z)/m(O,p,x)5(l+(z-x)
/Rp(Z))
,
and for any j,
1 5 j5p,
(Z-x)p-jm(o,j,z)
R, (z)
J
m(O,p,z)
(4 .4)
- - - - - , - - - : x - - - - x
m(O,p,x)
(z-x) j
R
(z)
m(O,p,x)
p
-1
sinee sUPx~OX(l+X)
=1.
(4.2),
(4.3)
and
(4.4)
Part i)
together ensure
Part ii).
Here are our results for A (l,k,t).
n
Theorem 4.1. Let FEr. Suppose that
(2.4)
holds,
then
23

---

:
1
. '
,
-88-
)
'\\
)
'\\
z
z
1
where T(k,t)
= nk- J n J n 1-G(t) dt - R (x
z )-R
(x )
in
X
y
2
n'
n
2
n
n
2
probability and 0'2 (ér)
= 6 (0+1) (0+2) / {(o+3) (o+4)},
0<0:S+00.
-2
Remark 4.1.
If FED(~ ) ,O<a, R (x ) and T(k,t)
tend
a
to a
so that
2
n
-1/2
1/2-~
An (l,k,t)
p a, as (tk
.,)(/n)~ (0,0) .A (l,k,t)
is then an asympto-
n
tically consistent estimator of a.
It will be studied elswhere.
\\
\\
Remark 4.2.0ne can weaken the assumptions on
(k,t)
by replacing
(2.4b)
by
-1/2
lim
nk - 1/2
n1 +1/ 0
( k 1/2 - ~ +1 / 0 )
sup {.
<00 or {.
=0
for
FED (1/1 ),
by tk
logk-~O
o
n~+oo
for
FED(~) and by t=o(k1/2-~(logk)2~+1) for FED(A). cf. Lines following
(4.18)). Also,
T(k)
can replace T(k,t)
in Theorem 4.1 under the same
assumptions.
Now,
put
*
1/2
v
(O){A (l,k,t)-c
(O)}=A (l,k,f),
v
(0)
= k
/R
( x ) ,
c
(O)=T(k),
n
n
n
n
n
2
n
n
-1
e
(o)=0
(0+ 2 ),
K(o)=(o+1)/(o+2),
O<o:s+oo,
3
2
4
3 2
{ 2
}
0'3(0)=(5 0 +11 0 +4 0 +7 0 +12)/ 0 (0+ 3 ) (0+ 4 )
,
0<0:S+00.
Theorem 4.2.Suppose that the assumptions of theorem 4.2 are satisfied.
We
have
*
-1{
2
2 -1/2
A (l,k,t)=N
(4,k,e)+e
(o)N
(-00) (1+0
(l))+(2K(o))
e
(o)
N (2,k)
k
n
n
4
n
p
1
n
-1/2
2
2
-1/2
}
+2k
e
(o)e
(o)N
(2,k)N
(-00) (1+0
(l))+e
(o)
N
(-00) k
(1+0
(1)
1
2
n
n
p
1
n
p
24

---

.. ";
,
,
-89-
J
' ' \\
x (1 +0
(1) +0
( 1) ,
P
P
where
2
N (4,k,l)-N(0,Œ
(0)). Then,
n
3
i)
*
d
2
A (l,k,l)
~ N(m,Π(0))
iff N (-00)
~ mie (0)
n
3
n
p
4
*
ii)
d
2
.
d
2
A
(4,k,l)
~ N(m,Œ
lff N (-00)
~ N(m/e (0), Œ(-oo)
)
n
c
n
with
4
'
limn~+ooŒ~(0)+Œ(-00)2~2e4(0)COVNn(4,k,l)Nn(-00) )=Œ~
*
1
iii)
A (4,k,l)
~ + 00 (rep. -00)
iff N (-00)
~ + 00 (resp.-oo).
n
p
n
p
Remark 4.3.
The characterizations are identical in Theorems 3.2 and 4.2.
We pointed out in Section III that we have the asymptotic normality
whenever
(1.7)
holds. The following examples concentrate on the case where
uf' (u)
has not a limit Js u~O.
Examples 4.1.
Let f(u)=u sin(l/u).
f'
exists and uf' (u)=u sin(l/u)
- COS(l/u)
does not converge as u~O.
But if there exists a sequence of integers
(p)
1 such that
(l/u )-
n n~
n
2rrp ~b,
-rr<bsrr,
then u f' (u )~ -cosb = a.
Returning to the proof of
n
n
n
1 4
Corollary 3.2 and remembering that
IU 1
_k/nl s n-
/
,
a.s. one has
k + ,n
for 0<2a<1/4
:
if Pn = [na], kn=[n/(2rrPn+b)],osb<2rr and cos b = a, then
T * (2,k,l)=N
(0,k,l)+(e
(0) -ae
(0))N
(2,k)+0
(1)
n
n
1
2
n
p
and
A* (2,k,l)=N
(3,k,l)+(e
(0)-ae
(0))N
(2,k)+0
(1)
n
n
3
4
n
p
2
In particular,
if b=O,
then T* (2,k,l)
(resp. A* (l,k,l))
~ N(O,Π(0)) (resp.
n
n
0
2
N(0,Œ
(0))
for FED(A)
U D(~). This fact occurs for FED(W ) iff osl with
2
o
cos b=l/ o .
25

---

.,
-90-
, ' 1
1
..
1
1
J
' ' \\
J
' ' \\
PRüüF8. We proceed as for T (2,k,t)
by general statements
(81.4),
(82.4),
n
etc ... First, use
(2.1)
and
(2.3)
to obtain
(4 .5)
R (X )-l k 1/2(A (l,k,t)--c(K,t)=N (3,k,t)+Q 1(k)+(t/k)1/2
2
n
n
n
n
-1/2
+ Qn1 (t) +tk
Qn2
(t) +Qn3 '
where
26

---

•
1
J
'~
-91--' 1
J
'~
- -
Z
Z
Qn2(i)=(n/i)1/2({J nJ n Bn (1-G(t) dtdy}-{J:n JZn 1 - G(t) dtdY})/R (X ),
2
n
x
y
x
y
n
n
and
-
-
Qn3
(n/k)
= 1/2 (JX:n
_
JyZn {an (1-G ()t )''-B (1-G ()}
t )
dtdy)/R
) ,
n
(X
2
n
n
We show that each of these error terms tends ta zero in probability.
(81.4) Qn3
-7 O.
P
If i~, we get, as in (3.5),
for sorne v,
0<v<1/4,
V
(4.6)
1Qn3r ~ 0p(k- ) R (X
2
n ,
zn' G1/2_V)/R2(Xn)
~ 0,
by Lemmas 3.2 and 4.2.
Now,
let i be fixed.
Thus
=:Qn3(1)+Qn3 (2)+Qn3 (3),
Where 0
stands for a
(l-G(t)) -B
(l-G(t))
dt dy.
n
n
One quickly obtains for sorne v, 0<v<1/4,
V
(4.8) Qn3(1)
=
0p<k-
)
~ 0, as k-7 + 00.
Next, by Lemma 3.2,
with PNOW as n and À are large,
with by Lemma 3.2,
27

---

.'
,
--92-
J
. "
By Lemma 1 in Lô
(1990)
and Lemmas 4.1.2 and Formulas
(3.13)
and
(3.15)
above,
FED(q>~) implies that R2(xn)-7~-2, R2(xn)-7~-2, (tn(t/~)-tn(t))
2
*
/R2 (zn)-7(log i\\)
so that IE.Qn3 (3)
-7 O.
By the same arguments IEQn3 (3)-70
1
when FED(A)
since
p-1(0)=
k 1 / 2 R (x )/R :(Z ) ~ k 1 / 2 - Ct C /2
n
2
n
4 n
'
for any C,
0<C<1/2, as n ïs large enough (use SVZ functions properties!).
For FED(l/J),
R (z )-K(~)R1(Z )2_K(~)(~+1)-2(y -G- 1 (1-t/n))2 and
~
2
n
n
0
hence
(t
(t/i\\)-t
(t))2/ R2 (Z )-7(~+2)-1(i\\-1/~_1)-2,
all
that
by Lemmas
n
n
n
,3.4
and 4.1 which also implies for O<c<l/~, p-1(0) ~k1/2 (k/t) as n is
n
*
*
. large.
In all cases,
IEQn3\\ (3)-70 Finally, Qn3 (3)
pO and hence Q~3 (3)pO.
- To finish,
with PNOW,
as n and i\\ are large,
*
(4 . 11)
1Qn3 (2) 1~Qn3 (2) ,
with
t
(i\\k) -t
(t)
t
(t/i\\) -t
(t)
R
(z )
*
n
n
n
n
1
n
1 / 2 - 1
(4.12)
lE Qn3 (2)~4t
K(~)
X
- - - - - - x
X - - - - - -
1/4
One shows exactly as above that k
R (X )/R
(Zn)
-7 +
1
n
1
00
and
(tn(t/i\\)-
1 4
t
(t ) )/R
(Z ) is bounded as n-7+OO whenever FEI. Also,
k-
/
(t
(i\\k) -t
(t))
n
1
n
n
n
/R
(x )-70 obviously when FED(l/J~) by Lemmas 3.4 and 4.1 If FE(q»
1
n
U D(A), GE(A) and, using (1.4) and (3.15) and SVZ functions properties,
one has any c,
O<c<1/4,
1 4
(4.13)
k- /
(t
(i\\k)-t
(t))/R
(X
)~{(l+c)s(t/n)/s(i\\k/n)-l+(l+c)
n
n
1
n
k c
i\\k
-1/4
X(7)
log(e-)}k
.
-1/4
-1/4
As in the preceding, k
s(t/n)/s(i\\k/n)-70,
k
R (Zn)/R
(x )-7o.one
1
1
n
28

---

" ,
" ,
)
''\\
)
''\\
-93-
concludes that
*
~ Qn3
(2)~O and hence (81.4) holds.
(82 .4)
Qn1 (k)
I~'
P
We have
/
x
x
x
Qn1 (k)
{(n/k)l 2J_n J n B (l-G(t)) dtdy+(n/k)1/2J_n
B
(l-G(t))
n
n
x
y
x
n
n
dt dy}/R
(x
2
)=:Qn1 (k,l)+Qn1 (k,2).
n
It follows that fbr any À>l,
one has with PNOW as n is large,
(4 . 14)
*
*
1Qn1 (k,l) I~ Qn1 (k,l);
1Qn1 (k,2) I~Qn1 (k,2)
with
(4.15)
~ Q*1(k,l)~3K(r)-lÀ1/2(t (k/À)-t (Àk))2/R1 (X )2,
n
n
n
n
and
where À'=À fort~+oo and À'is taken large for t fixed.
Arguments given
in
(3.24)
show that that ~
*
Qn1 (k,l)~O and a combination of these same
arguments and Lemmas 7 and 8 in Lô
(1990)
and Lemmas 3.2 and 3.3 above
*
*
*
ensure that ~ Qn1(k,2)~O. We conclude that Qn1(k,l)+Qn1(k,2)pO and
thus,
by
(4.8),
(82.4)
holds.
(83.4)
(t/k) 1/2 Q 1 (t)
~ O.
n
p
One has with PNOW as n is large
(4.17)
1 (t/k)1/2 Qn1 (t) I~Qn1 (t, 1) + Q~l (t,2)
with
(4 . 18)
~ Q*1(t,l)~3K(r)-1(Àt/k)1/2(t (k/À )-t (k))
n
n
1
n
(t
(t/À ) -t
(t)) /R
(x
) 2
n
2
n
1
n
and
where À >1 and either À
= À ( for t~ + 00)
or À
is taken large
(for t
1
2
1
2
29

---

;";
,-94-
"
1
J
''',
J
''',
fixed).
By the arguments many times used above,
one has ~Q~l (t,l)~O and
*
1/2
lEQn1 (t, 2) ~o and consequently,
(t/k)
Qn1 (t) ~O whenever
(4.20)
p
(l,0)=(t/k)oR
(Z
)/R
(x )~O,
n
1
n
1
n
for aIl 0>0 and for aIl d.f.
FEI. But,
GED(A)
U D(W) and by Lemma 3.4,
À+o-c
1
(4.21)
P
(l,o):S 2(t/ k)
, '
n
for O<c<o,
À = l/r for FED(W!)
or À
o FED(A)
D(~). This ensures
(2.20)
r
and completes the proof of
(83.4).
( S 4 . 4 ) tk - 1/ 2 Q 2 (t)
~ O.
n
p
One has with PNOW(t):sn is large,
-1/2
*
*
(4.22)
Itk
Qn2(t)!:s Qn2(t,1)+Qn2(t,2)
with
1 2
(4 .. 23)
Q*2(t,1):SÀtk- /
(t
(t)~t (À k))t (t/À )-t (t))/R (t (k))
n
n
n
1
n
2
n
2
n
and
1 2
(4.24)
Q*2(t,2):S tk- /
(t
(t/À )-t
(t))R
(Z )/R (t
(k)),
n
n
2
n
1
n
2
n
where À >0,À =À
or À
is taken large.
1
2
1
2
Always by the now familiar arguments used above and by general
properties of SVZ functions,
one shows that
(4.25)
Q * 2(t,2)
~ 0,
n
p
and
*
(4.26)
Q 2(t,1) ~ 0,
n
p
i)
for FED(W ) whenever lim sUPn~+oo or there exists ç,
0<ç<1/2,
such
r
that e1 + 1 / r = o(k1/2-ç+1/r)
1 2
1/2
-1
ii)
for FED(~ ) whenever tk- / logk ~osinceR1(t) -R (t)
~a
as
ex
2
a
t~+oo and t
(t)-t
(À k)
= O(logk)
(see
(3.7a)
in Lô
(1990)
where k_n ,
n
n
1
o<a<l)
;
iii)
and for FED(A)
whenever there exists n, O<n<1/2,
such that
1
t=o(k / 2 - n (log k)2 n +1))
since for any c<O,
as n is large,
30

---

,
,
.. ,
l
' ' \\
-95-
.. '
(4.27)
Os{tn(l)-tn(A k) }/R (X )S2{S(l/n)/s(k/n) }+l+(k/l)C log(A k/l)}.
1
1
n
1
AlI these conditions on
(k,l) _are implied by
(2.4c).
This completes the
proof of
(84.4)
Tt remains to prove this important result.
Lemma 4.3.
Let Fer,
k~+oo, k/n~O, l/k~, then N (3,k,l)
is a Gaussian
n
r.v. with mean zero such that!EN
(3,k,l)'2_ 6R4 (X,z )/R (x )2_<T (o),
n
n
n
2
n
2
0<1<+00.
Proof of Lemma 4.3.
That N (3,k,l)
is Gaussian follows
from Theorem C.
n
Its variance
Z
Z
Z
J n J n J n h(s,t) dt d s dP dq )/R2 (x )2
n
x
p
q
\\
n
where h(s,t)=min(l-G(t),
1 -G(s))-(l-G(t)) (l-G(s)).
Using the symetry of H(o,o)
and considering the cases pst and p>q yield
(4.29)
S~(3)-{ (2n/k) JZn JZnH(p,q) dp dq}/R (x )2.
2
n
x
p
n
Further, cutting the integration space into {sst} and {s>t} gives
(4.30)
s~(3)-2(2R4(xn'Zn)(1+0(1))+(~JZnJZn(q_p)JZn1-G(t) dt dq dP]
_
x
p
q
n
The second term in brackets is - R (x ,z ) by an integration by parts
4
n
n
with
Z
Z
V=q-Pi u=J nJ n 1-G(t) dtdy. Finally,
q
y
2
2
(4 . 3 )
s
(3) - 6 R
(x
, Z
) / R
(x
)
.
n
4
n
n
2
n
Lemmas 4.1 and 4.2 thus complete the proof of Lemma 4.3.
Theorem 4.1 is proved by
(81.4),
(84.4)
and Lemma 4.3.
..
..
•
1
31

---

"
1
1
" ,
J
' ' \\
J
' \\
-96-
The first part of Remark 4.2 follows from the lines just below
(4.26).
The second part of Remark 4.2 follows by remarking that
1 2
1 2
(4.32)
k /
1{-C(K,t)--C(K)}/R
(X) l:Stk-
/
(t
(t)-t
(i\\k))R
(t
(t)/R (t
(k))
2
n
n
n
1
n
2
n
+tk- 1 / 2R (t
(t) ) /R
(t
(k) ) ,
2
n
2
n
for i\\>l,
with PNOW' as n is large.
Both terms at right tend to zero
exactly as in
(4.24)
and
(4.25).
Remark 4.2 is now completely justified.
To prove THEOREM 4.2.,
remark that,
by Theorem 4.1,
2
(4.33)
A:==N
('3,k,t) +k1 /
(RJ:nJZn 1-G(t)dt dy - R JZn JZn 1-G(t) dtdy)
n
x
y
x
y
n
n
/R
(X
)+0
(1)==: N (3,k,t)+Q 2(k)+0
(1).
2
n
p
n
n
p
But,
as in
(3.41),
x
x
/
Z
Z
1
(4.34)
Q
(k)
== k1/2{(~J n 'f n 1-G(t)dt + k
2(R f_nJ n 1-G(t) dtdy)}
n2
\\
k SC
Y
lx
x
n
\\ n
n
/R
(x ) = (2k1 / 2 K (0)) -l ç (k) 2 (1) ) +K (0) -l ç (k) (1+0 (1) ) ,
2
n
p
where çn(o)
is defined in Lemma 3.6 by which
(4.35)
A*=(2k1 / 2 K(0))-1
{e
(o)N
(2,k)+e
(0)N
(-00)(1+0
(1))}2+
1
2
e4 (o)N
(-00)
n
n
n
p
n
+N
(3,k,t)+e
(0)N
(2,k)+0
(1).
n
3
n
p
It remains
for x :St:Sz
(see
(4.28))
and thus
n
n
This easily implies that
We have proved the first part of Theorem 4.1. The characterization is
1 2
obvious now when we remember that k-
/
N (-00)
= f(U
1
)-f(k/n)
~O.
n
k + , n
p
32

---

".
:
1
1
•
1
)
''',
-97-
v' -
Lirnit laws for ë
= y
-y
=2 -x .
- - - - - - - - - - - n - - n - t , n -n-k,n -n- n
We assume from now on that the regularity co~ditions (1.7) or
(1.8k,t)
or
(1.9k,t)
hold for for sake of simplicity.
But it will
appear i~ the proofs how optimality results may be obtained. Notice
that
(1.8t)
or
(1.9t)
is required only when t~+oo. Put C =z -x
n
n
n
Theorem 5.1. Let F=(f,b)El satisfying the regularity conditions.
Let
(2.4a)
holds.
1)
If FED(A)
U D(~), then
(ë -C )/Rl(Z )=2 -z )/Rl(z )+0 (1)
=-logE
(i) +0
(1)
~ -logE (e)
n
n
n
n
n
n
p
n
p
\\
\\
when is t fixed and
l
2
l
2
t /
(ë -C )/Rl(Z )=t /
(z -z )/Rl(Z )+0 (l)=el(o)N (2,t)+0 (1),
n n
n
n
n
n
p
n
p
wwhen t~+oo and t/k~O.
2)
Le t
FED (~ ).
o
a)
If 0>2,
then
(ë -C )/Rl(Z ) =(2 -z )/Rl(Z )+0
(1)=(0+1) (l-E (t)l/o)+o
(1)
~
n
n
n
n n
n
p
n
p
(0+1) (l-E (e) 1/ 0 ), when t is fixed and
l
2
l
2
t /
(ë -C )/Rl(Z )=t /
(2 -z )/Rl(Z )+0 (l)=el(o)N (2,t)+0 (1),
n n
n
n n
n
p
n
p
when t~+oo, t/k~O.
b)
If 0<0<2,
then
l
k / 2 (ë -C )/Rl(X )=el(o)N (2,k)+0
(1)
~ N(O,e (0)2),
n
n
n
n
p
l
in both cases where t is fixed and t~+oo while e/k~O.
Remark S.l.These results notably extend earlier results by de Haan and
Resnick (1980)
and by Lê
(1986b).
33

---

)
''\\
-98-
Remark S.2.
Z dominates X in these results since C follows the law
n
n
n
provided,by z
except when FED(~ ),
0<1<2.
This fact will occur many
n
1
times in the
sequel.
Corollary S.l.
Let F~(f,b)Er satisfying the regularity conditions and
If FED(A)
U D(~), then
\\
(nuT
(B)-l- c
(B) )/R
(Z ) Q -logE(t) when e is fixed and
n
n
1
n
.
u
-1
d
v (B) (n T (8)
-c
(B))
= -N (2,e)+o
(1)
~ N (0,1) when e~+oo, e/k~o.
n
n
n
n
p
n
fixed and
v
(B)(nuT
(B)-l- c
(B))=e
(1)N
(2,e)+o
(1)
when
e~+oo, t/k~o.
n
n
n
1
n
p
1 2
If FED(~1)' 1>2, then for V~(B) = k / /R (x ),
1
n
V~(8) (nU T '("B)-l - cn(B))
e
(1)N
(2,k,)
+ 0p(l),
n
1
n
when t is fixed or t~+oo while e/k~O,
If FED(~ ),
0<1<+00,
then
1
R (X ) (T
(9)
- c
(9))/R
(Z
)
Q (E(t)1/1 - 1) when e is fixed and
1
n
n
n
1
n
1
v
(9) (T
(9)
- c
(9))
=
1- N (2,e)
+ 0
(1)
when t~+OJ
while e/k~O.
n
n
n
n
p
Remark S.3.
For 1 = 2,
the results depend crucially on b(.)
(see Lemma
6,2 and Remark 6.1 below), Mixture cases are signaled in these examples,
34

---

..
1
1
.. '
J
'4,
J
'4,
-99-
PROOFS.
First,
the following lemma lS a direct consequence of properties of
p-RLZ functions.
p~O.
Lemma 5.1. Let p
(2,1/2)=k- 1 / 2R (X ) (z ),p
(3,1/2)=t1/2k-1/2R1(X )/R
(Z ).
n
1
n
n
n
n
1
n
If FEI,
then p
(2,1/2)~0 as n~+oo, k~+oo, k/n~O, t being fixed and
n
Pn(3,1/2)~ as n~+oo~ t~+oo, k~+oo, k/n~O, t/k~O, for 2<O'oS+oo. For 0<0'<2,
both limits are inflnite.
1/2
a)
Now,
let t~+oo.1Using Lemma 3.6 and the fact that m
a (m)=o (1)
for
,n
p
m=k or m=t, we get after routine considerations,
1/2
(5.1)t
(C -C )/R (z )=e
(0')N
(2,t)+p
(3,1/2)e
(0')N
(2,k)+0
(1),
1
1
1
n
n
\\n
n
n
n
\\
p
which combined with Lemma 5.1 yields
1/2 ~
(5.2a)
t
(C -C)/R (Z )=e
(0')N
(2,t)+0
(1)
n
n
1
n
1
n
p
for FED(A) U D(~) U D(W ), 2 <O'oS+oo,
and
a
(5.2b)
v * (8) (è -C )=e
(0')N
(2,t)+0
(1),
n
n
n
1
n
p
for FED(W ), 0<0'<2. This completes the proofs of Theorem 5.1.
a
b)
Let t be fixed.
For FED(W ),
one has by Part i)
of Lemma 4.1 and Formula
(1.6)
a
-
y
-z
±c +1/0'
(5 . 3 )
(z
-2 ) /R (z ) - (0'+1) (
0
n 1) - (0'+1) { (nU
1
n 1
ft)
n
-1} (1+0
(1)),
n
n
n
y 0 - Z
{.+
,n
p
where
c = sup
jb(t) I~ O.
n
OoSuoSU
P
t+1, n
This and Theorem C together prove Part a)
i)
of Theorem 5.1.
For FED(~) U D(A), GED(A). Using (1.4) gives
1
(5.4)
Z
-2 =s(U n 1
)-sU/n)+J(n/
U
) s(t)t- dt,
n
n
{.+, il
{.
n,
n
1
{.+
, n
Set
El ={s (t) /s (t/n),
t/noStoSUn 1
or
Un 1
oStoSt/n}.
Tt
is
easily
shown
n
{.+ , n
{.+ , n
35

---

-;,'100--
, 1
,
J::.,
J
'::.,
from
(1.3)
that infe
~ 1 and supe ~ 1.
It follows that
n p
n p
-
( 5.5 ) z
- z
logE
(t)
+
n
n
0
(1).
n
p
We do remark that -logE(t)
is the Gumbel extremal law for the
(t+1)th
maximum.
Finally,
(5 .6)
(C -C )/R (Z )=-logE (t)+p
(l)N
(2,k)+o
(1)= -logE
(t)+o
(1).
n n
1
n
n
n
n
p
n
p
This proves Part a)
ii)
of Theorem 5.1.
The results related to T
(8)
in Corollary 5.1 are immediate.
To prove'
n
those related to T
(9),
chek that
n
-1
(5.7)
T
(9)-c
(9)=((y
-x ) (y -X))
((y
-x ) (z -z )-(y -z ) (x -x ))
n
n o n
0
n o n
n
n o n
n
n
and hence by Lemmas 3.2 and 4.1,
-
z -z
\\
-1
n
n
-1/2
(5.8)
R
) (T
(9)-C
(9))/R
(Zn)-('1+ 1 )
(R
(Zn)-e
('1)N
(2,k)k
).
1 (Xn
n
n
1
1
n
1
And this proves the results related to T
(9)
in Corollary 5.1 following
n
the lines just above.
In the next section,
we deal with all the ratios of statistics
already studied including the major element of the FACSEXT which is
VI - Limit law for ratios from the FACSEXT.
A)
Case of T
(1, k, t) .
- - - - - - - n - - ' - - - - ' - - - -
First, we obtain a general result.
Theorem 6.1. Let F=(f,b)El.
If
(k,t)
satisfies
(2.4),
then
k 1 / 2 (T
(l,k,t) _«(K)/T(K)1/2)
(2VK('1))
-1(2N
(ü,k,t)-N (3,k,t))+o
(1).
n
n
n
p
d
2
~ N(ü'~4('1)), ü<'1~+oo,
36

---

".
-101-
,
1
"
1
)
'~
3
2
2r +10r +32r+24
2
where EN (O,k,t)N
(3,k,t)-3(r+1)/(r+3)
so that Π(r)
n
n
4
4 (r+1) (r+3) (r+4) ,
Theorem 6.2.
Let F=(f,b)Er and
(k,t)
satisfy
(2.4).
Under the
regularity conditions, we have
v
(1)
(T
(l,k,t)-c
(1))=2VI<.(r))
-1(2N (l,k,t)-N
(4,k,t))+0
(1)
n
n
n
n
n
p
d
2
~ N(O,Œs(r)), O<r~+oo.
3
2
r
+r +2r
4 ('1+1) (r+3) (r+4) ,O<r~+oo.
PROOFS. We need only to prove Theorem 6.1 as a corollary of Theorems
3.1 and 4.1 Theorem 6.2 follows from Theorems 3.2 and 4.2 by the very
same arguments: By Theorem 3.1.
(6.1)
T
(2,k,t)
n
so that
(6.2)
T (2,k,t)2 == «(R)2 + 2k- 1 / 2 N (O,k,t)
+ 0
(k- 1 / 2 «(k)2).
n
i l
p
Since «(R)-«(k)
in probability,
one has
(6.3)
T
(2,k,t)
==
«(R) (1 + 0
(1)),
n
p
Furthermore,
by Theorem 4.1,
(6.4)
A
(l,k,t)
n
so that
(6.4)
A (l,k,t)
==
T(K) (1 + 0
(1)).
n
p
Nowa straighforward calculation based on
(6.1) - (6.5)
and on the fact
2
that T(k)-K.(r)«(k,t)
(see Lemma 3.2 and 4.1)
yields
37

---

".
, '
-102-
)
.~
1/2
-1
(T
(l,k,t)-«(K))(r(K)
)=(2VIc(1))
(2N (O,k,t)-N
(3,k,t))+o
(1).
n
n
n
p
This partly proves Theorem 6.1.
It remains to compute
~N (O,k,t))N
(3,k,t)
= s
(0,3)
which is
n
n
n
where h(s,t)
is defined in
(4.28).
Cutting the space integration into
s~y and s>y and using the first
(resp.
the second)
expression of
s
(0,3)
for s~y (resp.
s>y) , we obtain
n
Considering now the cases s~t and s>t in the second term of
(6.8)
gives
(6.9)
s
(O,3)=(R
(X) (1+0
(1))+2R
(X
) (1+o(1))/R
(X
)R
(x ),
n
3
n
8
3
n
1
n
2
n
where we used Lemma 3.2. Now Lemmas 4.1 and 4.2 imply
-3
(6.10)
s
(0,3)-3(1 + 1) (0 + 3)
,O<1~+oo,
n
from which we derive Π(0).
Theorem 6.1 is now entirely proved. Theorem
4
6.2 is proved by the same arguments but we must say a few words on
Π5 ('1).
One has
(6.11)
2N
(l,k,t)-N (4,k,t)=2N
(O,k,t)-N
(3,k,t)+
N (2,k),
n
n
n
n
n
which,
together with
(3.43)
and
(6.10), permits to compute Π(1).
5
B)
Case of T
(3,k,t).
n
We already noticed in Theorem 5.1 that when z -x
intervenes,
the
n
n
contribution of z (normal or extremal) dominates that of x
(normal).
n
n
Here again,
this is the case except when FED(W ) where each of z
and
'1
n
x may get the better of the other with possibilities of a mixture of
n
38

---

1 ",:1 1
: ,
) '4,
-103-
bath. We begin with.
1
-1/2
-1
Lemma 6.1. Let Fer. Put
1
2
p
(4)
= k
(z - x )R (z ,G)
, p
(5)=e
/
p
(4),
1
n
n
n
1
n
n
n
with k-HOO, e/k~O,
k/n~O
i
e is fixed in p (4) and e~+oo in p (5).
n
n
1
1)
If FeD(i\\) U D(/f», then (p
(4),
p
(5) )~(O,O).
n
n
2)
Let FeD (l/J'a') , 'a'>0.
1
a)
If 'a'<2, then
(Pn(4) ,Pn(5))
~
(+00, +(0) .
b)
If 'a'>2,
then
(Pn (4) 'Pn (5))
~
(0,
0) .
J
c)
If 'a' = 2,
any limit is possible.
1
1
proof of Lemma 6.1.
Let FeD(i\\) U D(/f». Thus GeD(i\\)
and by
(4.27),
for any c,
O<c,
as n is
large,
(6.12)
(z -x )/R
(Z ) s3{s(k/n)s(t/n) }+3{(k/e)C log (k/e) }+1.
n
n
1
n
This and
(4.21)
together ensure that P (4)~0
(t fixed)
and P (S)~O
n
n
(t~+OO) . Now let FeD(l/J ),
i.e., GeD(l/J ). By Lemma 4.1 and Formula
(1.6),
'a'
'a'
n
(6.13)
1/
rk /
1
(z
- x
) / R1 (z ) - ('a' + 1) (k / e)
'a' exp
(
b (t ) t - dt) .
n
n
n
tin
a)
Let 'a'>2. For any c,
0<c<min(1/'a',l/2-1/'a'),
one has for large values
of n,
(6.14)
P
(4)s2('a'+1)eC-1/'a')k-(1/2-C-1/'a')
i
P
(S)s2('a'+1) (e/k)1/2-C-1/'a'
n
n
which bath imply that
(p
(4) ,p (5))
~
(0,0).
n
n
b)
Let 'a'<2. For c,
O<c<min(l/'a',
-1/2 + 1/'a') , one has as n is large,
which both imply that
(p
(4) ,p
(5))~ (+00,
+(0).
n
n
c)
Let 'a' = 2.
-1/2
a)
If b(t) =t,
p
(4) ~ ('a'+1)t
, p
(5)~('a'+1).
n
n
V
(3)
If b(t)",,1/log1og(1/t), k_n ,O<V<l,
(p
(4) ,p
(5)) ~ (0,0).
n
n
39

---

: ,
, ' 1
1
-104--
J~
}
.~
u
0)
If b(t) =-1/10g10g(1/t),
k_n
,
O<U<l,
(p
(4),p
(5))
~ (+00, +(0).
n
n
a), ~), 0)
prove that any limit is possible when 0=2.
Here are our results for T
(3,k,e,u).
n
Theorem 6.3.
Let F=(f,b)El and
(k,e)
satisfy
(2.4).
Suppo~e that the
regularity conditio~s hold.
1)
Let FED (A)
U D (4)):.
a)
If e is fixed,
then
-
z -z
n
n
d
R
(
)+0
(1)
~ -log E(e).
1
Zn
p
b)
If e~+oo,
then
e1 2
/
(Z -z )
n
n
v
(3) (nuT
(3,k,e,u)-c
(3))=
+0
(l)==-N
(2,e)+0
(l)~N(O,l),
n
n
n
R
(zn)
p
n
p
1
1 2
where v
(3)
== e /
R (X )/R
(z
),
c
(3)
==
(z
-x )/«(k).
n
1
n
1
n
n
n
n
2)
Let FED(W ) with 1>2,
then
1
R
(x
) (nUT
(3,ke,u)-c
(3))/R
(Z
)
~ (1+1) (1_E(e)1/1) when e is
1
n
n
n
1
n
fixed and
v
(3) (nuT
(3,k,e,u)-c
(3))=e
(1)N
(2,e)+0
(1)
when e~+oo.
n
n
n
1
n
p
1
2
3)
Let FED(W ) with 0<1<2,
then for v* (3)==k /
,
1
n
*
U
v
~3)(nT (3,k,e,u)-c (3))==-(1+1)N (O,k,e)-N
(2,e)+0
(1)
n
n
n
n
n
p
~ N (0, (1 + 1) 2 (51 + 8) / (0 + 2)),
in both cases where e is fixed and e tends to infinity.
40

---

,'1
t
J
' ' \\
Remark 6.1. We may obtain a mixture case for 0=2.
For instance,
put
b(t)=t a , a>l,
in
(1.6).
Then,
p
(2,1/2)~-1/2=b , P (4)~3l-l/2=b and
n
0
n l
Pn(5)~3 = b
so that.
2
-1
v
R
(Z)
R
(X) (n T
(3,k,l,v)-c
(3))
3b N (2,k)/2-b N (l,k,t)
1
n
1
n
n
n
o n
l
n
+bl(l-E
(l)1/2)+0
(1)
n
p
and
ll/2 Rl (Z )-lRl(X ) (nvT (3,k,t,v)-c (3))
=3N
(2,k)/2
n
n
n
n
n
- 3N (1, k , l, v) +0
(1).
n
p
*
v
PRüüFS. Set T (3)
= n T (3,k,l,v)
-
c
(3). By Theorem 4.1,
n
n
n
(6.1.6) T*(3)=«(k)-l(_(Z -2 )+(x -x )-vk(z -x)N (l,k,l)+o ((z -x )/Vkl).
n
n
n
n
n
n n n
p
n
n
\\
Hence,
- p
(4) N
(1, k, l) +0
(p
(4)).
n
n
p
n
From this step,
Lemmas 5.1 &
6.1 and the fact that
(z -x )-(0+1)R (X )
n
n
1
n
give all the possibilities listed in Theorem 6.3.
C)
Case of T (6).
- - - - ' - - - - - - - n - -
Theorem 6.4.
Let F=(f,b)El and
(k,l)
satisfy
(2.4)
with l~+oo.
If
the
1/2
regularity conditions hold,
then for v
(6)
= l
R
(Z)/(Z-x),
n
1
n
n
n
v*
(6)
=ll/2(z
-x )/Rl(Z),
c
(6)
=(z
- x
) / « (e) ,
n
n
n
n
n
n
n
v
(6) (T
(6) -l_ c
(6))
=-N (l,l, 1) +0
(1)
for 2<0:S+00,
n
n
n
n
p
and
41

---

, ' 1
1
"
1
1
.'"
'"
-106-
)
)
v* (6) (T
(6) -l_c
(6))
-N (l,i,l)+o
(1),
for 0<0<2.
n
n
n
n
p
PRüüF.
It i~ a simple case of the next proof.
D)
Case of T (7)
n
Theorem 6.5,
Let F:(f,b)El and
(k,t)
satisfy with t~+oo. If the regularity
l
2
conditions hold,
then for v
(7)
= t /
R (Z )/(z -x )2,
,
n
2
n
n
n
l
v*(7)
~ t / 2 (z -x )2/R2 (Z ), c (7)=(z -x )2/r:(t),
n
n n
n
n
n n
v
(7) (T
(7) -l_c
(7))
=-N (4,t,1)
+ 0
(1)
for 2<0:500
n
n
n
n
p
and
v* (7) (T
(7) -l_c
(7))
N (4,t,1)
+
n
n
n
0
(1)
for 0<0<2.
n
p
PRüüF.
By (5.1),
ln the case where 2<0:5+00 for instance,
l
2
2 -x
l 2
=(z -x ) -Rl(Z )t- /
e
(0)N
(2,t)+0
(Rl(z )t- /
)
n
n
n n
n
l
n
p
n
Thus
l
2
l
(2
-x )2=(z -x )2_2(z -x )Rl(Z )t- / e (0)N (2,t)+0 ((z -x )Rl(z )t- / 2 )).
n n
n n
il
n
n
l
n
p
n
n
n
l
2
The term 0
((z
- x )Rl(z )t- /
))
is justified by the fact that
p
n
n
n
Rl(z )/(z -x )~O
(See Lemma 4.2).
Hence,
using Theorem 5.2,
we arrive at
n
n
n
( -
- ) 2
z
-x
(z
-x )
n
n
n
n
A
(l,k,t)
1:" U)
}
n
z -x
n
n
-1 {
2
=K (0)
- 2 R
(z
)
el(o)N (2,t)-((z -x )/Rl(Z ))
N (4,t,1)
il
n
n
n
n
) +0
(
P
which ln turn implies
42

---

-107-
" ,
J
' \\
J
' \\
(z -x ) -2 R2 (Z )t1 / 2 (T (7) -l_ C (7))=-N (4,t,l)+0 (1).
n n
n
n
n
n
p
Using again
(5.1)
for 0<1<2
(that is p
(i,l/2)~+oo,
i=2,3)
gives the
n
result for the last case. The proof is now complete.
Finally,
we give the multivariate version of all that precedes.
VII - Multivariate asymptotic normality of the characterizing vectors.
We neglect T
(4)
since its asymptotic law is extremal. Remark that by
n
Corollary 3.3,
( 7 . 1)
v
( 5) (T
(2, t, 1) - c
(5)) =N
( l, t, 1) +0
( 1) ,
n
n
n
n
p
1 2
where v
(5)
= t /
/R
(Z
),
c
(5)
= <<(t) ,
F=(f,b)Er,
(k,t)
satisfies
n
1
n
n
(2.4a)
and the regularlty conditions
(with respect to t)
holk and
(t,t/k)~(+oo, 0)
Now set
v
(1) (T
(l,k, t)
c
(1))
n
n
n
v
(0) (A
(l,k,t)
c
( 0) )
n
n
n
v
(2) (T
(2,k, t)
c
(2))
n
n
n
v
v
(3) (n T
(3,k,t,v)
c
(3) )
n
n
n
v
(5) (T
(5)
c
( 5) )
n
n
n
v
(6) (T
(6)-1
c
( 6) )
n
n
n
v
(7) (T
(7)-1
c
( 7) )
n
n
n
-v
-1
v
(S) (n
T
(S,v)
c
(S)
n
n
n
*
-1
T
(1)2)
is defined by replacing v
(S,v)
-c
(S))
by v
(9) (T
(9) -c
(9)).
n
n
n
n
n
n
*
*
T
(1<2)
is defined from T
(1)2)
by replacing v
(3),
v
(6)
and v
(7)
by
n
n
n
n
n
*
*
*
v
(3),
v
(6)
and v
(7)
respectively.
n
n
n
43

---

,
,
', , -
)
'~
)
'~
- 108-),~
We obtain
Theorem 7.1.
Let =(f,b)el and
(k,t)
satisfy
(2.4).
We assume that
the
regularity conditions hold.
a)
If FeD(~) U D(A), then T* (00)+' (00)+0 (1),
where
t,*
,*
L
n
Ln
p
(oo)=(L
(00),
n
n
,**
8
3
L
(00))
is an ~ -Gaussian vector slich that L* (00)
(an ~ -
r.v.)
and
n
n
,**
L
(00)
(an ~5_R.v.) are asymtotically independent with respective
n
limiting covariance matrices.
o
1
*
**
-1
and
(00 )
Ln (00)
5
Ln
o
2
\\
o
-2
2
5
o
o
o
with t
denoting the transpose of the matrix M.
M
b)
If FeD(W ),
0>2,
then T * (0)2)
o
n
,**
8
,*
Ln (0)2))
is an ~ -Gaussian vector such that Ln(0)2)
(an ~3 -r. v)
and
,**
5
Ln (0) 2) (an ~ - r . v . ) are asymptotically independent with respective
limiting covariance matrices
3
2
o + 0 + 20
4(0+ 1 ) (0+ 3 )(0+ 4 )
4
3
2
0+ 2
0 ( 0 - 5 )
50
+110
+40
+70+12
-1/2 (
) 1/2
0+ 1
(0+3) (0+ 4 )
2
o (0+ 3 ) (0+ 4 )
3
2
3
2
0+ 2
20
20
+40
+180+18
o +0+ 2
1/2
1/2 ( - - - )
- - - - - -
0+1
(0+ 1 )(0+ 2 )
2
2
o (0+ 3 )
o (0+ 2 )
44

---

..
1
..
1
J
'~
J
'~
-109-
and
'jy+1) l'jy
3
2
2
'jy +'jy +2
-
< 'jy+1)/'jy
- - 2 - - - - - -
'jy +<'jy+2)
**
3
2
3
2
Ln
'jy +'jy +2
'jy +'jy +2
('jy>2) =
2
<'jy +1) l'jy
2
2
'jy
<'jy +2 )
'jy
<'jy +2 )
3
2
3
2
4
3
2
2'jy +4'jy +18'jy+18
2'jy +4'jy +18'jy+18
S'jy + 1 1 'jy +4'jy +7'jy+12
2
2
'jy+U'jy
2
2
2
'jy
<'jy +3 )
'jy
<'jy +3 )
'jy
<'jy+3) <'jy+4)
2
- 2
-2
-
< 'jy+1) l'jy
'jy
- 2
-'jy
-2'jy
c)
If
FED(I/J ) l
'jy>2 ,
then T* ('jy<2) =\\
('jy<2) +0
(1)
where
'jy
n
Ln
p
*I
**
\\
*
t
('jy<2)=(L
'L
r
('jy<2)
('jy<2))
is an ~8-Gaussian vector such that
Ln
('jy<2)
n
n
Il
**
(an ~4_r.v.) and L
4
(an ~ -r.v.)
are asymptotically independent
n
*
with respective covariance matrices
L ('jy<2)
n
3
2
'jy +'jy +2
4<'jy+ 1 ) <'jy+ 3)('jy+ 4)
'jy<'jy-S)
4
3
2
'jy+2
112
S'jy +11'jy +4'jy +7'jy+12
-112 ( - - - - - )
'jy+1
<'jy+3) <'jy+4)
2
'jy
<'jy+3) <'jy+4)
3
2
3
2
'jy+2
2'jy +4'jy +18'jy+18
'jy +'jy +2
-112 (
) 112
'jy+1
<'jy+3) <'jy+4)
2
'jy
2
<'jy+3)
'jy
<'jy+2)
2
2'jy
2
2
2
+12'jy+12
2'jy -4'jy -12
'jy -2'jy-4
1/2
2 S'jy +8
-112 <'jy +2 ) <'jy +1) )
-<'jy+1)
-<'jy+1) - - - - -
<'jy+1)
_
<'jy+2) <'jy+3)
'jy<'jy+3)
'jy<'jy+2)
'jy +2
45

---

: '
..
1
1
}
''\\
-110-
**
and In (1<2)
3
2
1 +1 +2
- 2 - - - - - -
1
<1+ 2)
3
2
3
2
1 +1 +2
1 +:1 +2
2
2
1
<1+2)
1
<!1+ 2)
3
2
3
2
4
3
21 +41 +181+18
21 +41 +181+18
51 +111 +41+71+12
2
2
2
1
<1+3)
1
<1+1)
'1
<1+3) <1+4)
-2
1
- 2
-2
-2
-1
-21
1
PROOF.
Putting together Corollary 3.2,
Theorem 4.1,
Corollary 5.1,
Theorems
6.3,
6.4 and 6.5 and Formula 7.1,
one gets
*
L
1+2
(00)
O.5( '>'+1)1/2
(2N (O,k,t)-;\\T
(3,k,t)+N
(2,k)),N
(3,k,t)
o
n
n
n
n
n
+e
(1)N
(2,k),N (O,k,t)+e
(1)N
(2,k))
3
n
n
1
n
**
L (00)=(e (1)N (O,t,1)+e (1)N (2,t),-N (O,t,1)-e (1)N (2,t),
1
n
1
n
n
1
n
n
-N
((3,t,1)-e
(1)N
(2,t),-N
(2,t)),
n
3
n
n
**
*
*
with 1=+00 iL (1)2) = L (00) with 2<1<+00 and L
(1)2 )
lS
obtained from
n
n
n
**
L (00) by replacing the last line by N (2,t)/1
n
n
*
*
L (1<2) is obtained from L (00) by adding -(1+1)N (O,k,t) - N (2,k) as a
n
n
n
n
**
fourth line
i
finally,
one forms
(1<2)
by dropping the first
line
Ln
**
of L (1~2). Now simple computations show that if L (1) (resp. L (2))
n
n
n
**
**
*
**
is any coordinate of L (00)
(resp'L
(00)),
L (1)2)) (resp'L (1)2) or
n
n
n
n
46

---

, ' 1
1
-111-
, ' 1
' )
}
''\\
}
''\\
*
L
**
(r<2)
(resp'L
(r<2)),
one has
n
(7.2)
lE L
(l)L
(2)-p
(1,1/2)
or
(2K(r))-l p
(1,1/2)2
or
(e/k) 1/2
n
n
n
n
(See Computations that led to
(6.10)).
Renee,
by
(4.20)
(7.3)
lE L (l)L
(2)~ O.
n
n
This together with
(3.34),
(3.43),
(4.22), (4.27),
(6.10),
Lemmas 4.1
and 4.3 yield the covariance matrices by routine computations.
For T * (r = 2),
several possibilities can happen depending on how
n
b(t)
converges to zero as t
tends to zero. Anyway Formuas
(6.17)
and
(6.18)
include a11- the possibilities of limiting laws.
47

---

..
-112-
'
, ' 1
1
)
.~
.~
REFERENCES.
Balkema, A.A. and de Haan, L.
(1972). On R. von Mises'
condition for
-x
the domain of attraction of exp(-e
). Ann. Math. Statist. 43,
~352-1354. "
Csorgo, M.,Csorgo,S.,Horvàth,
L. and Mason, D.M.
(1986). Weighted
~~pirical and quantile processes. Ann. Probab. 14, 31-85.
Csorgo, S. and Mason,
D.M. (1985). Central limit theorems for sums of
extremevalues. Math. Proc.
Cambridge Philos.
98,
547-558.
DE
Haan,
L.
(i970). On Regular Variation and: its Application to the
Weak Convergence of Sample Extreme. Mathematical Tracts,
3~, Amsterdam.
i
DE
Haan i,
L.
and Resnick,
S. 1.
(1980). A simp:le asymptotic estimate for
t~e index of a stable law. J. Roy. Stadist. Soc. Ser.B, 42, n 0 1,
1
1
83-87.
:
DeheuvJls,
P., Haeusler, E. and Mason,
M.
(]989). Laws of the iterated
l~garithm for sums of extreme values in the domain of attraction
_
'of a Gumbel law. Bull. Sci. Math., 'to appear.
(#aeusler, E. and Teugels, L.
(1985). One asymptotic normality of Hill's
?r
estimate for the exponent of regular variation. Ann. Statist.,
;
13,743-756.
L~all, P.
(1982). One sorne simple estimates of an exponent of regular
~
v~riation J. Roy. Statist. Soc Ser.B, 44,37-42.
_~all, P.
(1978). Representation and limit theorems for extreme value
- dtstributions J. Appl.
Probab.,
15,
639-644.
'Lô G.sl
(1986a). Asymptotic behavior of Hil]'s estimate.J.Appl.
Probab.
, 23,
922-936.
Lô, G.S.
(11986b). The weak limiting behavior the Haan-Resnick estimate
of the exponent of a stable distribution. Technical report,
49,LSTA, Université Paris VI.
Lô, G.S.
(1989). A note on the asymptotic normality of sums of extreme
values. J. Statist. Pann.
Inf.,
22,127-136.
Lô, G.S.
(1990). Empirical characterization of the extremes 1: A
family of characterizing statistics. Technical report,
LSTA,
Paris
VI.
Leadbetter,
M.R.
and
Rootzén,
H.
(1988)
Extremal
theory
stochastic
processes. Ann.
Probab.,
16, 431-478.
Resnick,
S.I. (1987).
Extreme
values~
Regular
variation
and
Point
Processes. Springer, New York.
Smith,
R.L.
(1987).
Estimating
tail
probability
distributions.
Ann.
Statist.,
15,1174-1027.
48

---

"
,
,
,
, ' )
1
) '4,
-113-
EMPIRICAL CHARACTERIZATION OF THE EXTREMES III
:
LAWS OF THE ITERATED LOGARITHM FOR THE CHARACTERIZING STATISTICS.
Gane Samb LO
Université Saint-Louis & Université Paris VI
(LSTA).
Abstract. Let Xl'
X ' · · .be a sequence of identical and independent random
2
variables
wi th
P (Xl :Sx) =F (x) ,
l:Sx< +00,
F (1) =0
and
let
Xl
:s ... :sX
be
,n
n,n
the order statistics based on the n first of these random variables.
A
familyof statistics including Hill's one
k - l"i = k
, (1
l
)
T =
L'
n
1 l
ogX
, 1
- ogX
,
n
l=<-+
n-l+
n
n-l,n
and this estimator of the index of a Pareto law,
-1 i==k
'=i
'
:
"' -1/2
k
S,
\\~
] a (logX,
-logX,
\\) (logX,
- l o g X , ) r
{,
Ll=t+1L]=t+1
ij
n-]+l,n
n-],n
n-l+1,n
n-l,I1)
,
(where l:st<k<n are integers
such that
2
t /k~O, k/n~O as n~+CXJ and
2(1-a .. )
is the Kronecker symbol)
which characterizes the limit laws of
1 J
the sample extreme X
, was
recently introduced in
[13].
The
n,n
multivariate asymptotic normality of these statistics was proved in
[14].
We characterize here
the distribution functions
F within the
domain of attraction for which the laws of the iterated logarithm
corresponding to the central limit theorems in
[14]
hold.
Key words and phrases. Extreme value theory,
domain of attraction,
order statistic,
Brownian bridge,
law of the iterated logarithm.
Mailing and
permanent address.
UER Maths & Info,
BP 234,
Université
Saint-Louis,
Sénégal.
Research affiliation:LSTA, URA CNRS 1321, Université Paris VI.
T.44/55,
4, Place Jussieu,
75230 Cédex 05 France.
1

---

1
,,,
:
1
"
,
1
4,
-114-
1
1 I- INTRODUCTION.
Let X,
X, ... be a
sequence of independent random variables
(r.v.'s)
1
2
1 with common distribution function (d.f.) F such that F(l)=O and for
each n~ l,
Xl
~ ... x
denote the order statistics based on the n first
1
,n
n,n
of these random variables.
Tt is shown in Lô
(1990)
that
the following
1 statistics
-1/2
T
(l, k, f.) =T
(2, k, e) A (l, k, e)
i
n
n
n
1
-1
i=k
j=i
,
A
(l, k, f.) = k
L. n 1L' n 1 J (1 - cS " /2) (y
.
1
- Y
.
) (y
.
1
- Y
.
)
n
1= L+
J = L+
l J
n -1 + ,n
n -l, n
n - J + ,n
n - J , n
-1 i=k
,
T
(2, k, f.) =k
L' n 1 1 (y
.
1
- Y .
)
n
l=L+
n-l+
n
n-l,n
1
-v
T D,k,f.,v)
= n
(Y
n
-y
k
)/T
(2,k,f.)
n
!
n-L,n
n-
,n
n
1
-1
1
T (4)
Y
i
T
(5)
=
T
(2,f.,l)
i
T
(6)
(Y
n
-y
k
)
T
(2,f.,l)i
n
n,n
n
n
n
n-L, n
n-
,n
n
-2
_
-v
-1
T
(7)
(y
n
-y
k
)
A
(l, f., 1)
T
(8) =n . (y
n
-y
k
)
i
n
n-L, n
n-
,n
n
n
n-L, n
n-
,n
1
T
(9)
= (Yo-Yn-f. n)/(Yo-Y
n
n - k n)
,
,
(only defined when y
= log(sup{x,F(x)<l})<+oo),
o
1
where
YI
~ . . . ~Y
are
the
order
statistics
of
Y =logX , ... ,
,n
n,n
1
1
Y =logX ,
k
and f. are
integers
such that
l~<n, v
is any real and
n
n
positive number,
(5..
is Kronecker' s
symbol,
characterize the asymptotic
1)
limiting laws of sample extreme X
. That is,
i t is possible to answer
n,n
from the values of the limits of these statistics these two questions
Does F ly in the domain of attraction of sorne non degenerate d.f.
or
not?
l
F is attracted to a non degenerate d.f.,
what is this d.f.
The marginal and joint asymptotic
normality of these
statistics
(except
T
(4)
which
is
extremal)
are
estabished
in
lô (1991) .
il
This familiy of statistics is quoted as a Familiy of Characterizing
Statistics for the Extremes
(FACSEXT).
In
statistical
terminology,
Part
l
of
this
work
treated
the
2

---

: ,
-115-
: '
J
''',
J
''',
asymptotic consistency while Part II focused on statistical tests based
on them.
Generally,
asymptotic normality
implies
laws of
the
iteratd
logarithm
which,
from
the
statistical
point
of
view,
determine
the
strong
accuracy of
confidence
intervals
deri ved
from
the
asymptotic
consistency.
This
motivated us
to determine
the
strong
laws
of
the
iterated
logarithm
related
to
the
central
limit
theorems
given
in
the
cited
paper above. Our best achievement is the characterization of the d.f.'s
within the domain. of attraction,
for which such laws hold.
Our results are closely related to results by Haeusler and Masan
(1987a-b),
Deheuvels-Haeusler-Mason
(1989),
Kuelbs-Le doux(1987)
etc ..
The
reader. is
referred ta Leadbetter and R60tzen
(1988)
and
Resnick
(1987)
for\\detailed references on extreme value \\theOry and ta
Lô
(1990,1991)
as a general introduction to this paper.
We only recall
that F is attracted ta sorne non degenerate d.f.M
(written FED(M))
iff
there
exists
two
non-random
sequences
(a >0)
land
(b)
1
such
that
n
n~
n n~
for any real x,
F(a x+b ) ~ M(x)
as n~ or equivalently (X
-b )/a
n
n
n,n
n
n
converges in distribution
(Q)
ta M.
It is well-known that M is
necessarily the Gumbel
type of distribution
(1\\. (x) =exp (-e -x),
XEIR,
the
Fréchet type of d.f of parameter 0>0
(~ (x)=exp(-x-O), x~O) or the
°
Weibull typeofd.f. ofparametero>O
(!/J
(x)
exp
(-(-x)O),
x:sO).
°
We now classify the elements of the whole domain of attraction ï
in
our convenience. First define the quantile function of F
and the endpoint of F
X
(F)
= sup {x,F(x)<l}.
a
Through'out, G is associated with F by G(x)=F(eX)=p(y1:sx).
We have ï=D(I\\.)
U D(~) U D(!/J), D(~)=U > D(~ ),
D(!/J)
=U
D(!/J)
and
0-0
°
0>0
°
3

---

1
:
1
1
."
}
J
. "
-116-
(see Theorem A and Formulas
(1.4-6)
in Lô
(1991))
1
1
1
(l.la)
FED(A)
~ GED(A) and G- (1-u) = d-s(u)+J
s(t)t- 1dt,
O<u<l
u
1
Where c is a real constant and
(l.lb)
s(u)= c(l+f(u))
exp
(Ji b(t)t-1 ),
O<u<l
u
1
Where c
is a real constant,
max
(If(u) l, Ib(u) I)~O as u~O;
(1.2)
FED(~ ) ~>O, iff GED(A) and,
1
~
G-l(l-u)
= -(logu)/~+logc+log(l+f(u))+ fl b(t)t dt-l,
O<u<l,
u
1
where c,
f,
b are as in (l.lb);
( 1. 3)
FED (r/J ),
~ >0,
if f y = x (G) <+00 and
~
0
0
1
Y -G-1 (1-U)
= c(l+f(u) ul/~exp(J~ b(t)-l dt ), O<u<l,
o
where c,
f and b(t)
are as in
(l.lb).
1
We see that each d.f.in r
is associated with a couple of
1
\\
functions
(f,b)
which defines a class of d.f.'s only distinguishable
1
by constants. A d.f. belonging to the class defined by (f,
b)
will be
1
denoted F=(f,b).
To finish,
we introduce this notation.
R (X,2,G)=
(l-G(x) )-lJ2 l-G(t)
dt,
X~2~y .
l
x
0
-1 2 2
2
R (X,2,G)=
(l-G(x))
f J
. . . . J
l-G(t)
dt dYl"
.dYp_l'
p
x Yl
Yp - l
for X<2~Y, p~2
o
p~l;
G-l(l-t/n)
;
If the d. f.
is not
specified ln R (.),
it
is assumed that R (0)=
p
P
We shall need these assumptions on
(k,t).
4

---

1
-117-
..
1
1
, 1
1 )
' ' \\
J ''',
(K2)
lim inf 10gk /log n > 2.
n
2
1
n~+oo
(K3)
k(n)-k' (n)1'+co, i(n)-i' (n)1' '
i(n)/n-(3 -JrO,
k(n)/n-a -Jr,0 as n +00.
n
n
1 (K4) 3 P >0, V O<p<p , lim sup
(k(n(r+1))/k(n(r))
< l+p,
o
0
r~+oo
for n(r)=[(l+p)r],
where
[0]
denotes the integer part,
r
is integer.
1
-2 1-v
(L1 ) 1:si<k,3v>0,i
k
10g2n~0 as n~+oo.
1
(L2)
lim sup
e/log n<+00 or lim
i/log n=+00.
n~
2
n~+oo
2
1
Here 1092 stands for the iterated logarithm and an-Jr,
for instance,
means that a
is nonincreasing as n increases.
It may be easily
n
1
derived from
(K4-3)
that
,
(K5)
lim
lim
inf
(resp.
sup)
a n (r+1)/an (r) (resp. (3n(r+1)/(3n(r) )=1
p~O
r~+co
1
Because of the relatively high number of statistics to be
studied,
we shall divide Section II into parts A,
B,
etc ...
In each of
them,
1092-laws related to one element of the FACSEXT are stated and
proved.
We finally make this convention.
If a
function of 0 is related ta
FEr,
the
specification
0<0<00 means that
FED(~ )
while 0=+00 means that
o
FED( A)UD( cp).
Unless the contrary lS spectified,
the second case is obtained from the
first by letting
O~+oo.
II
- Results and proofs.
A L_og -laws for Tn~(_2~,_k~,_e~).
2
5

---

1
:
1
"
1
1
-118-
J \\
J
' \\
Put «(k,t)= R JYo
1-G(t)
dt.
Our mains results for T (2,k,t)
are
1
n
y n-k,n
a/ the results.
1 Theorem lA. Let F=(f,b)El. Let (Kl-2-3-4) and (Ll-2) be satisfied.
Then,
1
k
a.s
lim sup
( - - ' - - - 2 - - - ) 1/2
(T
(2, k, t) -« (k, t) )
n~
n
2R
) 1092 n
1 (Xn
1
and
k
a.s
lim inf
(
)1/2
1
(T
(2,k,t) -« (K,t)
n~+oo
2
n
n
2R
(X
) 10g2
1
n
2
where Π(0)=2(0+1)/(0+2),
O<o~+oo.
1
o
Remark lA : This 1092-law is general withno condition on f
neither on
1 h. But a random noise k persists. It is natural, from a probabilistic
point of view to ~ry to replace it with a non random syquence.
1 This gives theorems below. We need
Definition 1A.A
sequence
of
random
variables
f,
,
n=l, 2, . "
satisfies
1
n
the
law of
the iterated logarithm LIL(a,b)
iff
a.s
1
l ,
c
/(21
)1/2 a.s b
lm sup
Sn
og2 n
, and lim inf
f, /(21092n)1/2
a,
n~+oo
n~+oo n
1
2
3
2
Now
put
~1 (0)=(0 +0 +2)/(0(0+ 2 )) ,O<o~+oo,
o (.) =f (U
1
) - f ( . In) ,
n
. + ,n
1/2
N (-oo)=k
0
(k),
where U.
denotes the j-th minimum among the n first
n
n
J, n
values of a sequence of independent uniform r.v.'s on
(0,1).
T* (2,k,t)=R
(x )-l k -1/2(T (2,k,t) -«(k))
n
1
n n
We obtain thhe following characterizations.
Theorem 2A.
Let F=(f,b)El.
Let
(Kl-2-3)
and
(Ll-2)be satisfied. Then,
*
a)T
(2,k,t)
satisfies
the
LIL(-Œ
(0),
~1(0))
iff
N (-00)
satisfies
the
n
1
n
LIL (0, 0) .
*
b)
T (2,k,t)
satisfies a LIL(.,+oo)
(resp.
LIL(-oo,.))
iff N (-00)
n
n
satisfies a LIL ( . , +00)
(resp. LIL
(-00,.)).
c)
T * (2,k,t)
satisfies
some
LIL(a,b)
with
-oo<a~b<+oo iff
N (-00)
n
n
6

---

",
-119-
,
1
J
'4,
satisfies sorne LIL(a' ,b')
with -oo<a'~b'<+oo.
It is very easy to derive useful corollaries from Theorem 2A
applicable for instance to all the models studied in Haeusler and
Teugels
(1985),
Hall(1982),
etc ...
Corrollary lA.
Suppose that the assumptions of Theorem 2 A hold.
1)
If,
in addition,
f admits a derivative f'
such that uf' (u)
has a
limit as u~o, then T * (2,k,f)
satisfies the LIL(-Œ
(0) ,Π(0)),
O<o~+oo.
n
1
1
2)
If for all sequence u -(k/n)
as n~, one has
n
1/2
1/2
k
f(un)/(21092n)
~O as n~+oo,
\\
i
Remark 2A.
The full-form of the characterization in Theorem 2A is
*
T (2,k,f)=N
(l,k,f)+e
(0)N
(-00) (1+0) (1) )+0(1),
a.s,
n
n
2
n
where
N (l,k,f)
satisfies
the
LIL(-Œ
(0),Œ
(0)),
e
(0)=0+l,
n
1
1
2
0<0<+00
and e
(00)=1.
2
This
sharp resul t
includes all
the LIL' s
of
subsequences
of
T* (2, k, f)
n
Cf.
for instance Remark 4.1 in Lô
(1991).
b/ The proofs.
They constitute
simple
cases
of
results
for A (l, k,f)
n
which are more complicatated and require double integration. Therefore,
for
sake
of
shortness,
we
reserve our efforts
for
the
next part
and
omit these proofs. We only show how Corollary lA may be derived from
Theorem 2A.
1)
It may be seen that if uf' (u)
has a limi t,
this limit lS necessarily
zero.
Further
-1/2
1/2
N (-00)/(210g n)-
=
(1+0(1))
n
2
(unf' (un))
(nU ,n- k )/(2klog n)
,
k
2
2
where by Theorem 6 of Kiefer
(1972),
k - 1 / 2 (nUk , n - k) (21 og2 n) 1 /
sat i s fie s
7

---

1
".
,
,
1
-120- ,";
,
'
J
' \\
J , \\
the LIL(-l,l)
whenever k/log2n~+00 and min(U
1
,k/n)~u ~max(Uk 1
,k/n)
k + ,n
n
+ ,n
1
and
nu /k~l,
a.s.
This
proves
the
n
1 first part of Corollary lA.
2)
The second part is obvious since U
-(k/n),
a.s.,
as n~+oo and
k ,n
1
INn(-00)/(210g2n)1/21~k1/2If(k/n)/(210g2n)1/2+k1/2f(Uk,n)/(21og2
n ) 1/2 1 .
1
1
-
n
Put L(k,l)=k JYo
y
JYo
Y
1-G(t)
dt dy.
The results obtained for A
(l,k,e)
n
1
n-k,n
are of the type of those related to T (2, k, e) •
n
1
Theorem lB.
Let F=(f,b)El.
Let
(Kl-2-3)
and
(Ll-2)
be satisfied.
Then
\\R {Xn:)-lk 1/2(A (1,k,e)-L(k,e))
satisf:j-es the LIL(-<T
(0),<T
(0)),
2
n
2
2
1
with <T; (0) =6 (0+1) (0+2) / (0+3) (0+4),
O<o~+oo.
1
*
-1 1/2
Let A (1,k,e)=R
(X)
k
(A (l,k,e)-L(k)). We have the following
n
2
n
n
1
characterizations:
1
Theorem 2B.Assume that the assumptions of Theorem lB hold.
Then,
a)
*
A (l,k,e)
satisfies the LIL
(-<T
(0) ,<T
(0))
iff Nn(-oo)
satisfies the
n
3
1
3
LIL (0,0) ,
2
4 3 2
where <T (0)=(So +11 0 +4 0 +70+12)/(0(0+3) (0+4)),
O<o~+oo;
3
b)
*
A (l,k,e)
satisfies a LIL(o,+oo)
(resp. LIL(-oo,o)
iff N (-00)
n
n
satisfies a LIL(o,+oo)
(resp. LIL
(-00,.)).
c)
A* (l,k,e)
satisfies a
LIL
(a,b)
with -oo<a~b<+oo iff N (-00)
n
n
satisfies a LIL(a' lb')
with -oo<a'~b'<+oo.
Remark lB.
Reals a,a' ,b and b'
in Part c of Theorem 2B may be known if
f(')
is sufficiently regular
(See Example 4.1 in Lô(1991)).
Corollary lB. Under the assumptions of each point of Corollary lA,
8

---

,
,
-121- ,";
)
''',
*
An(l,k,t)
satisfies the LIL
(-0'3('1),
0'3('1)),
0<'1::S+00.
W,e note that Corollary lB is derived from Theorem 2B exactly as
Corollary lA from Theorem 2A.
b/ Proofs.
We first have to prove a general LIL we shall use in all the sequel,
Lemma lB . Let {Zn(r) ,n(r)=[(l+p)r],
r=1,2, .. ,},
O<p<l,
be a sequence
of normal r.v.'s with means zero and such that
2
2
~ Zn(r)=O'n(r) * 0,
ii)
3C >0,
3T,
0<T<1,3j (l)~l,Vj~j (1) ,Vi~l,
o
~(Z (' ,)Z (')/0' (' ')0' (')) ::s C Ti,
n J+l
n J
n J+l
n J
0
then lim sup
(resp inf.)
Zn(r)/(20'n(r)10g2n(r) )1/2=1
(resp.
-1),
a.s.
r~+oo
PFOOF OF LEMMA lB.
It suffices to show\\ the "lim sup" part since Zn(r)
lS
a symmetrical r.v
. Recall this well-known expansion of the tail of the
d.f.,
say ~,
of a stantard normal r.v.,
2
2
-x /2
-x /2
e
e
l)Vo,
0<0<1,
3x (0»0,
Vx~x
::s l-~(x)
::s
o
0,
(1-0)x(2rr)1/2
x(2rr)1/2
Let A, (c)
= { Zn(j)~(2()~(j) (l+C)10g2 n (r))1/2 L c>O. We obviously have
J
-1/2
-l-c
,-l-c
(B2)
3j (2)~1, Lj~j(2)P(Aj(c))::s(2rr)
(log(l+p))
Lj~j(2) J
<+00.
Applying the Borel-Cantelli lemma and letting c~O, we get
(B3)
lim
r~+oo
The other inequality is always the most difficult to get in 1092-laws.
Here again,
i t requires long and heavy computations.
Keeping the same
notations and using always
(Bl),
one has
(B4)
Vc,
0<c<1/2,
\\, 1rP(A, (-c) )=+00 i
\\ .
lrP(A, (-c) )2<+00.
LJ ~
J
LJ ~
J
Using the exact expression of bivariate Normal vectors,
one has
9

---

-122-
..
1
1
: '
)
''\\
l
+00
(B5)
P(A,
,(-c)
n A, (-c))= --2rr--(1---Œ~2~(-,--,-)
J +1
J
11/2JI211-el1092n(j) )1/2
n J,l
2
2
2
- (u +v
- 2 UVŒ
('
')) / 2 (1 - Π('
'))
e
n J , l n J,l
du dv,
where Π('
')=[(Z
('
')Z
(,))/Œ
('
,)Œ
( , ) ) ,
Now using
n J , l n J +1
n J
n J +1
n J
2
2
2
(B6)
exp
(-(u +v -2UVŒ
('
,))/2(1-Œ
('
'))
n J,l
n J,l
2
1-Œ
('
')
n
J,l
1-Œ
('
')
n
Œ
( '
' ) (u-v)
J,l
2
n J,l
2
=exp (-
u /2)
exp(-
2
v
/2)
exp(-
2
2
1-Œ
('
')
n J,l
1-Œ
('
')
n J,l
2(1-Œ
('
'))
n J,l
\\
2
2
~exp ( - u /2 (1 +Π('
'))
exp ( - v /2 (1 +Œ
('
,))),
n J,l
n J,l
and using the change of variables
U'=U/(l+Œ
('
,)1/2
n J,l
V'=V/(l+Œ
('
,))1/2,
we get
n J,l
*
*
2
(B7)
P(A,
,(-c)
n A, (-c) )~f (' ,)IP(A, ,(-c) )IP(A, (-c)), where f (' ')
J+l
J
n J,l
J+l
J
n J,l
2
(l+Œ
('
, ) ) / ( l - Œ
('
,)
and
n J,l
n J,l
*
2
'
1/2
1/2
A,(C)={Z
(,)~(2Π(,)lo92n(J))
«(l-C)/(l+Œ
('
,))
},
0<C<1,j=l,2, . . .
J
n J
n J
n J , l
Define accordingly with Assumption
(ii)
of lemma lB,
i
-2}
(B8)
ih=min
{i~l,iE~, COL «2(1-c) lo92n(h))
Tt
is clear that
la

---

1
-123-
"
1
J
'~
J
'~
Noticing also that by Assumption
(ii)
of Lemma lB,
1
1 (BIO) 0 <
inf
f
('
')
sup
f
('
')
< +00
j2:j (1),
i2:l
n
J, 1
j 2: j (1),
i2:l
n
J, 1
and using
(BI)
and
(B9)
gives
1
2 (l-c)
(Bll)
'tIJ'
2:J' (1),
\\'~==~ \\' .. !P(A, . (-c) n A (-c))~0(10g2h) I~:~ j- l-T
o
LJ==J
Ll2:1h
J==l
j
o
o
1
==0 (log 2h) ,
1
(where f(t)==O(g(t))
as t~to means that lim sup t~t
If(t)/g(t) /<00) 1
o
1
whenever c
is chosen so that 0<C«1-T)/2
>0.
But
\\,j=h
\\,j==h
,-l+c
h
-l+c
c
(B12) L'
. !P (A. (-c) ) 2: Const. L' _. J
2:Const. J.
X
dX2: Const.
h
,
J==J o
\\ J
J-J o
Jo
i
1
for h large enough.
Hence,
{
j==h I
} {Ij=h
}2
-2c
(B13)
2 . .
. ,
lP (A . . (-c)n A. (-c))
/
'
. !P (A. (-c)
=0 (h
10g2h)~0,
I J=J
l:Slh
J+1
J
J=J
J
o
0
as h~+oo. Now,
using the Lemma of Chung-Erdos(1952)
j=h
Ij=h
2
Ij=h
!P(U . . A.(-c)2:{
. . lP(A.(-c))}
/{
.
, lP(A.(-c))
J==J
J
J==J
J
J=J
J
0 0 0
+\\' ,
" , h lP (A. (- c ) nA, , (- c) } , .
LJ
<J *J:S
J
J
o
and taking
(B4),
(B7)
and
(B13)
into account,
we get
(B14) :
j ==h
.
h
j =h
lP(
UA,(-C))2:l/{0(1)+(2\\':=. \\',
.
f
('
.)lP(A~ ,(-c)))/( \\'!P(A,(-C)));
.
, J
LJ==J L1>lh n
J,l
J+1
.L.
J
J==J
0
J=J
o
0
as h~+oo. Suppose that j
is large enough to realize
(BI),
we have for
o
I l

---

1 , ,
. ,
J
'~
J
~
1
-124-
1
2 ( - c ) log 2 n (h) 0'
( '
' )
1
n
J , l
X exp ( 2
) .
2 (1 +0'
( '
' ) )
n
J , l
1 But
1
1/2
i h /2
-1/2
ih/2}~0
(B16)
0<2 (l-c) 10g n(h)cr
('
. ):S{2C
(l-c) 10g n(h)l;
}{e
l;
,
2
n J , l
0
2
..
0
1
1
0'
( '
. )\\~O,
f
('
. )~l,aS h +00, uniform1y ln j\\o:sj:Sh, i~ih.Henee,
1
n ] , l
n
J , l
1
j=h
j=h
l=oo
(B17)
P( U A.(-C))~1{0(1)+(2(1-o)-1(1+0(1)) l
IIP(A.(-C))
j=j
J
j=j
i=i
J
1
o
0
h
j =h
1
X
IP(A . . (-c)))/( l
IPUA. (_c))2}.
J+l
j=j
J
o
It mat be quiek1y verified that
j=h
j=h
2
-2c
(B18) {\\"
\\" .
. II' (A. (- c))
II' (A.
. (- c) ) } / (
\\" II' (A. (- c))
= 0 (h
10g h) ~O ,
.L. Ll:Slh
J
J+l
.L.
J
2
J =J
J =J
as
h-700.
. -h
2
Expanding
(I~-· IP(A. (-c)))
and taking
(B4)
into aeeount,
one gets
J=J o
J
(B19)
j=h I
Ij=h
2
{2
. . . .
IP(A.(-c))IP(A . . (-C))}=(1+0(1))(
. . !P(A.(-c))
,
I J=J
l>lh
J
J+l
J=J
J
o
0
12

---

.. '
) '4.,
J
4.,
-125-
as h~+oo. Combining
(B17)
and
(B19) gives
U+oo
B ( 2 0)
IP (
,
, fA, (- c) ) ~1- 0 .
J =J
J
o
Since j
~j (1)
isonly required to be such that for all j~j
,
i~ih'
o
0
2
-1
.
1/2
one has
{2 (l-c) (1+<J"
('
'))
lo92n(J)}
>x
(0),
we see that
n J,l
0
(B21)
IP(U,oo'(l)A.(-C))
=1
J =J
J
and by doing the same for any j1~j (1),
we arrive at
(B22)
Vj. ~j (1), IP (U~
, A. (- c))
= l.
1
J =J 1 J
This completes the proof of Lemma lB.
We
shall
demonstrate
steps
denoted
(Sl),
(S2),
etc ... from
which
the final proofs of the theorems will be derived.Using
(1.4)
and setting
(B24)
U (l-G(t))
dt dy,
n
where U (s)
=#{i,
U.~s}, O~s~l, is the uniform empirical function based
n
1
1
U
U , X =G-
(1-U
)
-
-1
on
1"'"
n
n
k,n'
z =G
(l-U n
) .
Recall
n
L,n
Lemma 2B.
(K6mlos,
Major, Tusnady,
1975). There exists a probability
space carrying a Brownian bridge B and a sequence of independent r.v 's
uniformly distributed on
(0,1)
denoted U , U , . . .
such that
1
2
1/2
-1/2
sup o~s~lln
(Un(S)-S)-B(S) I=O(n
logn) ,
a.s.,
as n~+oo.
"
PRüüF.See
Csorgo-Révèsz
(1981),
pp.
133-135.
Removing the
subscript
"n"
in B
is possible since the sequence {B ,
n~l} in their construction is
n
n
13

---

: ,
";
, -126-
J
4,
J
'4,
arbitrary and of course,
it might be choosen so that B = B,
n~l.
n
Trough'out our proofs, we assume that we are on this probabi1ty
space. The derived strong 1aws do ho1d on every probabi1ity space.
First,
we prove
(81)
1/
Z
JZ
1/2
*
-)
2J
1
-_ { (n
n
n B (1- G (t)
dt d
}/ (R
(
) (21
)
)
(1)
Z
(1 )
k
Y
2 x n
og2 n
+0
=: n +0
,
X
Y
n
as n~.
PROOF OF (81). We have
*
*
(B26)
A (l,k,t) =Z +Z
(1) +Z
(2) +Z
(3),
n
n
n
n
n
where
-
/Zn(l)
<_nk1/2IJZn zn U (1 G(t))
dt d
I/(R (
) (21
)1/2)
1
n
-
y
2 x n
og2 n
,
Z
y
n
Let t be fixed.
By Theorem 2 ln Kiefer
(1972)
and Barndorff-Nie1sen
(1961)
respective1y,
14

---

) '4,
, )
4,
-127-
1 and lim
n(logn)l+8 Un
=+00,
a.s.,
n~+oo
<-, n
for any 8>0.
It follows from
(K2)
that exists a sequence of integers
(p)
l ' satisfying
n n~
1
Setting t
(.) =G
(1-. In)
and combining
the
two
last
facts
give
with
n
probability one
(w.po)
for large values of n,
n/~ (~)
e
e+1
(B28)
\\Z
(1) 1<
sup(U ( - ,
n
) (t((logn)-';)-t(2log n))2
n
~/2
n
n
2 (2klog n)
2
~(2kp -2log2n) -1/2(t ((logn)-';)-t (2log n) )2/(2R (x )),
n
n
n
2
2
n
Where ';>1. At this step, we recall lemmas proved in Lô
(1991)
concerning
the properties of G(·)
and R (·,G)
when FEr.
(Cf.
its Lemmas 3.2-3-4
and Lemmas 4.1-2).
-1
Lemma 3B. Let Fer. Then lim
O(l-G(G
(l-u)) )/u=l.
Furthermore,
u~
-1
a)
If FED(W ),
y -G
(l-u)
is Regularly Varying at Zero with exponent
o
0
-1
-1
'
-1
-1
1/ 0
o
(0
- RVZ )
l . e . ,
lim
0 (y -G
(l-Àu)/(y -G
(l-u))=À
, for all 0>0.
u~
0
0
1
b)
If FED (<p ) UD (A) , G-
(1-U)
is Slowly Varying at Zero
(SVZ),
that is,
O-RVZ.
Lemma 4B. Let FEr. The following limits hold as x~yo' X~XO'
(l-G(x))~O.
15

---

1
-128-
'''1
, ' 1
,
.
J
. 4,
b)
V p~l,R
(x,z,G)/R
(x,G)~l.
1
P
P
Lemma SB.
1
j=p
1
a)
Let FED(W ) ,then y <+00 and R (x,G)-(y -G-
(1-u))P
n (~+j)-l, as x~y .
~
0
p
0
j =1
0
1
b)
Let FED(~) U D(A), then R (x,G)-R (X,G)-Rl(x,G)p, as x y .
p
P
0
Now,
let FED(W ) and let K(~) = (~+1/(~+2) ,o<~<+oo.
By Lemma SB,
1
~
y -t
(log2n)
Y - t
(( log n) - € )
1
o
n
o n '
-1
-2
-1/2
_ _ _ _ _ _ _ _ _ ) 2 .
B (29) 1Zn (1) ISK (~)
(2kP
log2 n )
( - - - - - - -
n
y
-,t
(k)
y
-t
(k)
o
n
o
n
1
Since
(k/log n,
k/(logn)-€)
~(+oo,+oo), the terms in brackets tend to zero
2
1
in view of Lemma 2B.
Let FED(~) U D(A), ~ne has by Lemma SB,
for n
large enou~h,
1
t
(2102n) -t
(k)
n
n
2
1
k 1 / 2 R
(t
(k)))'
1
n
Lemma 6B.
Let FED(A)UD(~). Let 0>0,
(a)
Oand
(b)
0 such that a >0,
n n~
n n~
n
b
>0
for aIl n~ and a /n~O, b /n~O, b /a ~+oo, b ~+oo,
and such that
n
n
n
n
n
n
O-C
C
*)
3c , V o<C<C
, b
a
~+oo and
o
0
n
n
**)
3c
VO<c<c,
b-o(b /a )c(log(b /a ))
~ 0 as n~+oo.
0'
0
n
n
n
n
n
Then
-0
b
(t
(a )-t
(b ))~O, as n~+oo.
n
n
n
n
n
Proof of Lemma 6B.
FED(A)U D(~)
implies that GED(A).
Hence G admits the
de
Haan-Deheuvels-Haeusler-Mason
(1990)
representation
(l.la)
and
by
-1
Formula
(3.15)
in
Lô
(1991),
R (G
(l-u) )-s(u)
as
u~O.
Using
l
Karamata's representation
(l.lb)
yields,
as n~+oo,
O<c<o,
16

---

,
1
·
l
'J:",
-129-
t
(a
)
-
t
(b
)
s (a
ln)
n
n
n
n
n
-0
C
-0
b
(b la ) log(b la )+b
}
(B31)
0 < - - - - - - - - ~ 2{b- 0- - - - - +
n
n
n
n n
n
n
O
s(b ln)
b
R
(t
(b ))
n
n
1
n
n
and
s (a ln)
n
c
(B32)
b -o
~2/(bno-c a ).
n
s (b ln)
n
" n
(B31)
and
(B32)
conclude our proof of Lemma 6B.
Applying of :Lemma 6B to the second term of
expression in brackets
in
(B30)
is
obvious.
As
to
the
first
term,
i t
is
sufficient
to
impose
lim
inf
log/log2n>0
for
the
application
of
Lemma
6B.
Hence,
whenever
n~oo
(K2)
holds and FEr,
(b32 * )
IZ
(1) 1 ~ 0 as n~.
n
Also applyiJg Theorem 6 of Kiefer(1972),
(b33)
(nUk,n-k)/(2klOg2n)1/2
satisfies
the
LIL(-I,I)
if
lim
k/log n=+oo,
that is for all
À>I,
(k/Àn)~Uk
~(Àk/n) holds with
n~
2
,n
probability one as n is large enough.
Hence,
t
((logn) -ç) -t
(k)
t U ) -t
(2log n)
2
-2
-1/2
n
n
n
n
(B34)IZn(2)1~2K(o)(2Pnklog2n)
( R
(t
(k))
)(
tn(k)
) ,
1
n
a.s.,
w.p.o.
as n~. Lemma 3b
(for FED(~o)) and Lemma 6B
(for FED(A)UD(~))
together with
(K2)
imply
(B35 )
IZ
(2) 1 ~ 0,
a.s.,
as n~+oo.
n
Finally,
by Lemma 2B,
as n is large enough,
(B36)IZ
(3)I~K(o)-I(logn)(2klog n)-1/2((t (l)-t (k))2/
2
R1 (t
(k))2),a.s.
n
n
n
n
(K2)
and Lemma 6B,
together imply
(B37)
1 Zn (3) 1 ~ 0,
a. s.,
as n~+oo.
l
l+1
Remark that the inequality max
(U
(-),
) <U
a. s.
ln
(B34)
hold
n n
n
- p
n'
n'
17

---

..
1
..
1
130-
) '4,
also when lim sup
1/10g2n<+00 so that
(B32)
and
(B35)
are true under
n~+oo
this condition.
Now if lim
e/log n
= +00,
(B33)
implies for all 0>1,
n~+oo
2
(B38 )
(eji\\.n) ~Uo
~ (U/n
<-,n
holds w.p.o.
for large values of n.
Furthermore,
by Wellner(1978)
(B 39 ) U( 1 - G (z )) / (1- G (z ))
(l-G
(z
)/(l-G
(z ))~l,a.s iff
n
n
n
n
n
n
n
n(l-G(Zn) )/10g2n~+00
But Lemma 3B implies that n(l-G(Zn) )/10g2n-i/log2n.Hence,
replacing Pn'
(log n) -1;: and
(2log2n)
bye,
e/i\\.
and i\\.e respectively for an arbitrary
i\\.>1
and applying
(B3 8),
(B39),
and
(L1)
via Lemma 3B,
5B and 6B prove
(B32)
and
(B35)
whene/log2n~+00. Finally,
(B37)
holds whenever e/k~O.
In summary,
(B32*),
(b35)
and
(B37)
are proved under
(Kl-2)
and
(Ll-2).
This completes the proof of
(81).
(82)
Le\\t
p>O,
n =[(l+p)r],
r=1,2, . . . Define A(r)
= n
,k(r)=k(n ),
r~l,
n (r)
/ r z
z
r
r
Z
={ (
)1 2J n(r+1J n(r+1)B(1-G(t))d dt}/(R (x
) (210
n(r) )1/2)
n(r)
k(r)
y
2
n(r)
g2
x n (r)
y
r~l,
then
lim sup lim
*
sup
1 Zn
-
Zn (r) 1 = 0 , a . s .
p~O
r~+oo
n ( r) ~n~n ( r + 1 )
(We
shall
denote
lim
lim sup
sup
f
(r,p) =0
by
n
p~O
r~+oo
n(r)~n~n(r+1)
f
(r,p)=o
(1)
as r~+oo and p~O).
n
r,p
PROOF OF
(82) .We have
z
z
*
n
1/2J n (r+1) J n (r+1)
1/2
(B40)
Zn={k)
x
B(l-G(t))
dtdy}/R
(x ) (210g2 n )
)
2
n
n(r)
y
i=7
+
Iz (i),
i=4 n
Z
where Z
(4)=
{(~)1/2(x
-x )J n(r+1)B(1_G(t))}/(R
(x) (210g2 n )1/2)),
n
k
n(r)
n
2
n
x n (r)
18

---

: '
-131-
Xn(r)
Xn(r)
Zn(6)= {(~)1/2 Ix
I B(l-G(t)) dtdy}/(R (X ) (2l0g2n )1/2)
y
2
n
n
and
z
x
I
Z
(7)
{ (~) 1/2 I n (r+l)
n(r)
B(l-G(t))
dt dy}/R
(X ) (2l092n)1/2).
n
2
n
x
y
.
n (r)
Before studying each of these error terms,
we
point out that
(K3)
implies that
(B41a)
k(r+1)/k(r)
= 1+0
(1)
as r~+oo and p~O
r,p
1
and thus
(B41b)
n/k
(1+0
(1))
r,p
as r~+oo and p~O.
We shall need supplementary notation and lemmas.
Definition 1B.A
sequence
(a >0)
0
l S
balanced
by
an
other
sequence
n
n~
(b >0)
o (denoted a lib ) iff O<a~lim inf
(a /b )~lim sup
(a /b )
n
n~
n
n
n~
n
n
n~
n
n
~b<+oo.The latter inequalities are also quoted as
(a ,b )li(a,b).
n
n
Lemma 7B. Let FEI.
Let
(a ~1)
1 and
(b ~1)
1 and
(a ,b )~(O,O) as
n
n~
n
n~
n
n
n~+oo.
1
i) If
(a ub ),
then r(a ) =:R
(G-
(l-a ))
li r(b ).
n
n
n
1
n
n
ii)
If a /b
~O, a lia' and b li b', then for an arbitrary 0>0,
n
n
n
n
n
n
{aO r(a') }/{bO r(b') }~O as n~+oo
n
n
n
n
and
19

---

-132-,':
J
' ' \\
J
' ' \\
a
{b-~)o
1
1
(G-
(1-a' )-G-
(1-b'))/r(b')
~ 0 as n~+oo.
n
n
n
n
iii)
If
(a ,b )li(a,b)
and b ~a for aIl n. Then O<a~b~l and
n
n
n
n
when
FED (I/J ), (G -1 (1- a
) - G -1 (1- b ), r( b )) li ( (a +1) (1- b 1 / a) , (a +1) (1- a 1 / a) )
a
n
n
n
1
1
and
(r(a ) ,r(b ) )li(a /'O,b /'O)
n
n
when FED(A)
U D(~) ,
1
1
(G-
(1-a ) -G-
(b ),
r(b ) )li(-logb,
-log a)
n
n
n
and
(r(a),
r(b))
li
(1,1).
n
n
PROOFS OF LEMMA 7B.
AlI these parts are easy consequences of Lemmas
(3B)- (5B)
and
properties
of
regularly
varying
functions.
(B31)
and
(B32)
are pertinent examples of such demonstrations.
: We are now able handle the terms ~n (B40). By (K3),
for c>O,
\\
(B42)
(l-C)<Xn~~ «l+C)<X , (l-c)(3n~ Â ~(l+C)(3n' as n is large enough.
n
-1
-1
Set"g(c)=G
(l-(l+C)<X
(r))'
h(c)=G
(l-(l+C)(3n(r))'
n
Because of Lema 3B,
Formulas
(B41)
and
(B42),
(B43)
n(r)~n~n(r+1)'9 (1+0
(1)) (1-C)R
(g
1(-c))~R2(x)
r, p
2
r+
n
~ ( 1 +0
(1) ) (1 +c ) R (g
(c))
r,p
2
r
for r
large enough.
And applying Lemma 5B,
2
2
(B44)
R (gr+1 (-c) ) -Const.
RI (gr+1 (-c))
, R
(gr (c) ) -Const. RI (gr (c) )
2
2
as r~+oo. Now,
for n(r)~n~n(r+1), r
large enough,
h
1 (-c) -h
(c)
n(r)
1/2
r+
r
(B45)
\\2
(5) I~Const. (1+0
(1)) (-k())
(_--=----:-__--,,----,---,----_)2
n
r,p
r
R (gr+1(c))
1
1/2
x sup 0< < (1
) (3
lB (s) 1/ (210g 2n (r) )
,
-s-
+c
n(r)
where "Const."
"
depends only on rand p.
By Theorem 1.4.1 in Cs6rgo
and Révèsz(1981) , as r~,
1 B (s) 1
(B46)
sup
1/2 =1+0(1),
a.s.
O~s~(l+C)(3n(r)
(2 (1+c) (3n (r+1) log (1/ (l+C) (3n (r) ) )
From
(B41)
and
(K3-5),
(3n(r) =(3n(r+1) (1+0
and
r ,p (1))
20

---

..
1
1
, ) '>,
-133-
)
r:
(1+0
(1))
sa that
r,p
(B47)
sup
!B(S) l=o({(t(r)/n(r))10g2n(r))}1/2) (1+0
(1)))
O~s~(l-C)~n(r)
r,p
as
r~+oo and p~O, and thus
h
l(-c)-h
(c)
2
(B48 )
IZ
(5) I~Const. (1+0
(1)) {~i(~))} ( r+
(g
~-c)))' a.s., r~+oo.
R
n
r, p
n
r
1
r+1
Lemma 7B-(iii)
implies
from
(B48)
and ,(KS)
(B49)
sup n(r)~n~n(r+1) IZ
(5)
n
1 ~
0,
a.s.,
as
r~+oo.
Next,
using also the continuity modulus of B.via Theorem 1.4.1 of
Il
Csorgo-Révèsz
(1981),
for n(r)~n~n(r+1),
(BSO)
1 Z
( 6) 1 ~ Cons t . (1 +0
( 1) )
n
r,p
t(r)
1/2
.
2
X(ii1rf)
(h +
(-c) -gr(-c)) (h +
(-c) -hr(ct) /R
(gr+1 (-c))
,
r
1
r
1
1
1
which by Lemma 7B
( i i ) - ( i i i )
and
(KS)
implies
(BS1)SUp n(r)~n~n(r+1) 1 Z
( 6) 1 ~ a a. s.
as r~+oo.
n
Further,
using Theorem 1.4.1 of Csorgo-Révèsz
(1981),
for n(r)~n~n(r+1),
(BS2)
IZ
(7)I~Const.(1+0
(l)l{g
l(-c)-g
(C))/R
(g
1(-c))}2
n
r, p
r+
r
1
r+
R(g
(c))
2
g
(-c) -g
(-c))
~Const.(l+o
(1))
{ r
}
{r+1
r
}2.
r,p
R (g
(-c)
1
r+1
R
(gr(C)
1
By
(K3 -4) ,
(BS3)
O<p<po~ (l+p) -l~lim inf an(r+1)/an(r)~limsup an(r+1)/an(r)=T(p)~1.
r~+oo
r~+oo
Applyind Lemma 7B gives
(BS4)
or
((l+C
)1/1, ((l+p) (1+C))1/1)
1-c
1-c
and
(l+p) (l+c)
(BSS)
(lg +
(-c)-gr(c)l,
R
(gr+1(-C)))11(log((1+C)/(1-C)),10g
1-c
)
r
1
1
o
or
((1+1) (l-{(l-C)/(l+C)}l/
,
(1+1) (l-{(l-c)/((l+C) (l+P))}l/ o ,
21

---

..
1
t
-134-
,
1
"
1
1
J
' \\
(B56 )
sup
( )
( 1 ) 1 Z
(7) 1-70, a. S . as r-7+00.
n r
~n~n
r+
n
Finally,
for n(r)~n~n(r+l),
gr+l(-C)-gr(-c)
ITn(r) 1
(B57)
1 Zn (4) 1 ~Const. (1+0
(1) ) R
(
(-c) )
X ' - - - - - - - - - - -
r ,p
1 gr+l
RI (gr+l (-C))V2log n(r)
2
z
n(r)
1/2J n(r+l)
. . ,
where Tn(r)=(k(rf)
B(I-G(t)
dt.
Tn(r)lS a GaUSSlan r.v.
wlth
xn(r)
mean zero and whose variance is ŒT~(r)-2R(Xn(r),zn(r+l))-2K(o)R (X
1
n (r))2
as n-7+OO by Lemma 3.5 Formula
(3.33)
both in Là
(1991)
so that by
(B43)
1/2
(B58)
lim sup r-7+OO
IT
(r)I/{2R
(Xn(r))
n
1
(2K(o)
10g2n(r))
}~1.a.s. n-7+oo.
Hence,
using again
(B43),
\\
R
(gr(C))
gr+l(-c)-gr(C)
1
(B59) IZ
(4) I~ Const. (1+0
(1))
X
----=----,----,------,--
n
r,p
RI (gr+l (-c))
RI (gr+l (-c)
From this,
(B54)
and
(B55)
permit again to conclude that
(B60)
limp-70 lim r-7+oosup n(r)~n~n(r+l) IZn (4) 1 = O.
Returning back to
(B40)
and taking
(B41)
into account,
we see that i t
remains to prove
and
z
z
n(r)
1/2 {JxX(r+l) Jyn(r+l)B(l-G(t) dt dy}
(B62)
Zn(r)=(k(r))
n (r)
1/2
/R
a, a.s.n-7+oo.
2 (xn (r) ) (2l092 n (r) )
(B61)
follows easily from Lemma 7B-(iii)
and Formulas
(B42)
and
(B42)
and
(B54)
(B62)
will follows from the fact that Zn(r)
is a Gaussian r.v
22

---

.";
"
1
1
J
'4,
-135-'
J4,
with ~Zn(r)=O and,
by Lemma 4.3 in Lô(1991),
putting
1/2
Wn (r)=Zn(r)/(210g2 n (r))
,
so that by
(843)
(B64) 'dp,
O<p<l,
lim
r-7+OO
that is
(B65) 'dp,
O<p<l,
limpsup
IZn(r)I~(T2(o), a.s., O<o~+CXJ.
r-7+OO
This proves
(861)
and concludes the proof of
(82).
To finish,
we have to prove
1
where W (r)=2
(r)
(2l0g2n(r))1/2.According to Lemma lB,
i t suffices
n
n
to prove its point
(ii).
But by Lemma 3.4 of Lô
(1991)
2 2 2
(866) (Tn(r)= ~(n(r) )-(T2(o)
as r-7+CXJ ,
O<o~+oo.
Hence, we have only to show
(B67)
3C,
3T,
O<T<l,
3j(1)~1,'dj~j(1), 'di~l, I~(Z (' ,)2 (.))I~c Ti.
a
n J+l
n J
a
We have
n(j)n(j+i))1/2JZn(j+i)JZn(j+i+1)JZn(j+1)
(B68)
1 ~(Zn(j+i)Zn(j)) 1 {
(
~
k(j)k(j+i)
xn(j)
xn(j+i)
p
Z
n(j+i+1)h(S,t)
ds dt dp d}/
(
)
(
)}
J
q
R 2 x n (j) R 2 x n (j +i)
.
q
23

---

"
1
1
'"
, )
-136-
where h(s,t) = mlp {l-G(t),
1-G(s)}- (l-G(t)) (l-G(s)) .It should be noted
that h(., .)~O. Thus for c>O,
one has for j
large enough,
-2
-1/2
-1/2
-2
(B6 9)
I[[(Z (' ,)Z (,))I:SK(-3")
(l+c)a
('
,)a
(')
R1 (g· l(-c))
n J + l n J
n J +l
n J
J +
h,
1 ( - c)
h,
. 1 ( - c)
h,
1 ( - c)
h,
. 1 ( -.c )
-2. J+
J+l+
J+
J+l+
R1 (g'+i+1 (-c))
r
J
J
J
h(s,t)
ds dt dp dq.
J
Jg.(c)
g,(c)
p
q
J
J
Put m (x,z)=(l-G(x))
R (x,z,G). Routine calculations based on
p
.
p
*)
cutting the first integrand of
(B69)
into
[g, (c),
g,
l(c)]
and
J
J +
[g,
l(c) ,h,
l(-c)].
J +
J +
**)cutting the third integrand into
[p,
g,
.(-c)]
and
J +l
(g,
. ( c ) , h ,
l(-c)],
J +l
J +
\\
***)replacing h.
l(-c)
by h,
, l(-c)
and
J +
J +l+
****)Formulas
(4.28) -(4.30)
in LB (1991)
give
-2
{
-1/2
-2
(B70)
IlE(W
('
,)W
(,)I:SK('3")
(l-c)
a
('
,)a
( ' ) }
R1 (g"
_(-cl)
n J+l
n J
n J+l
n J
J+l+.L
-2
{
2
XR
(g,
1 ( - c) )
x
6m (g,
,( c) ,y ) + (g.
. (c) - g, (c))
m (g,
,( c) 1 Y )
1
J +
4
J +l
0
J +l
J
2
J +l
0
+(g.
, l(c)-g. (c)) (h,
, l(-c)-g,
. (c))m
(g,
,(c) ,y )+(g,
,(c)-g, (c))
J+l+
J
J+l+
J+l
2
J+l
0
J+l
J
2
x (h.
,
1 ( - c) - g,
,( c))
ml (g,
,( c) ,y )}.
J+l+
J+l
J+l
0
The necessary calculations for handing these terms are exactly similar
to those that proved
(82).
For sake of conciseness,
we remark only that
we
finally
get
by
Lemmas
SB
and
7B
for
any
«,
0««1/2,
for
j
large
enough,
24

---

:
1
"
1
1
-137-
J
' \\
J
' \\
-1
Finally,
by
(K4),
lim sup a
('
l)a
(.)=~ <1. This and (B71)
together
j~+
n J+
n J
0
prove
(B67)
which conclude the proof of
(83).
PROOF OF THEOREM lB.
(SI),
(S2),
and
(S3) yield
*
(B72) VO<p<p
, n(r)~n(r+1)~ Z
=z
()+o
(1),
a.s.,
as r~+oo and p~O,
o
n
n r
r,p
where n(r)
is the integer part of
(l+p)r for r=l,
2, ...
and
(B73) A~ (l,k,e) = Z~+O(l), a.s. n~+oo.
(S3),
(B71)
and
(B73)
together prove Theorem lB
PROOF OF THEOREM 2B.We have
Z
++
*
1/2
*
-1/2 JZn
(B14) An (l,k,e)=A (1,k,e)/(2l092n )
=ZIi+ nk
'{_
J n 1-G(t) dtdy
n
y
x n
z
J n 1-G(t) dt èy}/(R (x ) (2l0g n)1/2)+0(1) , a.s. n~+oo,
2
n
2
y
A
straighforward
computation
exactly
as
in
Lô
(1991)
but
based
on
nU
n/k e, a.s.,
(See its Lemma 3.6 and Formulas
(4.33)
-
(4.34))
yield
k ,
Z
Z
Z
z
1 2
n
(B75)
nk- /
{J_nJ n l-G(t)
dt dy -Jx
J n 1-G(t) dt dy}/R (X )
2
n
x
y
n
y
n
a.s.,
as n~+oo.
where,
as n~+oo,
1 2
1 2
(B76)ç
(k)=k /
(x -x )/R (x )=e (ô)nk- / (U
-~)+e2(2)k1/2ô (k)
n
n
n
1
n
1
k ,n n
n
(1+0(1) )+0(1) ,a.s.,
as n~+oo.
By Lemma 2B,
since U. (U
) =k/n,
r:
k , n
1 / 2
_
log n
k
1 / 2
(B77}
!B(Uk,n)+n
(Uk,n-k/n) 1= O(
1!2)0((n 1092 n )
),
a.s.,as n~+oo.
n
because of
(K2). Also,
by Theorem 6 of Kiefer
(1972)
(8ee B33)
and by
using the continuity modulus of B ( .)
(See Theorem 1.4.1.
in
Il
Cs6rgo-Révèsz
(1981)),
25

---

r-
, ,
:-138-
}
''\\
.. '
l
2
(B79)
Çn (k) =-e
(0) (R) 1/2 B (k/n) +e
(0)k /
0
(k) (1+0 (1) +0 ((l092n) 1/2) ,a.s,
I
l
2
n
as n~ with el (0)=(0+1)/0,0<0~+00,e2(0)=0+1for 0<0<+00 and e
(00)=1.
2
1
Now,
i t is not diffieult to see from the eontinuity modulus of B that
1
(80)
lim
lim sup
sup,
'l(f)1/2B(~)-(~i~~)B(~i~~)I/(210g2n(r))1/2
p----,;O
r----,;+oo
n(r)~n::s~(r+l)
i
==0,
a.s.
and sinee n(r)/k(r)
is u~timately deereasing in r beeause of (K4),
n(j)n(j+i)
(B81) [(({k(j)k(j+i)
X(l-k(j) /n(j)),
for large values of
j,
for aIl i~l.
\\
\\
Always,
beeause of
(K3),
there exists j (2)
and Tl'
O<Tl<O sueh thô.t
(B80)
and
(B81)
together with Lemma lB imply
(82)
lim
sup(resp.
inf)
(R)1/2B(g)/(2l092n)1/2=1
(resp.
-1),
a.s.
n----,;+oo
Putting together
(B72) ,
(B74)
(B75 ) and
(B82),
f
one has
A++
**
*
(B83 )
=z
+ R +0(1)
n
n
n
where
**
*
i l
1/2
k
1/2
(B83a)
2
Z
- e 3 ('0) (k)
B (il) / (21 og 2 n)
satisfies
i l
n
**
(B83b)
n(r)::sn~(r+l) ~ 2
-2' ()=o
(1),
a.s.,
as r----,;+oo and p----,;O,
n
n r
r,p
with
26

---

'"
.. ,
)
.",
-139-
k 1 / 2 0n (k)
1/2
( 2l092 n )
2
X(1+0(1))+0(1)}
+0(1),
a.s.,
as n~.
2
Z'
is a Gaussian r.v.
whose variance is -
~3 (0)
See Formulas
(4.36)
n(r)
and
(4.37)
in
Lô
(1991)),
satisfies
Point
(ii)
of
Lemma
lB.
Precisely,
(B71)
holds
for W (r) = Z~ (r) (2l092n (r) ) 1/2. The details of
n
this computation are omitted.
Hence,
by Lemma lB,
1
**
(B84)
lim
sup
(resp.
inf)
Zn
=~3 (0)
(resp. -~3 (0)) .
Put ting together
(B83),
(B83c)
and
(B84)
conc ludes the proof of
Theorem 2B.
The
proof
of
Theorem
lB
would
be
identical
but
less
complicated.
z
(B86)
Putting Q
={(n(r))1/2J n(r+1)
n(r)
k(r)
x n (r)
1/2
X(2log2 n (r))
).
where n(r)
is defined as below,
we have
*
lim
lim sup
sup
1Qn - Qn(r) 1=0,
a.s.
p~O
r~+oo
n(r)~n~n(r+1)
1/2
(B87)
For Vn(r) =Qn(r) (2log2 n (r))
,
2
2
a)
lEV
( ) = 0 and lE V
( )
~ (0)
an r~+oo.
n r
n r
0
27

---

-140-
..
1
, ' 1
1
J
' \\
1
b)
3j
,
V«,
1
0<<<<-,
Vj?:j
,
i?:l,
-<
{an (j + i) } 2. - jJ.
(
o
[
V
('
') V
(')) -Const .
2
0
n
J+l
n
J
an(j)
Part
a)
of
(B87)
lS
Lemma
3.5
in Lô
(1991).
Part
b
of
(B87)
is
the
analogue of
(B71).
Next,
let e
(o)=(o+1)/0
for
O<o~+oo, e
(o)
= 0+1 for
1
2
0<0<+00 while e
(oo)
= 1 and put
2
l
l
-
-
n 2
2
(k) /(210g n)
and
2
1
,
n(r)z
k(r)
2.
Qn (r) =Qn (r) -el (0) ((k (r)) B (n (r) ) / (210g2 n (r))
.
We also have
**
*
2.
**
**
(B88)
T
(2,k,t)=T
(2,k,t)/(210g2n)
=Q
+R
+0(1),
a. s.,
as n-.7+oo,
n
n
n
n
1
1
\\
1
with
**
(B88a)
n(r)~n~n(r+1)~ Qn
=Q~(r)+ 0r,p(l), as r-.7+oo and p-.70
and
1
l
* *
2.
2.
(B88b)
Rn =e
(o)k 0n(k) (l+0(l))/(210g2 n )
+0(1),
a.s.,
as n-.7+oo ,
2
and
(B88e)
lim
sup
(resp.
inf)
Q~(r)=O'1(o) (resp.
-0'1(0)),
a.s.
r-.7+oo
As a final remark,
replaeing «(k,t)
by «(k)
does effeet the results on
T * (2,
k,t)
sinee for any v,
O<v<l,
n
1
--
z
J
(B8 9)
{nk 2
n 1-G (t)
dt
1
x n
beeause
of
(LI)
and
properties
of
p-regularly
varying
funetions
(See
(B32)
for
example).
Replaeing
""[ (k,
t)
by
""[ (k)
in
results
established
28

---

: ,
} '4,
-141-
..
1
1
+
for A (l,k,t)
is also possible by using the same methods.
n
We shall use these two regularity conditions
,(RC(k)) 1 f (.)
admits a derivative f' (.)
such lim
uf' (u)
exists
U-70
-
-
(RC (k) ) 2 for all sequence un-(k/n),
k2f(un)/2l092n)2-70,
a.s.as n-7ro.
1/2
(RC (k) ) 3 Nn(-oo)/(2l092n)
-70,
a.S.
~evertheless, it is possible to use sharp results for A+ and T+ to have
n
n
complete characterizations for T (l,k,t).
n
'Theorem 1C.
Let F~(f,b)Er.Let (Ll-2)
and
(Kl-2-3-4)
hold.
Then,
1
-
-
~k2(T(1,k,t) _«(~)/~(~)2) satisfies thé LIL (-U (7), U (7), where
4
4
n
3
2
1
2
27 +107 +327+24
where U 4 (7) = 4 (7+1) (7+3) (7+4)' 0<7:S+oo .
Theorem 2C.
Let F~(f,b)Er. Let
(Ll-2),
(Kl-2-3-4),
(RC(k))l or
(RC(k))2 or
(RC(k)3 hold.
Then,
1
-
-
k2(Tn(1,k,t)-«(~)/~(~)2) satisfies the LIL (-U (7) ,U (7)), where
S
S
3
2
7 +7 +27
2
where crs
(7)
oo
4 (7+1) (7+3) (7+4)'
0<7:S+ .
PROOFS. Calculations based on
(SI),
(B83),
(B83a-c),
(B8S),
(B88 a-cl
and
(B89)
yield
1
k 1 / 2
(T
(1,k,t)-«(~)/~(~)2)=(2K(7)1/2)-1(2Z* -Q* )+0(1), a.s,
n
n
n
(C1)~====
v'2l092 n
as n-7ro for an arbitrary function f(.).
Now,
if k and satisfy
(RC(k))2
or
(RC(K))3,
or f(.)
satisfies
(RC(k))l,
29

---

:
1
)
'\\
-142-
"
1
1
1
1/2
-
-
k
2
2
-1
**
(C2)
(T
(l,k,i)-«(k)/T;(k) )=(2K(0))
(22
-
**
Q
) +0 (1) ,
a.s.
1
n
n
n
(2l092n) 2
According to Theorems 6.1 and 6.2 ln Lô
(1991),
1
Quot *
2 -1
*
*
2
( 2 K (0))
( 2 2
-
Q
)
-
N ( 0 , 0"4
( 0) ) ,
n
n
n
,n
, and
1
*
2 -1
**
**
2
Quot=(2k(0))
(22
-Q
)
-N(O,O"s
(0)),
n
n
n
,n
where o"~,n(o) ~ O"~(o) and o"~,n(o)~ d~(o) as n~.
1
-
2
That
(2log2 n (r)) Quotn(r)
and
*
Quotn(r)
satisfy
the
. assumptions of Lemma lB, follows from computations as in
(870)
and,
ln
the preceding,
we conclude with the \\helP of Lemma lB.
D- Log -laws for C = z -x
- - - - =2----:...~---=----=----n- n- n
Put C =2 -x .
n
n
n
Theorem D.
Let FEr.
Let
(Kl-2-3-4)
and
(L'l) i~+oo, i2/k~O and i/log n ~+oo,
2
hold.
If
one
of
(RC(k))l,
(RC(k))2
and
(RC(k))3
and one
of
(RC(i))l,
(RC(i))2 and
(RC(i))3 hold,
then
-
2
i (C -C )/R
(zn)
satisfies the LIL
(-el (0),
el (0))
n
n
1
for 2<0~+oo and
k1/2(Cn-Cn)/R1(xn)
satisfies the LIL
(-e
(0),
e
(0))
for 0<0<2.
1
1
Remark 1D.
For FED(~2)' the results crucially depend on b(.)
in
(1.3).
Examples related ta the asymptotic normality for this case are given in
Lô
(1991). As it was said in that paper,
Formulas
(Dl)
and
(D2)
below
include all the possible limits. One has only to have enough information
on b(·)
for deriving the corresponding LIL's.
30

---

",
. ,
: ,
J
'~
- 1<$3-
PROOF.
Put
N
(2,')
~).
n
The
regularity
assumptions
(RC('))i,
i=l,2,3 yield
1
1
-
-
2
2
m 0n(m)/(2l02n)
~O, a.s., for m=k or m=e
so that by
(B76)
1
1
1
-
2
(Dl)
e (ë -
Cn) /R
(Zn) (2lo,g2 n )2)
=-e
(o)N
(2,e) /2log2n)2
n
l
l
n1-
1 ;
2
+Pn(l'2)el(o)Nr~(2,k)/(2log2n) +0(1), a.s., as n~+CXl,
1
-
where Pn(l,~) =(e/k)2 Rl(xn)/Rl(zn)~O for FED(A) U D(~)UD(~o)'
2<O~+CXl.
If FED ( ~ ),
0<0 < 2 ,
1
0
1
1
-
-
(D2)
k 2 (ë - C ) / (R
(x ) (2log2 n ) 2) =+e 1 (0) N
(2, k) / (2log2 n ) 2) - P
(l, ~) -1
n
n
l
n
n
n
1
X
e~ (o)Nn(2,e)/(2l0g2n)2+0(1), a.s., as n~+CXl,
1
where P (l,-)~+CXl as n~+CXl.
(Dl)
and
(D2)
along with Theorem 6 Kiefer
n
2
(1972)
(See B33)
conclude the proof.
Remark 2B.
A direct application of this theorem gives 10g2-laws for
T
( 8 )
and T
(9).
n
n
E - Log -laws for T
(3),
T
(6)
and T
(7).
2- - - - - - n
n
n - -
Theorem E.
Let the assumptions of Theorem D be satisfied.
I)
If 2<O~+CXl, then
1
Z
-x
n
n
2
e R
(X ) (nVT
(3,k,l,v)- Rl(Xn))/R(Zn)
satisfies the LIL(-el(o),el(O))i
l
n
n
1
Z
-x
2
-1
n
n
eRl(Z
)(T
(6)
-
R
(
))/(Z -x)
satisfies
the
LIL(CT1(o),CT1(o));
n
n
1
zn
n
n
31

---

: ,
)
'~
)
'~
-144-
2
(z
-x )
2
-1
n
n
2
ER
(z
)(T
(7)
-
R
(
)
)/(z
- x )
2
n
n
2 zn
n
n
2
II)
Let 0<0<2 and put ~6
(0)
(0+1) 2 (5 0 +2).
Then
-
2
V
k
n T (3,k,V)-(zn-
(x ))
satisfies the LIL(-~6(0), ~6(o));
n
X n)/R 1
n
)/~ (ti ) satisfies the LIL(-~ (0), ~ (0)).
PROOFS. The results fol~ow from direct expansions from Theorems2A,
2B and D.
32

---

."
1
: ,
.'
,
-145-
.'"
}
.'"
}
REFERENCES.
1 [1] -Chung, K.L and Erdos, P. (1952). On the application of the
Borel-Cantelli lemma,
Transactions of the American Mathematical
1
Society,
72,
176-186 76~186
1
"
[2]
- Csorgo,
M.
and Révèsz,
P. (1981).
Strong Approximations in
Probability and Statistics.
Academie Press,
New-York.
1
1
i
[3]
- de Haan,
L. (1970) .On Regular Variation and its Applications to
,
the Weak Convergence of Sample Extreme.
Mathematical Centre Tracts,
1
32,
Amsterdam
1
[4]
- Deheuvels,
P., Hauesler,
E and Mason, D.M.
(1990).
Laws of
iterated logarithm for surns of extreme values in the domain of
1
è
~
.
attraction of a Gumbel law.
Bull.
Sc.
Math.,
2
serle,
114,
61-95.
\\
[5]
- Haeusler,
E.
and Mason,
D.M. (1987). A law of the i~erated
1
logarithm for sums of extreme values frorn a
distribution with a
1
regularly varying uppertail.
Ann.
Probab.
15,
932-953.
[6]
- Haeusler,
E.
and Teugels,
L. (1985).
On asymptotic normality of
1
Hill's estimate for the exponent of regular variation.
Ann.
Statist.,
13,
743-756.
1
[7]
- Hall,
P. (1982).
On simple estimates of an an exponent of regular
1
variation.
J.
Roy.
Statist.,
Ser B,
44,
37-42.
[8]
- Kiefer,
J. (1972).
Iterated logarithm analogues for sample
1
quantiles per p ~O. Proc. Sixth Berkeley Symp. Math. Statist.
n
Probab.1,
227-244.
University California Press.
[9]
-
Kuelbs,
J.
and Ledoux,
M. (1987).
Extreme values and the laws of
the
iterated logarithm.
Probab.
Theory Relted Fields,
74,
319-240.
[10]
- Leadbetter,
M.R.
and R6otzen,
H. (1988).
Extremal Theory for
stochastic processes.
Ann.
Probab.,
16,
431-478.
33

---

-146-
1
"
1
1
:
1
.,,-
.,,-
}
}
[11]
- Lô,
G.S. (1986) .Asymptotic behavior of Hill's estimate and
1
applicat~ons., J.Appl.
Probab.,
23,
322-936.
[12]
~ Lô,
G.S. (1989). A note ont the asymptotic normality of sums of
1
extreme values.
J.
Statist. Plann.
Inference,
22,
127-136.
1
[13]
- Lô,
G.S. (1990). Empirical characterization of the extremes I:
A family of characterizing statistics. Technical Report LSTA-CNRS,
1
Université Paris VI.
[14]
- Lô,
G.S. (1991).
Empirical characterization of the extremes II:
1
. .~
The asymptotic normality of the characterizing vectors .
1
Technical report,
LSTA-CNRS,
Université Paris VI.
~15]
- Resnick,
S.I. (1987). Extreme Values~ Regular Variation and
1
Point Processes.
Springer Verlag,
New-York.
1
1
1
1
1
1
1
34

---

J
.,,"
J
.,,"
-147-
SUR UN PROCESSUS GAUSSIEN LIMITE DE SOMME-PRODUITS DE VALEURS EX-rREMES.
GANE SAMB LO
UNIVERSITÉ DE SAINT-LOUIS ET UNIVERSITÉ DE PARIS VI-(LSTA)
RESUME .. Soit P(p,h)
l'ensemble des partitions ordonnées de l'entier p
en h entiers strictement positifs avec O<h~p. Soit YI
~ .... ~Y
la
!
,n
n,n
statistique d'ordre associée à l'échantillon aléatoire YI'.' .,Y
avec
.
n
G(Y)=~(Yl~Y)=F(eY). Pour tout p~l et pour tout triplet d'entiers (k,n,e)
tel que o<e<k<n>3,
nous ~ntroduisons la classe suivante de statistiques:
i
( Y .
_Y
.
) si
h
n-l+l,n
n-l,n
n
i=i
s. !
1
l
Elle contient la statistique de Hill(1975).
Nous montrons d'abord
que
pour chaque p,
(T fP))-l/ P est un estimateur de l'index\\d'une loi de
n
Pareto si 1-F est à variation régulière. En fait,
la plupart des
propriétés
connues
sur
la statistique de
Hill
pour des
fonctions
de
distribution F appartenant à un domaine extrêmal d'attraction,
sont en
effet adaptables à chaque marge T (p),
p~l. Enfin et surtout, nous
n
étudions la loi limitedu processus {T
(p),
l~p<oo}. Le processus gaussien
n
limite est entièrement décrit grâce à sa fonction de covariance. La
détermination de celle-ci fait apparaître trois classes remarquables de
nombres entiers qui seront décrites selon des méthodes de calculs
combinatoires.
Mots clés.
Domaine d'attraction extrêmal,
statistique d'ordre,théorie
des valeurs extrêmes, processus gaussiens,
fonction de covariance,
convergence vague de mesures de probabilités.
Addresse. UER de Mathématiques Appliquées et d'Informatique,
Université de Saint-Louis. BP 234,SAINT-LOUIS,
SENEGAL.
Fax (221)611884.
Email:
lo@louis.univ-stl.sn.
1

---

.,
1
"
1
1
. '
,
-148-
} '4,
1 -
1NTRODUCT 1ON.
1
Soit Xl'
X ,
. . . . une suite de variables aléatoires
(v.a.) .indépen-
2
danteset identiquement distribuées avec P(X s x)=F(x) , xeR et F(l)=O et
1
1
soit Xl
s ... sX
la statistique d'ordre associée aux n~l premières v.a.
,n
n,n
1 Soit l, k et n trois entiers tels que O<~ksn. La statistique de Hill(1975
1
T (l,k)
= k
j (logX
. 1
-logX
.
)
(1. 0)
n
n-J+
,n
n-J,n
1
l+lsj sk
a certainement joué dans les dix dernières années un des rôles les plus
1
1 importants dans le traitement statistique des domaines d'attraction des
extrêmes ainsi que dans l'estimation de la queue d'une v.a.
(voir,
par
1 exemple, Hill(1975), Hall(1982) , Mason(1982) , Davis et Resnick(1984) ,
Il
Cs6rgo-Deheuvels- ,Mason(1987) , LO(1989-1991),
Dekkers et al. (1989),
1
etc .. ) .
1
Rappelons pour fixer les idées que si une fonction de distribution
(f.d.)
F est attirée par une f.d.
non-dégénérée M,
c'est-à-dire qu'il
1 existe deux suites (a >0) 0 et (bn)n~o telles que
n
n~
VxeR,
lim
Fn(a x + b )=M(x).,
(1. 1)
1
n
n
x~oo
alors M est nécessairement du type de Fréchet de paramètre ~>O,
(cp
(x) =
~
1 exp(-x-~), x~O), ou du type de weibull de paramètre ~>O, (W (x)=exp(-(-x)~)
~
x
xsO)
ou enfin du type de Gumbel,
(A(x)=exp(-e-
), xeR).
1
Si
(1.1)
a lieu on note F e D(M).
1
Dans le but de trouver une caractérisation statistique de ces trois
domaines d'attraction,
Lô(1992)
a
introduit la statistique T (2,k)
suivant
1
n
j=k
i=j
\\' i (1-8 .. /2) (logX
. 1
-logX
.
) (logX
. 1
-logX
.
),
(1.2)
1
kL
L
1J
n-1+ ,n
n-1,n
n-J+
,n
n-J,n
j=l+l i=l+l
où
8 .. est le symbole de Kronecker,
et a montré que le couple
(T
(1),
1
1J
n
T
(l)/T
(2)1/2) caractérise chacun des domaines d'attraction.
n
n
1
Il est apparu lors de ce travail que les deux statistiques T
(l,k)
et
n
2
1
1

---

l,
-149-
"
,
J
'4,
T (2,k)
sont deux marges d'un même processus statistique.
En effet,
si,
n
pour l~l~h~p, P(p,h) ,désigne l'ensemble des partitions ordonnées de p en
h entiers strictement positifs,
i.e.,
(1.4 )
et si nous posons Y.
=logX.
, i = l , ... ,n, nous pouvons définir le proces-
l,n
l,n
sus de paramètre p,
l~p~oo,
s.
y
.
_y
l
p
( n-l+l,n
)
n-l,n
T (p)=
- - - - - - - - -
(1. 5)
lh
n
~ l
l
s, !
h=l(sl"
.Sh)EP(p,h)
il =i+l ih=i+l
l
;11 est aisé maintenant de vérifier que les estimateurs T (l,k)
et T (2,k)
n
n
sont les premières marges du processus {Tn{p),
l~p~oo}. Pour des raisons
~e commodité, nous désignerons ce proqessus sous le nom de statistique
généralisée de Hill.
Notre principal résultat est la convergence de la statistique généra-
lisée de Hill dans ~~. Nous déterminons entièrement la fonction de
covariance du processus gaussien limite à l'aide de calculs combinatoires.
L'étude des nombres entiers apparaissant dans la fonction de covariance
occupe la part la plus importante de cet article car les résultats concer-
nant la f.d.
F et devant être utilisées pour obtenir les lois limites sont
suffisamment developpées dans lô{199l-II).
Dans la suite,
nous décrivons les classes de nombres obtenus et les
processus gaussiens qui leur sont associés dans le section II.
Toujours
dans la section II,
nous exposons la loi limite de la statistique de
Hill. La section III est consacrée à la mise en évidence et à la démons-
tration de l'existence des nombres décrits au II. Enfin,
les preuves
de nos lois limites sont données dans la section III.
3

---

: ,
-150-
)
''\\
Il - PRINCIPAUX RESULTATS.
Nous décrivons d'abord les classes de nombres qui interviennent
dans la suite.
A: NOMBRES DE TYPE 1, Il ET III·
Définition 2.1.
Les nombres entiers positifs ~(v,r), v~O,r~l, définis par
i i ) Vr~3, ~ (1, r) =~ (2 , r:-1) +~ (1, r -1) .
iii)
Vr~2, ~(O,r)=~(l,r-1)
iv)Vv~2,r~3, ~(v,r)=~(v+1,r-1)+~(v-1,r)
sont dits nombres de type 1.
\\
\\
Remarque 2.1.
Il, est aisé de calculer tous les nombres de type l
grâce
aux deux premières règles de sabot suivantes.
1
v
u+v
V
u
u+v
(1.7)
1
u
U
u+v
v
!
CR
correspond à ii)
tandis que CR
est relative à
iv).
En répétant
1
2
1
l'application de CR
et de CR , on obtient tous les nombres de type 1.
1
2
Par exemple,
on a pour v~10, r~ll:
1
1
1
1
1
4
1
1
1

---

",
"
\\
1
"
1
1
--151-
J
, "
J
, "
r
1
2
3
4
5
6
7
8
9
10
11
V""
0
1
1
1
2
5
14
42
132
429
1430 3862
u+v=
1
1
v=l
5
14
42
132
429
1430 3862 11156
2
2
1
u=l
3
9
28
90
297
1001 2432 7294
3
1
1
4
t=14
48
165
572
2002 6072
t+s;=
4
1
1
5
75
275
1001 3640
20,
5
1
1
s=6
27i
110
429
1638
6
1
1
7
35
154
637
-
,
7
1
1
8
44
208
8
1
1
9
54
:
9
1
1
10
\\
!
10
1
1
i
Tab.
2.1
Remarque 2.2
. A titre de comparaison,
CR
est la règle du sabot des
2
nombres ~* (v,r) de chemins de l2 joignant (0,0) à (v+r,r) à l'intérieur
du parallélogramme
[(0,0),
(v,O), (v+r,r), (r,r)].
Ces nombres sont déter-
minés par la formule ~* (v,r)= (v~2r)-2(~:~r)+(v~;;)' r~v+1. (voir Kreweras,
1984) ).
Ces nombres diffèrent des nôtres par le domaine de définition
(les
*
~
(v,r)
sont seulement définis pour r~v+1, les ~(~,r)
l'étant sur ~ X ~)
et par les deux premières lignes.
Définition 2.2.
Les nombres positifs entiers ~
(O,v,o),
l~O~T, verifiant
T
i)
Vv~O, VT~l, ~
(O,v,T)=l
T
ii)
Vo,
l~O<T,
~T(O,O,O) = 1
iii) Vo,
1~0 <T,
Vv~l, «
(O,V,O)=«
(O,v-1,0)+«
(O,v,o+l)
T
T
T
sont appelés nombres de type II(T).
Remarque 2.3.
iii)
est associé à
la règle du sabot CR
de
(1.7).
Ces
3
nombres sont encore aisément calculés. Voici quelques exemples.
5

---

1
, ' 1
1
" ,
J
'4,
J
'4,
-152-
1:"=1
1:"=2
1:"=3
1
~ 0 1
~o
1
2
V~o
1
2
3
1
0
1
1
0
1
1
1
0
1
1
2
1
1
3
2
1
1
1
1
'2
3
1
2
6
3
1
2
1
,
3
4
1
3
10
4
1
3
1
1
1
4
1
5
4
15
5
1
4
1
,
;
5
21
6
1
5
1
5
6
1
1
Tab.
2.2
Tab.
2.3
Tab.
2.4
1
Enfin,
nous avons la dernière classe de nombres.
1
Définition 2.\\3. ~Les nombres de type III (1:")
sont l~s nombres entiers positi 1
notés M1:"(l,v,o),
v~o, o~l, 1:"~1 satisfaisant aux conditions ia), ii) iii)
et iv)
de la définition 2.1 pour o=r et satisfait en lieu et place de ib)
1
la condition suivante
v+1
«
(l,v,2)
=
«
(o,k,l).
1
1:"
1:"
k=l
1
Remarque 2.4.
Pour 1:"= 1,2 or 3, Tab.
2.2,
2.3 et 2.4 permettent de
calculer «1:"(1,v,2).
On pourra donc déterminer les M1:"(l,v,o).
1
1
1
1
1
1
6
1
1
1

---

",
,'1
1
" ,
-153-
,
)
''\\
1
1:"=1
1:"=2
1
~O
1
2
3
4
5
~O
1
2
3
4
5
0
1
1
2
5
14
0
1
1
5
14
42
1
1
1
2
5
14
42
1
1
5
14
42
132
2
1
3
9
28
2
1
9
28
90
1
3
1
4
14
3
1
14
48
,
:
1
4
1
5
,
4
1
20
5
1
1
5
1
1
Tab 2.5
Tab.
2.6
1:"=3
1
r""'O
1
2
3
4
5
4
5
11
29
70
0
1
1
9
28
90
1
3
4
9
20
1
1
9 ~
\\28
90 ."
297
2
3
6
2
1
19
62
207
1
1
2
3
1
34
117
1
r/
4
1
55
1
2
3
4
P
5
1
1
Tab 2.7
Tab 2.8
Notons que ces tableaux sont immédiatement remplis par les logiciels
1
tableurs classiques.
1
Nous sommes en mesure d'introduire les processus gaussiens de
"
Csorgo-Mason-Lo
[CML]
"
(Csorgo-Mason(1984)
et Lô(1989))
que nous définissoll
à l'aide des nombres introduits ci-dessus.
1
1
1
1
7
1
1
1

---

'"
,
,
,,1
"
,
J
'4,
-154-
B- LES PROCESSUS CML.
1
Définition 2.4.
Le processus gaussien {I(r),r =1,2, ... } est dit
de type CML ssi ~(I(r))= 0 pour tout r~l et s'il existe deux fonctions
1
positives Cl (.)
et C (.,·)
telles que ~(I(r)2)=~2(r) et
2
1
~(I(r)I(p))=~(r,p) satisfont aux conditions suivantes
i)
a(O)=l, a(l)=2
; j=r
1
ii)
Vr~l, ~2(r)= C (r)a(r) et a(r)=~.I ~(l,j)a(r-j)
1
, J-1
j =r
: -
iii) V1~r<p, ~(r,p)=
1
C
(r,p)
I j.L
Cl,l,j)
a(r-j)
2
j =0 p-r
avec la convention que «
(1,1,0)=1 pour tout L~l.
L
1
A titre d'illustrations,
le tableau 2.8 ci-dessus présente les premiers
1
élé~ents de ~2(r) et de ~(r,p) pour C1(r~=c1 (r,p)=l. Ce tableau montre
~(2,1)=3+9 ~(3,2) 1
clairement que ce processus n'est pas stationaire car
=
+29 = ~(4,3).
1
Définition 2.5
. La série temporelle {I(p)+e(p)l, p=l,2, ... } est
1
appelée processus gaussien CML réduit si
1
1)
e(·)
est une fonction réelleà valeurs dans
[1,00[,
2)
l est un processus de type CML
1
3)
et si l e s t unev.a. gaussienne centrée réduite telle que
~(I(p)l)=-l pour tout p~l.
1
Les nombres définis ci-dessus et les processus décrits auparavent
1
permettent de décrire entièrement les lois limites de la statistique
1
de Hill généralisée.
1
1
8
1
1
1

---

: '
-155-
1
J
'4,
c: LOIS LIMITES DE TN(P).
1
Nous avons besoin des notations supplémentaires suivantes.
pour une fonction de répartition quelconque G(·):
1
2 2 2
2
m (X'2,G)=J J
... J (1- G(t)) dt dy
ml (x,
p
1 · .dYp _l' p~2;
J
2 , G) =
x (1- G (t) )
dt 1
x Y1
Yp - 1
avec m (x'~o,G)~m (x,G), p~l, oa Y
= sup{x, G(xJ<l};
p
i
P
O
1
1
1
1
1
X
= G-
(1- l k/n),
x =G- (1-U 1
),
2
=G-
(1-t/n) ,
Z =G-
(1-U
k
o
1
),
n
.
n
+ ,n
n
n,
c..+ , n
oa U
s ... sU
est la statistique d'odre d'un échantillon de taille
1
l,n
n,n
n~l issu .d'une loi uniforme sur (0,1)
qui sera spécifiée plus tard.
Soit
1
n
enfin,
pour p~l, -r
(x ,2 )= k- m (x ,2
,G).
Dans toute la suite,
m (0,0,0)
p
n n
p
n
n
p
ne seront utilisées que pour la f.d.
G.
1
\\
\\
Les théorèmes ci-dessous décrivent complètement la loi limite de la
1
statistique généralisée de Hill pour toute distribution F attirée une
une distribution non-dégénérée, .i.e.
FE l=D(A)U{Ua>o D(~a)}U {U'1>OD(W )}, 1
'1
ou U
0 désigne l'union indexée par a>O.
a>
1
Dans le reste de l'article,
nous ferons la convention que si une
1
propriété est satisfaite pour O<'1soo,
la double inégalité 0<'1<00 concerne
le domaine d'attraction D(W ) et l'égalité '1=00 concerne les deux autres
a
1
classes.
D'ailleurs,
à moins que le contraire ne soit spécifié,
les cas
correspondants à '1=00 s'obtiennent par passage à la limite '1~.
1
Théorème 2.1.
Soit FEl. si t est fixé et si k satisfait à
la condition
1
(K)
O<k=k(n)~+oo,
k(n)/n~O quand n~+oo,
1 2
alors {k /
(T
(p)--r
(x-))/-r
(x ),
1 s p<+00},
converge en distribution vers
n
p
n
p
n
ul
processus gaussien CML dans la topologie canonique de ~oo avec
r
'1+j
j=r
'1+j
Cl (r)
=
II
{ - } , r~l; c (r,p)
= ,II
{
. },
r~l, p~l,
1
2
'1+p+J
j=l
'1+r+j
J =1
1
9
1
1
1

---

1
Le second théorème consiste à remplacer le coefficient de centrage
1
aléatoire T (x)
par le coefficient non aléatoire T (x ).
Cela exige
.
p
n
p
n
des conditions de régularité caractérisées dans Lô(199l).
Rappelons
1
que toute d.f.
FEI peut être représentée par des constantes c>O,
d et yo
1
et par des fonctions f(u)
et b(u),
O<u<l,
tendant verz zéro quand U~O,
de la manière suivante:
1
-1
/
JI
-1
G
(l-u)=logc-(logu)
0+
b(t)t
dt,
O<u<l,
(FED(rp )) ;
o
u
l'
-1
1/ 0
J
-1
yo-G
(l-u)=c(l+f(u))
u
exp (
b(t)t
dt),
O<u<l,
(FED (I/J )) ;
1
o
1
u
G-l(l-U)=d-s(u)+J s(t)t-ldt,
O<u<l,
(FED(A)),
u
1
1
avec s(u)=c(l+f(u))
exp(J b(t)t-ldt),
O<u<l.
u
1
Puisque nous sommes principalement intéressés par la mise en évidence des
processus CML
~ d .
\\l'
.
,
l
d , .
d
re UltS,
nous nous
lmlterons a
a con ltlon
e
~ l .\\
regu
~
arlte
1
la plus simple,
i.e.,
(RC)
f' (u)
existe dans un voisinage de zéro et lim uf' (u)=O.
1
u-l.-O
Cependant,
la caractérisation citée ci-dessus est encore valable lCl.
1
1
Théorème 2.2.
Soit FEI.Si e est fixé,
k satisfait à
(K)
et si
(RC)
est
l
2
satisfaite,
alors {k /
(T
(p)-T
(x ) )/T
(x ),
l~p<+oo} converge en distri-
n
p
n
p
n
1
bution vers la forme réduite du processus gaussien CML du Théorème 2.1.
dans la topologie canonique de ~oo avec e(p))= (o+p)/o,O<o~+oo.
1
Nous allons maintenant générer les nombres de type I,II(T)
et III(T).
1
111- GÉNÉRATION DES NOMBRES DE TYPE 1, IICT), IIICT).
1
N'oublions pas que pour tout j~l, m. (X,y )<+00 si FEI. Définissons
J
v
(qr-l-Pr-l)
1
m. (q
1 '
z ) dq
1 '
(v) !
J
r-
n
r-
*
où VE~, jE~ , r~2, q = Pl' P =x ;
o
1
0
n
10
1
1
1

---

-157-
..
l
'),~
)
1
'~
1
1
1
(qT-o-l-Po-l)
v!
mj (qT+o-l'Zn)} dqT+O-l'
1
où
*
VE~,TE~ , 0=2,3, ...
1
Les nombres de type l,ll(r)
et 111(T)
apparaissent lors du calcul de
ces intégrales.
Nous allons nous borner à montrer comment ils apparaissentl
par l'énoncé de trois lemmes. Les preuves de ces lemmes seront exposeés
l'annexe.
1
1
Lemme- 3. 1.
Pour tout j ~l,
r>2 v~O,
les rapports
v
(3(v,r,j)
=
h.
(r)/m.
2(
1) (x ,Z )
]
]+v+
r-
n
n
1
sont des nombres positifs entiers dépendant uniquement de
(v,r)
et sont
égaux aux nombres de type l,
i.e.,
1
Vj~l, Vv~O, Vr~2, (3(v,r,j)=(3(Il,r).
1
Lemme 3.2.
Pour tout 0,
l~O~T,V~O, les rapports
1
r~(O,T)/m.
~ l(x ,Z )=~ (O,v,o,j)
]
]+V+T-o+
n
n
T
sont des entiers positifs indépendants de j
et égaux aux nombres de type
1
II(T),
i.e.,
VT~l, VO,O<O<T, Vv~O, Vj~l, ~T(O,V,o,j)= ~T(O,V,o).
1
Lemme 3.3.
Soit TE~ * fixé.Pour tout j~l,
V~o et 0~2,
les rapports
v
1
r· (l,o)/m.
2(~) (x ,Z )=« (l,v,o,]')
]
]+V+T+
u-l
n
n
T
sont des entiers positifs indépendants de j
et égaux aux nombres de type
1
III(T),
i.e.,
1
11
1
1
1

---

-158-.";
,1
}
.~
}
.~
VT~l,
VO~2,
Vv~o,
j.l
( 1 , v, 0 , j ) =«
( 1 , v, 0) .
T
T
1
IV- DEMONSTRATIONS DES THEOREMES 2.1 ET 2.2.
1
Il
Il
Csorgo-Csorgo-Horvath and Mason(1986)
ont construit un espace de
1
Probabilité
(Q,S,~) portant une suite de Ponts Browniens {B (s) ,O:ss:sl}
n
n=1,2
et une suite de v.a.
indépendentes uniformément réparties sur
1
telles que pour tout 0<V<1/4,
- , -
1
-V+1/2
-v
1 Sup 1 Ivn
(Un(S)-S)-Bn(S)
l(s(l-s))
=op(n)
(4 . la)
1
-:ss:sl--
n
n
-
-V+1I2
1
-v
1 Sup 1 Ivn (s-Vn(S)-Bn(S) I/(s(l-S))
=0
(n
)
(4 . lb)
p
-:ss:sl--
n
n
1
où Un(o)
et Vn(o)
désignent respect~vement la f.d.
empirique et la fonti~nl
des quantiles de l'echantillon U , .. "U
dont la statistique d'ordre est
1
n
notée O=U
<U
:s ... :sU
<U
1
=1.
O,n
1 ,n
n,n
n+ ,n
1
Les démonstrations auront désormais lieu dans cet espace de ~robabilitéS, 1
Pour des raisons de commodité,
nous supposerons que F(l)=O,
l.e.,
X~l, p.s.
et nous noterons y,
=log X.
, i = l , ... ,n. Y1
:s .... :sY
est donc la
l,n
l,n
,n
n,n
statistique d'ordre d'un échantillon issu d'un v.a. répartie selon la
distribution ~(Y:SY)=G(y)=F(eY) et nous désignerons
par G (.)
la f.d.
n
Pour commencer, notons que des calculs directs aboutissent à
-
-
-
z
z
z
n
Tn(p)= k
J_n J n
... J n
U (l-G(t)
dt dy ·· .dy _ '
p~l.
n
1
P 1
x n
Y1
Yp - 1
Nous ommettons le détail des calculs. Néanmoins,
la vérification peut
aisément se faire pour p=l
(voir(l.0)), pour
(voir
(1.1))
et pour p=3.
12

---

"
,
" ,
) '4,
) '4,
-159-
1
1/2
Posons a
(s)=n
(U
(s)-s),
O~s~l.
n
n
l,
La démonstration se fera à partir des lemmes suivants.
1
Lemme 4.1.
Pour tout p~l, nous avons
1
1
j =p -1 (z - x ) j
-
n -
1
v'k (T (p) -1;" :(x ,z )) =W (p) +R + (R) 2
Ii __n__n__ m
. (z )
m1
(x) (k) 2
n
Pi
n
n
n
,.
n
,p
n
2 ,P-J
n
j =0
j
1
1
j =p-1
-
,
+ l
_m_1~,--=:j__(x_n_) m3 , p_j (xn,izn) .
j
1
j =1
où
1
-
-
-
-
JXn JXn
JXn
ml
( x ) = ,
. . .
B (l-G(t))dt
1
.
,p
n
n
ln Y1
Yp - 1
-
-x
1
-
(z)= JXn
m
Jn B (l-G(t)
2 ,p
n
z
n
Y
n
p - 1
1
-
m
( x , -
z
)
JZn
=
3 ,p
n
n
x
1
n
-
-
z
z
1
J n ... J n a (l-G(t))-B (l-G(t)) dt dy ·· .dY _
n
n
1
p 1 i
Y1
Yp - 1
1
et enfin
1
1
Preuve.
Evidente. _
La suite de la démonstration consiste à décrire le comportement de chacunl
des termes du dévelopement du lemme 4.1. Mais les méthodes utilisées dansl
Il
Csorgo-Mason(1994)
et dans Lô(1991)
,pour les ordres un et deux,
sont très
aisément extensibles ici à l'ordre quelconque p,
grâce surtout aux lemmesl
13
1
1
1

---

1
l
,
"
1
,
)
''\\
-160-) ''\\
4.1 et 4.2 du dernier article cité résumé dans le lemme suivant.
1
Pour cela, posons pour tout p~l, .
1
\\f x<Z::5x
(F),
R (x,z,F)=m (x,z,F)/(l-F(x) )p; R (X'XO,F)~R (x,F).!
O
p p p
p
x
Lemme 4.2.
Soit FEI avec F(l)=O et soit G(x)=F(e ), x~l. Alors, nous a-0-ons
1
1
i)
Si FED(~ ), alors R (x,F)-(~O-x)p {rr~=P1 (o+j)-l}, quand x~xO.
o P :
J=
1
ii)
Si FE(A)U D(~), alors Rp(x,~)-R1 (x, G)p, quand x~yO'
iii)
(z-x)P/R
(x,G)~+oo and R (x,i,G)/R (x,G)~l
p p p
1
as x~xO' z~xO '
(l-G(z) )/(l-G(x) ~ o.
-1
iv)
(l-G(G
(l-u)) )/u ~ 1 quand u~O.
1
1
Une extension judicieuse des méthodes citées permet d'obtenir
\\
Lemme 4.3; Soit FEI. Alors pour tout p~l, nous avons
1
vk (T (p)
-T
(x )) Ir:; (x ) = W (p) Ir:; (x ) +0 (1), quand n~.
n
p
n
p
n
n
p
n
p
1
Ce lemme donne la composante gaussienne de la statistique généralisée de
Hill centrée sur r:;
(x ). Le lemme suivant permet le centrage sur le
1
p
n
coefficient non-aléatoire r:;
(x ) .
p
n
1
Lemme 4.4.
Soit FEI. Si
(RC)
est satisfaite alors, p.s.,
quand n~,
1
vk (r:; (x ) -"[ (x )) Ir:; (x ) = e (p) nk -1/2 (U
-~) +0 (1) =-e (p) (~) 1/2B (~) +0 (1)'1
p
n
p
n
p
n
k ,n n
p
k
n n
p
Preuve. Donnons une esquisse de la preuve. On vérifie aisément que
-
k
-
1
r:;
(x )-r:;
(x )=-
(x -x ) m
l(x )+R (2)
(4.5)
p
n
p
n
n
n
n
p-
n
n
où
1
1
p-1
x Ij
n
m
, ( x ) .
(4.6)
~ IRn (2) 1::5l xn - xn ) IP sup (~,l-G(Xn))+ L
j
P-J
n
j =2
1
14
1
1
1

---

"
1
J
1
-161- ,";
"
1
J
' ' \\
J
' ' \\
Or,
d'après le lemme 3.6 dans Lô(1991-II)
et le lemme 4.2 ci-dessus, on a l
vk
n
(x
-x )IR (x ) =- D+1
(-k ) l/2B
(15)
+
0
(1)
as n-Hoo.
(4.7)
n
n
1
n D
n n
p
Aussi,
en combinant
1
(4.7)
et le lemme 4;2 et en tenant compte du fait que
n
k(l-G(Xn))~ 1 quand n~+oo (voir Lemme 4.2 ci-dessus), on montre que
Vk R (2)/T (x ) ~ 0
1
as n~+oo. En conclusion
;
n
p
n
p
vk (T (p)-T (x , Z )/T (x ) et Vk(T (p)-T (x ))/T (x )
n
p
n
n
p
n ;
n
p
n
p
n
1
se; comportent asymptotiquement,
respectivement,
comme
i
1
W ( p) 1T
(x
),
p~1
(4.8)
n
p
n
et
w
1
(p) 1T
( X
) - e (p) (~) l/2B
( 15) .
(4.9)
n
p
n
k
n n
1
11\\ est clair que ces processus de paramftre p ont des distributions
finies gaussiennes puisqu'elles sont deux tranformations linéaires
1
du Pont Brownien dans [(0,1).
Nous allons maintenent calculer les
1
fonctions de covariance pour établir la convergence vague dans ~~.
1
AI CALCUL DE LA FONCTION DE LA VARIANCE DE WN(R), R~1.
1
D'abord,
rappelons que pour O~s,t~l, on a
lE(B
(l-G(t)B
(l-G(s)) =min(l-G(t)), 1-G(t)) - (l-G(t)) (l-G(s)) =h(s,t).
n
n
1
Nous avons
Vr~l, lE W (r)
= O.
n
Ensuite,
1
(4.11)
1
1
Mais pour tout couple
(p,q)
tel que p<y , q<y , on a H(p,q)=H(q,p). Ainsi
o
0
en intégrant sur le demi-espace
(P1<q1)
puis sur son compémentaire
(P1~q11
15

---

"
1
1
•
-162-
1
}
''\\
}
''\\
on obtient
Z
Z
1
(r)=2 k f
2
n
n
CT
(4.12)
dPI
n
f n H(Pl'
x n
Pl
11
1
Or d'après Csorgo-Mason(1984)
et Lô(1989) ,
Z
Z
VXn~y~zn f nf in h(s,t) ds dt -_ 2( fZn fZn 1-G(s)
.
dt) )
1
(l+r
(1),
(4.13)
n
y
t
1
y
1
où
1 r
( 1) 1 ~ 1 +- G ( x ) .
1
n
i
n
1
i
1
Nous voulons p~ouver la formule
(4.13)
à
l'ordre p.
1
Procédons par recurrence et supposons que pour r~2, la formule
suivante est vraie. Alors,
1
Vj,
1~j~r-1, Vx ~y<z
,
(4.14 )
n
n
1
Z
f n h(s,t) ds dt=2a(j)m (y, zn) (l+r (j)),
1
2j
n
qj-1
où
Ir
(j) l~l-G(x ) et a(j)
est un entier ne dépendant ni de y ni de n.
n
n
1
Montrons que l'égalité dans
(4.14)
est vraie à
l'ordre r. Mais,
en
z
fq·
1
En décomposant
n . dpj+1 en
J
• d p
+
+ fZn
. dpj+1 dans
(4.12)
f
j
1
q.
Pj
Pj
J
1
sequentiellement de j=l à j=r-1 et en appliquant
(4.14),
le membre
de droite de
(4.14)
devient à l'ordre r,
j =r
1
(l+r
(j,r)),
(4.15)
2.L a(r-j) h~(r-j)+l(j)
n
J=l
1
1
En appliquant ensuite le lemme 3.1,
i l devient
1
j =r
2
a (r - j) (3 (1, j ) ) (1 +r
( r))
m
(x ,
Z
)
(4.16) 1
L
n
2 r
n
n
j=l
avec
1 r
(r) 1 ~ 1 -G (x ) .
n
n
1
16
1
1
1

---

1
, ' 1
1
:
1
J
~
J
.~
-163-
1
1
Nous concluons donc que pour tout r~l,
2
n
1
~n(r)=2 k a(r) m
(x ,
zn)
(4.17)
2r
n
1 Maintenant, en utilisant le lemme 4.2, on obtient
1
j=r
1
Lim
La(r-j)j3(l,j),
(4.18)
n~+oo
1
j =1
1
1
1
•
r
'1jJ
où Cl ('1)1 = 1 pour FED(A)UD(~), Cl ('1)
=.rr { '1+ r +j } pour FED(~'1) .
1
:
J =1
1
1
i
B: CALCUJ DE LA FONCTION DE COVARIANCE DE WN(J.
1
1
Nous avons pour '1<P,
~=P-'1,
t
z
z
z
z
[(Wn(R)Wn(P))= RJ n dq1 J ~.J n dq~J n
1
(4.19)
x n
q1
q~-l
x n
1
1.
1r
Zn
JZ
1
dP1 +
. dp
et
n .
dp.
en
1
JZn.
En décomposant
J
1
p.
J
x
q~
J
n
1
r-2,
1
et en utilisant
(4.17), nous
Jq~+j
JZn
• dp. +
.. dp j' pour j =l, ... ,
Pj
J
q~+j
obtenons
lE (W
(rW (p) )
n
n
r
1
+L a(r-j)'1~(r_j)+l(l,j) (l+r (r,j)),
n
j =2
1
où
Ir
(r,j) I~ 1- G(x ), pour l~j:sr.
n
n
1
D'après le lemme 3.3,
j =r
IE(W
('1)W
(p))
= {R L a(r-j)«~(l,l,j)} (l+r (r,p) m
(x ,Z ),
(4.21
n
n
n
r+p
n
n
j =1
où
Irn(r,p) l~l-G(Xn) et par convention,
«~(l,l,l)=g~(O,l,l).
1
Maintenant,
d'après le lemme 4.2,
1
17
1
1
1

---

: ,
-)64-
J
' ' \\
'\\
lim ~(w (r)W (P)!L
(x )L
(x ))
C
(r, p) a (r, p) ,
(4.22)
n
n
r
n
P
n
2
n-700
où pour l::sr<p,
j=r
a(r,p)
= \\' a(r-j)«
(l,l,j)
(4.23)
L
p-r
j =1
et
r
'1+j
C ('1,p)= ,II
} si FeD(il/J)
et c
(r,p)=1 Sl FeD (A) UD (<p) •
(4.24 )
2
{ '1+p+j
i'1
2
J=l
1
!
CI
Il est immédiat que
k
n
~(W (r)B (-) )=(l+r ) k- m (x
, 2 ) ,
où
Ir
l::Sl-G(x ).
(4.25)
n
nn
n
r
n
n
n
n
Pose W (r).
{W
(r)!L
(r)}_e(r)(~)1/2 B (~). Donc,
n
n
n
\\
k
n n
Lim
~(Wn(r)W(p))=c2(r,p)a(r,p)-e(r)-e(p)+e1r)e(p)
(4.26)
n-.?oo
tandis que
Lim ~(W (r)2)
2
2C
(r)a(r)-2e(r)+e(r)
.
(4.27)
n
1
Nous avons donc entièrement fini de calculer les fonctions de covariance
de W (r)!L
(x)
et de W (r)
à travers
(4.18),
(4.22),
(4.26)
et
(4.27);
n
r
n
n
formules qui montrent bien que W (r)!L
(x ) est un processus de type CML
n
r
n
et que W (r) est sa forme réduite.
n
REFERENCES.
Il
Il
Csorgo,
M.,
Csorgo,
S.,
Horvàth,
L. et Masan,
D. (1986). Weighted
empirical and quantile processes. Ann.
Probab. 14,
31-85.
de Haan,
L. (1970).
On Regular Variation and its Application ta the
Weak Convergence of Sample Extreme.
Mathematical
"
Center Tracts,
32, Amsterdam.
Csorgo,
S.
et Mason,
D. (1985). Central limits theorems for sums of extreme
values. Math.
Proc. Cambridge Philos. Soc.,
98,
547-558.
"
Csorgo,
S.,
Deheuvels,
P. et Mason,
D. (1987).
Kernel estimates for the
tail index of a distribution. Ann. Statist.,
13,1 467-1487.
Davis,
R.
et Resnick,
S. (1984). Tail estimates motivated by extreme
value theory. Ann. Statist.,
12,
1467-1487.
18

---

-l65-
" ,
)
'\\
Dekkers,
A.L.M.,
Einmahl, J.H.J. et de Haan,
L. (1989). A moment estimator
for the index of an extreme-value distribution. Ann.
Statist.
17, 1833-1855.
Hall,
P. (1982). On simple estimates of an exponent of regular variation.
J.
Roy.
Statist. Soc. Ser. B.,
44,
37-42.
Kreweras,
G.
(1984).
Cours delDEA : Combinatoire.
ISUP,
Paris 6.
Unpublished.
Lô, G.S. (1989). A note on the asymptotic normality of sums of extreme
values.
J.
Statist.:Plann.
Inference,
22,
127-136.
Lô, G.S. (1991).
Empirical characterization of the extremes l
& II.
Technical report,
#143,
LSTA CNRS-URA 1321,Paris 6.
1
Mason, D.M. (1982).
Law of large numbers for sums of extreme values.
Ann.
Probab.,
10,754-764.
'
Resnick,
S.I. (1987).
Extreme Values,
Regular Variation and Point
Processes.
Springer~Verlag.
1
REMERCIEMENTS.
Nous remercions l'examinateur dont les critiques de forme et de
fond ont beaucoup contribué à améliorer le texte.
ANNEXE.
DETAILS DE LA GENERATION DES NOMBRES DE TYPE 1, IICc) ET IIICc).
PREUVE DU LEMME 3.1.
Traitons le cas r=2 d'abord:
h~(2)
dq1'
v~l,j~l.
(3.1)
J
En remarquant
m
(3 .2)
j (q1,zn)= - dm j + 1 (q1,zn)/ dq1' a.e.,
et en intégrant par parties,
on obtient
h~(2)
h~-ll (2), v~l, j~l.
(3 . 3 )
J
J +
On peut aisément vérifier que
o
h. (2) =m.
2
(x ,z ),
j ~1
(3.4)
J
J+
n
n
si bien que
(3.5)
En appliquant
(3.3)
jusqu'à ce que l'indice supérieur de son membre de
droite soit nul,
on arrive à
v
Vv~l, Vj~2,
h.(2)
=m.
2 ( x , z ) .
(3.6)
J
J+v+
n
n
Ce qui prouve le lemme 3.1 pour r=2,
i.e.,
19

---

-166-
,
,
, ' \\
1
J
''\\
J
' ' \\
v
1=h.(2)!m.
2(2 l ) ( x ,
z
)=(3(v,2)
(3.7)
J
J +v+
-
n
n
Dans le cas général, :on peut encore intégrer par parties en utilisant
(3.2)
pour obtenir
i
v
v+1
v-1
h.(r)=h.
l(r-1)+h.
l(r).
(3 .8)
J
J+
J+
Raisonnons par récurrence sur r
et supposons que les propositions énoncées
1
1
dans le lemme 3.1 soAt vraies jusqu'à l'ordre r-1~2, i.e. i
1
v
!
Vj~l,V~~l, (3(v,r-1)=h. (r-1)!m.
2(
2) (x ,z!).
(3.9)
i
J
J +V+
r-
n
n
1
i
En répétant alors l'application de
(3.8)
jusqu~à ce que l'indice superleur
du _dernier membre de
(3.8)
soit égal à zéro,: on s'aperçoit que pour
tout,; j~l, pour tout v~O, ht:'(r)!m.
2(
1) ex ,z )=(3(v,~) ne dépend
J
J +v+
r - n
n
nurlement de j
et que la formule suivante est vraie.
h=v-1
V~ ~1, Vv~l, (3 (v, r) = l
(3 (v - h, r -1) .
(3.10)
h=-l
Ceci prouve que (3(v,r)
est bien entier pour v~O et r~3. En outre,
combiné,
avèc
(3.7),
i l prouve les propositions du lemme 3.1 par récurrence.
Tirons de
(3.10)
les formules de récurrence énoncées pour les nombres
de type 1.
La formule suivante est immédiate à partir de
(3.10).
Vr~3,Vv~2, (3(v,~)=(3(v+1,r-1)+(3(v-1,r)
(3 .11)
De plus,
on montre aisémént que pour r~3,
j~l, on a
a
1
h.(~)=h. l(r-1)
(3.12)
J
J+
si bien que
Vr~3, Vj~l, (3(O,r)=(3(l,r-1)
(3 . 13)
En appliquant maintenant
(3.2)
et en intégrant par parties,
on obtient
, 1
2
a
V~~3, h. (r)=h. l(r-1)+h. 1(~)
(3 . 14)
J
J+
J+
ce qui,
combiné avec
(3.12)
et
(3.13),
implique
Vr~3, (3(l,r)
= (3(2,r-1)+ (3(l,r-1)
(3.15)
Pour assurer la cohérence de ces nombres,
on pose par convention que
Vv~O, (3 (v, 1) =1.
(3 . 16)
En résumé,
(3.17),
(3 .11),
(3.13),
(3.15)
et
(3.16)
montrent que les
nombres de type l
sont générés par les rapports ht:'(r)!m.
2(
1) (x ,z ),
J
J+v+
r-
n
n
20

---

-167-
, ' 1
1
..
1
J
.",
J
''',
j ~1, v~O , r~l .
)REUVE DU LEMME 3.2.
~n intégrant par parties selon (3.2), on obtient
1
v
v-1
.
O' (O,T)
O'
1 (O,T),
J~1.
(3.18 )
J
J+
)n vérifie
aisément aussi que
a
o . (0, 0:) =m .
1:
1 (x ,z
)
(3.19)
J
:
J +T-o+
n
n
!
:eci à son tour implique
Vo,l~8~T, gT(O,O,o)=l.
(3.20)
i
~n combinant (3.18)
et
(3.20), jnous obtenons
1
Vv~O, gT(O,V,T)=l.
(3.21)
roujours un~ intégration par parties utilisant
(3.2),
donne
.
v
v
v-1
VJ~l,Vv~O, V1~0~T, O· (0,0)=0' 1(0,0+1)+0' 1(0,0)
(3.22)
J
J+
J+
~ partir de -là, les propositions du lemme 3.2 se prouvent par récurrence
sur T-r,
r
= 0,1, . . . ,T-1 en utilisant judicieusement
(3.22)
et en tenant
~ompte de la première colonne donnée au (3.21). Cette récurrence doJne
k=v
Vv~O, V1~0<T, «T(O,v,o) = L «T(O,k,o+l),
(3.23 )
k=O
~ui, à son tour, implique la règle du sabot suivante
Vv~l,V1~0<T,g (O,v,o)=«
(O,v-1,o)+«
(O,v,
0+1).
(3.24)
T
T
T
Nous avons déjà prouvé que les nombres de type II(T)
définis dans la
jéfinition 2.2 sont bien générés par les intégrales o~(O,o)
J
PREUVE
DE
LEMME
3.3.
En intégrant encore par parties selon
(3.2),
on a
v
v+1
v-1
0j (1,2)
0j+1 (0,1)
+ 0j+1 (1,2),
j~l, v~O.
(3.25)
En répétant
(3.25)
v fois et en utilisant
Vj~l, 0~(l,2) = o~ 1(0,1)
(3.26)
J
J+
et finalement,
en appliquant le lemme 3.2,
nous arrivons à
o~(l'2)=[ k=~+~ (O'k'l)] m.
2(x ,
z )
(3.27)
J
L
T
J+V+T+
n
n
k=l
Cela prouve les proposition du lemme 3.3 pour 0=2 avec
v+1
VV~O, g
(1, v, 2)
I gT(O,k,l)
(3.28)
T
k=l
A partir de là,
la preuve suit les mêmes étapes que celle du lemme 3.1.
21

---

".
"
,
..
1
1
-
) '4,
)
4,
TROISIEME PARTIE.
CONTRIBUTION A L'ETUDE DES ESPACEMENTS.

---

-168-
ON THE INCREMENTS OF THE EMPIRICAL k-SPACINGS
PROCESS FOR FIXED OR MOVING STEPS.
Gane Samb Là (*)
Abstract. Several strong limit theorems for the oscillation moduli of the empi-
rical process have been recently given in the iid-case. We show that, with very
slight differences, those strong results are also obtained for sorne representa-
tian of the reduced empirical process based on the (non-overlapping) k-spacings
generated by a sequence of inde pendent random variables (rv's) uniformly distri-
buted on (0,1). This yields weak limits for the last cited process. Our study
includes the case where the step k is unbounded. The results are mainly derived
from several properties concerning the increments of gamma functions with para-
meters k and one.
Key words. Oscillation modulus, empirical processes, increments of functions,
law of the iterated logarithm, order statistics.
\\
(*)-Rese~~ch address. L.S.T.A., T.45-55, E.3. Universfté Parîs VI. T.45-55.
4, Place Jussieu. F-75230, Paris Cédex 05. France.
~. (
1

---

-169-
- 195 -
1- Introduction and statement of the results.
Consider U , ... , Un a sequence of independent rv's uniformly distributed on
l
(0,1), and let U
=Ü~l
~... ~
~
1
=1 be their order statistics. The rv's
o , n
-
, n-
- n, n- n+ , n
D~ = U - -U(-_l)k ' l~~[n+klJ=N, where [x] denotes the integer part of x, are
l,n
kl,n
1
,n
called the non-overlapping k-spacings. Throughout, we shall assume that N and k
are given and that n is defined by n=inf {j, [j~lJ=N} and then we will be able
to study all our sequences as indexed by N since k will be either fixed or func-
tion of N.
The study of the properties of D~
was introduced by Pyke [8J and several re-
l,n
lated papers have appeared in recent years (see e.g. [3J). One of the problem
cocerning the k-spacings is the study of the empirical process associated with
NkD~ , 1~.
l,n
- -
In order ta give a comprehensible definition of that process, we recall the fol-
lowing representation which can be found in [lJ in the case where (n+1)/k is an
integer:
(11) {Dk
1
N}d {y /S
l<·<N} __ ·{("j=ik
E )/S
1: N}
.
i,n'
~i~
=
i
n+l'
=1=\\
-.
Lj=Ci-l)k+l
i
n+l'
~l~ ,
i
where ~ denotes the equality in distribution and S is the partialstim associa-
n
ted with El' ... , E , a sequence of independent and exponential rv's with mean
n
one, i.e., S =E + ... +E . Thus, it follws that, if (n+1)/k is an integer, the li-
n
1
n
miting distribution function of Nk D~
, for any i and k fixed, is
l,n
k-1
- t
t
e
(k-l)!
dt,
x~o.
Therefore the empirical process (E.P.) associated with NkD~
, l~i~N, may be
l,n
defined by
l
0.2)
2
BN(x) = N
{FN(x) - I\\(x)}, o~x<+oo,
where FN is the empirical distribution function (E.D.F.) of NKD~ , l~i~N, with
l,n
0.3)
"c. r
"c ( ,
1
.'

---

-170-
Straightforward mçmipulations from (1.1), (1.2) and (1.3) as given in [Q show
that even in the general case where (N-l)k~n+l~Nk, the reduced process aN(s)=
8N(H~I(s)), O~s<l, satisfies
0.4)
{aN(s), O~s<I}~ { N!{~N( 0nIÇl (s) )-s}+O(N-!), O~s<l}},
where ~1 is the inverse function of ~, ~N is the E.D.F. pertaining ta YI' ... ,
Y and 0 =S
I/Nk.
N
n
n+
The aim of this paper is to give the behavior of the oscillation modulus of
U (.) bath where k is fixed and where kt+oo. To this end, we define
N
AN(aN, RN) = sUPO<h<a sUPO<s<l_hIRN(s+h)-RN(s)1
==N
==
-1 ~
and
KN(a , RN) = AN(a , R )/(2a logloga
) ,
N
N
N
N
N
for any sequence of functions RN(s), O~s~1 and for any sequence (aN)N~I' O<a <I.
N
The properties of AN(a , ~), the oscillation modulus of RN' have been first
N
described by CsHrgo and Révèsz [21 and Stute ŒOJ when RN represents the E. P.
pertaining to a sequence of independent and uniformly distributed rv's with
-1
-1
(SI) NaN+~' (S2) (loga
)/(NaN)+O, (S3) (loga
)/loglogN ++00, as N++oo.
N
N
Later·, Mason, Shorack an,d Wellner (M-S-W) [7J dealt with the same for several
1
l,
\\
choices of (aN) and give among the results an ErdHs-Renyi law .
. The chief achievement of this paper is the extension of those limit results
ta sorne sequence of processes
aN
equal in distribution to aN for each N. From
there we derive weak characterizations of the increments of aN' In fact, the
fundamental role is pla~ed here by the properties of the tails of the gamma func-
tion Hk(.), the derivat~ve function of H . These properties are established in
k
Section 2 through technical lemmas and the proofs of the following r~sults are
given in Section 3.
'" (

---

-171-
TheoremA. Let k be fixed. Then, there exists a sequence of processes ~(s),
QSs<l, N=l, 2,
... such that
(0) Ji- rel, {~(s), O~s<l} ~ {~(s), O~s<l}.
(1) If (a )N>l is a sequence of non-decreasing numbers satisfying the CsHrgo-
N
Révèsz-Stute conditions (SI), (S2) and (S3), then lim KN(a N, ~)=1, a.s.
Nt+=
-1
-
1- +
(II) If aN=cN
10gN, c>O, N~l, then lim KN(a , ~)=(c/2) (B -1), a.s., where
N
Nt+<'"
+
+
-1
B (10gB -l)=c
-1.
l
l
(III) If a =(logN)-c, c>o, then c2~lim inf KN(a , aN)~lim sup KN(a , aN)~(1+c)2, a.s.
N
N
N
- N+-f-'X'
N4+00
-1
-1
-1
(IV) If aN=cNN
10gN, cN~ such that (cNlogN)=NaN~ro and (logN)
(logcN )log10gN
~ as Nt+oo, then
l
N2 10g0!c )
N
Hm sup
10gN
Nt+ro
Theorem B. If k=k(N)~ro such that for sorne 0>2 and for sorne NO'
k( 0-2)
0
O<aN~tk(6)=k
exp(-k 12), N~NO'
then Parts (1), (II), (III) and (IV) rernain true.
Remark A. If each ~ is the spacings E.P. based on a sarnple depending on N, say
\\
X(N), and ~f these samples X(N), N=1,2, ... are mutually\\ independent (this statis-
tica1 situation i5 quite conceivab1e, for instance when checking homogeneity)
the strong limi t results of Theorem A are also va1id for ~. One might seek other
conditions to get the same extensions. Here, we restrict ourselves to weak exten-
tions in the fol10wing
Corollary.' Let k be either fixed or \\c++OO. Let (aN)N>l be a sequence of posi ti ve
numbers suchthat ~aN~tk(o) when ~O' for sorne NO and!o>2. Then
1
~
1
'"
r
, :
1

---

-172-
(1) Under the assumptions of Part 1 of Theorem A, we have
1 in probability,
(II) Under the assumptions of Part II of Theorem A, we have
1.
+
lim
KN(aN'~) = (C/2)2(8 -1) in probability,
~
(III) Under the assumptions of Theorem A, we have
1.
lim KN(a ,
N aN) = c 2 , in probability,
N-++oo
(IV) Under the assumptions of Part IV of TheorernA, we have
1
N2 log(l!c )
N
lim
P( ----:-----'--
a for all 1::>0.
logN
N+-f'X'
Remark B. It appears from Theorem A and B that the oscillation rnodulus of uN and
that of the uniform ernpirical process are almost the same. In [sJ, we prove that
the exact strong bounds in (1) and (II) remain for aN when aNsatisfies further
conditions.
Rernark C. One might think that deriving the results of our corollary by using
invariance principles ( asgiven in [lJ land [SJ) and well-known results for the
1
Brownian bridge would be easier ( at least for sorne sequences aN)' This is not
true at aIl ( see Remark D).
2- Technical lernmas.
It will follow from Lemma 1 of Section 3 that the increments of aN behave
as the increments of yN(\\jJ(.)) and thoseof t!>(.) where YN(') is the E.P. pertai-
ning to Ul' ... , UN' I)J(S)=Hk(JJ IÇl(S)), ~s~l, \\.ln~n' n=1,2, ... , t!>(s)=Hk(H~~s))H-k~s)"
n
~~1, with Hk(x)=dHk(x)/dx, for aIl positive x. Then, since K (., Y ) is kIlown,
N
N
our study is reduced to describing the increments of I)J ( .) and that of <t> ( .), what
we do in this paragraph.
r,
"
\\
(
,,,'
\\

---

-173-
Lemma Al. Let k be fixed and a=aN be a sequence of positive numbers satisfying
1
(Ql) (n-lloglogn) 210g(1/a) -+ 0 as N-++oo and a-+O,
then, as N-++oo, we have the fo110wing properties
(l·) Sup
suPO
l hl,l,(s+h)-\\lJ(s)\\=a(l+o(l»), a
O<h<a
~s~ -
'l'
• s. ,
= =
- -
(ii) Uniformly in s, O~s~l-a,
1\\IJ(s+a)-\\lJ(s)!=a(l+q(a», where q(a)-+O, a.s., as a-+O.
Proof of Lemma Al.
We need several properties of gamma functions. First note that for a fixed k,
+ ik-l)!}
k-l '
x
and
(2.1b) x=H~l(l-s)=lOg(l/s)-log(k-l)!+ (k-l)logX+log(1+k~l + ... ik-kl_)l! )
x
from k-l integrations by parts. Next for a fixed k or for k-++oo, we have, asx+O,
k
(2.2a)
s=Hk(x)=x /k!(l+O(x»,
and
(2.2b)
X=H~l(s)=(k!)l/k(l+O(X/k»,
"vhere for any function g(.), g(x)=O(y) as x+O means that lim sup Ig(x)/yl<+ro.
i
xro
\\
To see this, use the fo11owing inequa1ities:
-x
- t
O<t<x => e
<e
<1
==
=
=
to obtain that e -x xk/k!~ Hk(x) ~xk/k! and the results follow. Now, we are able
to prove Lemma Al.
Proof of Lemma Al. (continued). Define
(2.3) ~h(s)=\\IJ(s+h)-\\lJ(s), O~~l-h, O~~, n=1,2, ...
Straighforward ca1cu1atidns give
(2.4) d'flh(s)/ds = u~-l {exP((Un-l)H~l(s+h»-exP((lJn_l)H~l(s»}.
Thus, for each e1ementary event w of the probabi1ity space, for each N (that is
to say for each n) and for each h, 'fI h(·) is non-decreasing or non-increasing
" (
~ " "

---

-174-
according ta the sign of U (w)-l. Thus we have
n
(2.5)
Computation of ~h{l-h). Using (2.1), with h=l-~(x), we have
-1
.
(k-l)1
U H
O-h)=jJ 10g(1/h)-U 10g(k-1)! + \\.l (k-l)logx +jJ 10gO+ ... + k 1')·
nk
n
n
n
n
-
x
-1
Now recall that ~h(l-h)=l-Hk(jJn~ (l-h»
and, using (2.1a), get
\\.l
jJ
k-1 (log(l/h»k-l h n «k-1)!) n -(k-1)jJ
(2.6)
~h(l-h)=jJn
(k-1)!
x
n (l+q1(h»,
where there exist A and Bk depending only on k (k being fixed) such that
k
-1
-1
(2.7) Iql(h) 1~Akx
logx + Bkx
,as h~ (i.e. as x7+oo).
These constants A
and Bk are provided by the approximation
k
Now since jJ +1, a.s., by the strong law of large numbers, we obtain from (2.6)
n
and (2.7) that
jJ
(k-l)(l-lJ )
-(k-l)lJ +1
(2.8) ~h(l-h) = h D(log(l/h»
n (1+q2(h»
n
(1+Q3(N»,
where Q2(') ratisfies (2.7) with the same constants A add Bk and 43(N)=0(1),a.s.,
k
-1
-1
independently of
h, O~~l, as N++a>. Since the functions x
logx and x
are
non-increasing as x-+t-oo, it follows from (2.6), (2.7) and (2.8) that
u
(k-1)lJ
(2.9)
Jf
1
0j1~, '}IhO-h)= (1+q(a»h n(log(l/h»
n
where q(a)+O as a+O and
N~ By convention, we shaH write g(h)=q(a) for O~h~a,
1
i f for aIl h,! O~h~a, g(h)=oO) where the "00)" depends o~ly on a, as a-+Û.
,
Computation of ~~.
1 We have ~h(O)= ~(jJn~l(h», and using (2.2), we obtain
1 (2.10) jJ 1
H-
(h)= jJ (k!)1/k h1/k (1+q(a», O~;;'a.
n-1<
n
1
1
"'r
'"
r ,
1

---

-175-
Use again (2.2) and get 1\\(~nIÇl(h))=h~~(l+q(a))=h(l+q(a)),a.s., 0.9l~-+O, sihce
k is fixed and ~ +1, a.s., as N++oo. Then,
n
~
1 (k-l)(I-~ )
(2.11) suPO~ sUPO~s~l_h ~h(s) = (l+q(a)) sUPO~h~amax{h, h n(logh- )
n },
a.s., as N+~. But,
~
(k-l) (l-~ )
~
1 (k-l) (l-~)
(k-l) (l-~J
(2.12) V- N~I,d{h n(10gh- 1 )
n }/dh = h n(logh- )
n {~n~
- 1 } '
log h
~
1 (k-l)(~ -1)
Thus h n(logh- )
n
is non-decr.easing for n sufficient1y large since
k(I-~ )+0, a.s., as N++oo by the strong 1aw of large numbers (k being fixed).
n
Then,
~n
-1 (k-l)(I-~n)
~n
-1 (k-l)(I-~n)
( 2 . 13) O~h~a
>
h
(logh
)
~ a
(loga
)
, a.s., as N++=.
Furthermore,
-1 (k-l)(I-~n)
-1
(2.14) (loga
)
= exp((k-l)(I-~ )logloga
) = 1+0(1), a.s.,
n
1
whenever (1-~ )10gloga- +O, a.s., but this is imp1ied by (QI). Indeed, we have
n
by the 1aw of the iterated 10garithm ( the 10glog law ) that
-1
-! 1
1
(2.15) 1im sup (2n
10glogn)
~n-l ~1, a.s.,
N-+t=
-1
which combined with (QI) implies that (l-~ )logloga
-+O,-a.s., as N++oo. In fact,
n
the 10glog 1aw ho1ds for 0 , that is
n
-1
-! 1
1
\\
(2.16) 1im sup (2n
10glogn)
0n-1 ~1, a.s.
N+t-<o
But (2.16)may be obtained from (see [5] , Appendix)
n +C 1
-1
L P( Pu {(2n 10glogn) IOn-ll~I+E/2})<
p6>
n p
where (n ) is an increasing and unbounded sequence of positive integers and [>0
p
.
is arbitrary. This and the equa1ity in distribution of 0
and ~
for each N imp1y
n
n
(2.15). The same 10g10g 1aw shows that (QI) imp1ies that
1-~
1
(2.17) a
n
eip(il-~ )10ga- ) = 1+0(1) as N++oo.
n
We fina11y get from (2.9), (2.13), (2.14) and (2.17) that
(2.18) sUPO~h~a sUPO~s~l_h l~h(s)l= a(l+o(I)), a.s., as N++oo)
\\ ' r
1
~ " .'

---

· '
'"
-176-
,
)
whicn proves Part (i) of Lemma 1. To prove Part (ii), it suffices to remark that
we may have trough (2.4) that min(<I> (0), <1> (1-a) )~<I> (s)Snax(<I> (0), <1> (1-a)), 0__
<
a
a
- a
-
a
a
~l-a, and the part in question fol10ws since the first part implies that ~ (0)
a
=a(l+Q(l)): a.s. and <l>a(l-a)=a(l+o(l)), a.s., as N++oo.
Lemma A2. ~et k be fixed, then we have as a+O, N++oo,
sUPO~a sUPO~s~l_hl<l>(s)-<I>(S+h)I=(alOga-l)(l+o(l)).
Proof of Lemma A2.
Consider ~h(s)=~(s+h)-<I>(s), O~s~l-h. Direct considerations yield that d~h(s)/ds=
-1
-1
H
(s)-~ (s+h), O~s~l-h. Then for each h, ~h(') is non-increasing and thus,
k
sUPO~s~l_h1~h(s)1 = max{ 1~h(0)1, 1~h(1-h) I} .
But, by (4.2), ~h(O)=Hk(~l(h))~l(h)=kh(l+q(a)),O~h~a~. Here we omit the
details concerning the uhiform approximations which provide q(.). These netai1s
are very simi1ar to those of the computation
of ~h(O). Using the same conside-
rations previously as guetting (2.6) from (1.1), we have
~h(1-h)=Hk(~1(1-h))H~1(1-h)=(hlogh-1)(1+q(a)),O~h~~.
Notice that Hk(H~l(l-h)) yie1ds something 1ike (2.6) whi1e H~l(l_h) yields
{logh~l)(l+q(a)), O~~~. We then obtain
- 1 - 1
sUPO<h<a sUPO<s<l_h 1~h(s) 1=(l+q(a)) sUPO<h<a max(kh, h10gh
)=(l+q(a)(aloga
),
= =
= =
= =
a+O, since k is fixed here. Hence Lemma A2 is proved.
Now, we concentra te on the case where k++w. First, we give the fo110wing
Proposition.
k(0-2)
0
Let O~s~tk(o)=k
exp(--!k ), 0>1. Then, as k+t<x>, we have
-1
-1
(2.19)
x=H
(l-s)=(logs
)(1+q4(s)),
k

---

-177-
:
1
1
, 1
1
):"
)
':"
where there exist A and k
such that Iq4(s)I~A l~~î for aIl O~s~tk(o), k~kO'
O
k
Proof. Integrating by parts, we get
k-l -x
k-1 -x
Jf ~O, (k-l)!
~+-Hk(x)~(k_1)!
Then,
k-1 -x
' k
1
k-1-x
(2.20)
x e ) {
}-
x
e
(k-1)!
~l~Hk(x~ 1- ;
(k-l)!' Jf ~O.
We are ab~e to see. that the expansion of Hk(x) is then possible if k/x+O. Now,
o
let O~s~sk=l-Hk(k ). Apply (2.20) and get
k-l -x
O~~sk >s=l-Hk(x)= (k-1)! {l+qS(x)},
-(0-1)
-(0-1)
with 1qS(x) 1~(l-k
.)k
for all O~s~sk·
\\
Then by Sterling's formula and sorne straighforward calculations, it is possible to
find a kr such that tk(O)=kk(O-2)exP(-~kO)~skfor aIl k~k1' Then for O~s~sk' k~k1'
-1
-1
(2.21) x=H
(l-s)=(logs
)-log(k-1) !+(k-l)logx + o( qS(s)).
k
Now since
O<s<s =-J (k-l)logx 1 <
logk
= O(k-(O-l)logk),
= = k ' 1
x
=
0-1
k
O<s<s =-J log(!c-l)! 1 < logk! = O(k-(o-l\\ogk),
= = k '1
x
=
kO
by Sterling's formula. Thus, these two facts and (2.21) together imply that
-1
-(0-1)
-1
-0+1
logs
= x(l+O(k
logk) = H (1-s) (l+O(k
logk)) ,
k
which was to be proved. We finally give two lemmas which correspond to Lemmas
Al and A2 in the case of infini te steps k.
Lemma A3. Let k satisfy
1
(K) kN- (10glogn)+O, and k=k(N)~ as N++oo,
and let
Then the following assertions hold.

---

-178-
(i) .. sUPO~h~a sUPO~s;g_h IW(s+h)-W(s)\\= a(1+o(1», a.s., as N-+t=.
(ii) IW(s+h)-W(s)1= a(l+q(à», for O~s~l-a, with q(a)+O, a.s., as N++ro.
Proof of Lemma A3. As in Lemma Al, we have
(2.22)
sUPO~s~l_h l~h(s)l= max{ l~h(O)I, l~h(l-h)D.
First we treat ~h(O)=Hk(~nH~l(h». Equations (2.2) yield
1
l k
IlnH~\\h)=(k!) /k h /
(l+o(H~l (a) /k) )Il , O~h~a.
n
Now we note that ~s~a implies that 0~~1(h)~Cla1/k(k!)1/kfor small values of
a, Cl being a constant. Sterling's formula then implies for large values of k,
O
0<H-1(h)<C
ko-l
(.!k O- 1)
~~tk'
==K
= onst.
exp -2
.
-1
Then H
(h)+O andwe are able to use (2.2) to get
k
-1
k
-1
.lf O%a => H (ll H
(h»= Il
h (l+o(H
(a»
as a-+O.
k
n k
n
k
1
The loglog law implies that k(l-Il )=O(k(2n- loglogn)2), a.s. Thus, whenever (K)
n
k
is satisfied, one has Il =exp(-k(l-W )(1+0(1»+1, a.s. Hence
n
n
We now treat ~h(l-h). By ~he Proposition, we get
\\
(224).Jf{)~~tk(o), X=:llnlÇl(l-h)=lln(1ogh-1)(l+O(1~~~) ), a.s.
k
Since X/k=ü(k-(O-l», a.s., one has
-1 k-1
Il
-(k-l)1l
Il
(2.25 )l-H (X)=ll k- 1 (logh)
(1+0( 1~_gk2 »h n x
n «k-l)!) n (l+q4( h» ,
k
n
(k-1)!
k\\J
-1
a.s., as N+~. Replace x by logh
in (2.25) . On account of (2.24) and of the
1-0
k-1
2-0
fact that (1+0(k
logk):) = (l+ü(k
logk» ,we get
1
(2.26)
Finally, by taking (K) and (Q2) into account, we find ourselves in the same situa-
1
tion as in the proof of Lemma Al ( see Statement (2.8». But in order to have
i
1
"
r
"
r
1

---

-179-
the same conclusion, i.e.,
(2.27)
suPO~~ 'P (1-h)=a(1+q(a)), a.s., as N+t-=,
h
we have to verifythat «k-1)!)1-lJ n =exp«(1-lJ )log(k-1)!)=:p -+1, a.s., as N-+t-=.
_
n
n
But the loglog-law and Sterling's formula together show that
l
l
k4 (loglogn)ï
-1
-1
p =exp( 0«
r
loga
) (logk/loga »)).
n
N1
Obviously the condition O<a~tk(0) implies that logk/loga-1 -+ 0 as N-+t-CXl, and as
k-++CXl • This fact combined with (Q2) clearly shows that p -+1 as N-++oo. Now putting
n
together (2.22), (2.23) and (2.27), we get
(2.28) (O<~tk(O), 6>2) - ) sUP09t~a sUPO~s~1_h \\'Ph(s) 1= a (1+q(a)), a.s .. as N-+4-00.
Lemma A4. Let 0<~tk(6), 0>2. Then as k++CXl , we have
suPO~ sUPO~s~l_hl~(s+h)-~(s)I=(alOga-1)(1+q(a)), q(a)+O as N-++=, a+O.
Proof of Lemma A4. If we proceed as in Lemma A2 and as in Lemma A3, we get
sUPO~a sUPO~s~1_h l~h(s)l=
1
max(ka, (aloga- )) (1+q(a)), as N-++oo, a++O.
From there, the conclusion is obtained by noting that the condition O<a~tk(o)
-1
\\
implies that (loga
)/k-+OO as\\k-+~.
3- Proofs of the theorems.
l
Throughout, we shall use the following representation which follows from
(5) ( see e.g. the study of R (x)).
N1
1 Lemma 1. Let k be fixed or k++oo as N+~, then
d
{
1.
1
-
ClN(s), 0~s<1l={YN(I\\J(s»+N2{Hk(lJnH~ (s))-s}, 0~S<1}:=:{ClN(s), O~s~l}, a.s.
1 Lemma 1 will be sytematically used. Then, if aN satisfies
2
-1
1 (QI) lim (loglogn) /NaNloga = 0,
N
N+~
l
1 we will be able to focus our attention on YN(I\\J(s))+N2(lJn-1)~(s) in the following
il'1
il
~
::1
1
id
", r
ri
•
'-o.
r , ,",'

---

-180-
a.s.
=:ANl(s)+AN2(s)+AN3(s).
-1 ~
with b =(2a logloga
) =b(a ). It follows that if (Q3) holds we have A
(s)=0(1),
N
N
N
N
N3
a.s., uniformly with respect to s, O~s~l.
Proof of Part l of Theorern A. By (2.29), we have
1.
and by Lernrna A3, we have for a fixed k, KN(a , AN2)~2Il-Wnlb(aN)(1+0(1», a.s.,
N
asNtro. Thus the loglog-law implies that
whenever
1.
1.
(Q4)
k- 2(210glogn)2b(a )+O as N++oo
N
is satisfied. On the other hand, Lernma Al and Theorem 02 of Stute D-OJ
together
implies that
Then if (QI), (Q3), (Q4), (SI) and (S3) are satisfied, we get
Now let
By Lernma A2, we have for large N that
-1
l
(3.5) bN sUPO<s<l_
IAN2(s+aN)-AN2(s)I~N2Il-Wnlb(aN)(1+o(1», a.s.
= =
aN
Thus if (Q3) and (Q4) are satisfied, we get
Furthermore it may be derived from Theorem 0.2 of Stute [j.OJ that (SI), (S2) and
..,.
f
.., (

---

*181-
(S3) yield
(3.7) b~l sUPO~~_a IYN(W(s)+o(aN))-yN(w(s)+aN) \\=0(1), a.s., as N++oo.
- -
N
It follows from (3.6) and (3.7) that (SI-2-3) and (Ql-3-4) together imply
b~leN(aN' aN)~b~l{suPO<s<1_a yN(w{s)+aN)-yN(w(s))}+o(l), a.s. as N++oo.
==
N
Since
W:(O, 1-a )---->(0, W(l-a )), is a bijection and since W(l-a )=l-a (l+o(l)),
N
N
N
N
a.s., we may use Lemma Al ( formulas (2.4a) and (2.5a) ) when (QI) holds ta find
for any E>O, for any elementary event w , an N (w) such that
o
-1
N>No
>b N eNCaN' aN) ~ sUPO~s~l-aN(l+E)(YN(s+aN)-YN(s))+o(l),
a.s.
Once again, we use Theorem 02 of
[OJ to see that, under (Sl-2-3), we have
l
lim suPO< <1-
(1 E){ IYN(s+aN+EaN)-YN(s+aN)I/bN}~E2,
a.s.
N+-t""
~= aN +
Thus, under (Ql-2-3) and for large values of N, we get
-1
-
-1
l
(3.8) b
)-(I+t)E2 , a. s. ,
N eNCaN' aN)~bN eNCaN' YN
Hence Lemma 2.9 in uO] and (3.8) together yield
l
l
(3.9) Vt>O, lim inf KN(a , ~) ~ lim inf eNCaN' ~)~(1-E)2-(1+t)E2,
a.s.
N
N++oo
N++oo
\\
finally (3.4) and (3.9) together ensure that
lim
KN(a , ~)=1, a.s.,
N
N++ex>
whenever (Ql-2-3) and (Sl-2-3) hold. But since 10gn~logN (k being fixed), onehas
(i) ( loglogn)~ 1
-1 ~ k1-(loglogN )1-( ~l
-l)~/(N! t)
n
oga N
-1
aN ogaN
aN '
10gaN
1
". 1'1'.) (loglogN
)2 1
-1
(1
1
N)2/(1
-1)2
\\
.
-1
ogaN ~ og og
ogaN
,
NaNloga N
1 (:"') (21 1 )( 1 -1) (loglogN)( 11 -1)3/2
' H l
og ogn
aN oga N ~
-ï
aN ogaN
"
10gaN
1 for large N. (i), (ii) and (iii)"show that (SI) and (S2) imply (Ql-2-3) and
this completes the proof of Part 1 of Theorem A.
1
1
\\:
r
\\:. r
1
~ " '
, .'
1

---

0'
,:",
•
-182-
Proof of Part II of Theorem A.
The proof is the same as that of the first part. We only notice that if a =
N
1
cN- logN, c>O, (Ql-3-4) are satisfied for a fixed k. To get Part II of Theorem
A, we use Theorem 1 (Part 1) of [7] for the inequality ";;;" and the Erd8s-Renyi
law for the increments of the uniform empirical process due to Komlos and Al.
[41 for the inequality "~". Similarly to the first case, we get an anolog to (3.8).
That is, for any E>O, for any elementary event w, we can find an N (w) such that
1
l-
+
+
where for any s, h(s)=(s/2)2(B (s)-l) and B (s) is the unique solution of the
-1
+
equation x(logx- l)+l=s
such that B (s)~l. Now, since for for any f(.), g(.),
K, sup
Kmax(f(s), g(s»=max(sup
K f(s), sup
K g(s»
and Ixj=max(x, -x) and
se
se
se
since (see e.g. the third formula that follows Statement Il in [J]),
l
VE~,
lim
inf \\yN(s+a(l+E))-YN(s) I/b = (1+E)2h((1+E)c), a.s.,
N
N-+-+oo
then (3.10) implies that
\\ l
l
l
VOo,
lim
inf KN(a , ~)~(l+d2h((l+E)C)-2 2h(CE)-E2 , a.s.
N
N-+-+oo
Thus it suffices to prove that: (i) for each fixed c, h((l+E)c) -+h(c) as E-+O,
l
and: (ii) for each fixed c, E2h(CE) -+0 as E-?Û. But these two points may be direc-
tly obtained by simple
considerations.
Proof of Part III of Theorem A. The proof is very similar to that of Part l of
Theorem A. It suffices to remark that Part III of Theorem 1 in [7J holds in the gene-
-c
ral case where aN=a(logN)
,c>O, a>O.
Proof of Part IV of Theorem A.
-1
-t
-1
-1
Here 8 =c N 10gN, cN-+O as N-++oo. Let dN=N (logN)
10gc
On the one hand, wehave
N N
N
~ ., .

---

-183-
1
~
2
Q) (10glogn)2 (1
-1)-1 '\\"
(loglogN)
( IJ
1
N
ogaN
-1
2
cN og
(logc
)cNlogN + cN(logN) (1+0(1»
N
1
-1
(10glogN)2
-1
10gcN
=
10gN
(cNlogN)
(1+ log N + 0(1» ,
1.
-1 1. 2
-1
-1
-1
2
(y) «10glogn)2(a 10ga
)2) =cN(logcN)N
(logN)(loglogN)+cNN
(1ogN) (loglogn)
N
N
(1+0(1» .
Obviously (a), (S) and (y) together imply that the conditions of Part IV of
Theorem A, namely
-1
-1
(Wl) cN~' (W2) cNlogN ~+oo, and (W3) (logN)
(logc
)(10g10gN)~, as N~oo,
N
themselves imply the conditions (QI-3-4). On the other hand, we have
1
~ ~
1
_
(aNloga- ) (1og10gn) 2N 2logc-N
1
N
fi. (
) < d f\\ (
)
O(
)+O( loglogn 1
- )
dN N aN' aN = N N aN,AN1 +
l
10gN
ogcN '
k 2 10gN
a.s., as N~~, where we have used Lemma A2 and (3.1). Furtheri
-1
-! ~
-1
-1
3
(aNloga
)(loglogn) N 10gc
(loga
)(loglogN)(logN)
N
N
N
(QS)
\\!-
' \\ . , . 1
\\cN~O, as N-+t=,
k:2l og N
(log N ) ( k 2 10gN)
N
by the definition of aN and by (Wl) and (W2). Thus, as N++ro, we have
d
~N(aN' aN)~dN
N
i\\N(aN, AN1 )+0(l), a.s.
At this step, we apply Part II of Theorem 1 of [7J by using Lemma Al which is
true on account of (QI).
Proof of Theorem B. We shal1 omit the detai1s of the proofs df the different parts
1
that are the same as those of the parts of Theorem A. The only problem concerns
1
the bounds depending on k. However, this problem is solved by Lemmas A3 and A4.
1
1
"", (
1
: , , '
1

---

-184-
Rence we only provide the following remarks.
(RI) In our different choices of (aN)' we have that Na 7+00, as N7+00.
N
(R2) If a~tk(6), 6>2, then for any y>O, there exists k
such that
y
JI. k>k '
k-YN~ (Ntk)-l~(NaN)-l-+o as N-+-t=.
y
(R3) (loglogn)=(loglogN)(l+o(l)) and logn = (logN)(l+o(l»), as N-+-t=.
With these rernarks, it is easily seen, as in the proof of Theorem A that the
conditions (K), (Q2), (Q3), (Q4) and (QS) are satisfied at the same time with
the specific assumptions of each part of Theorem B as follows.
(a) (Q2) and (K) are always satisfied if a~tk. Indeed,
(K) kN-1loglogn~(kN-t)(N-tloglogN)-+O as N++ro by (R2)
and
-1
-t
-1
-t
-t
(Q2) k(loglogn)(loga
)~(N
kloga
)(N
10glogN)(Na )
+0 by (R2-3).
N
N
N
(b) In Parts l, II and III of Theorem B, the implication {(SI), (S3)}
>{(Q3),
(Q4) } is true whenever loglogn~ 10glogN ( see the lines that fol1ow Formula (3.9))
and (NaN)++OO
as N++oo, which are derived from (RI), (R2) and (R3).
(c) In Part IV, (QS) is true independently of the behavior of k.
Thus we may use Lernmas A3 and A4 in places of Lemmas Al and A2 in the proofs of
\\
!
Theorem A to get the results of Theorem B in the same way.
Proof of the Corollary. This isa direct consequence of Theorems A and B and of
Lemma 1. For Part III, the methods used in Part l of Theorem A must be repeated .
.!
t
-1 -~
Remark D. By letting (NloglogN)4(10gN) (2a loga
)
+0, it would be possible to
N
N
,
derive Parts l, III and IV of Theorem A from invariance princip les such as in
Cl] or [sJ. But the necessary amount of work would be unchanged relatively to
our method.
'1;. (
,
1

---

-185-
j
References.
/17
Aly, E.E.A., Beirlant, J. and Horvath, L.(1984). Strong and weak approxima-
tion of the k-spacings processes. Z. Wahrsch. verw. Gebiete, 66, 461-484.
/27
CsHrgo, M. and Révész, P.(1981). Strong Approximation in Probabi1ity and
Statistics. Academic press. New York.
/37
Deveuvels, P.(1984). Spacings and applications. Techn. report 13, L.S.T.A.,
Université Paris 6.
/47
Komlos, J., Major, M. and Tusnady, G.(1975). Weak convergence and embedding.
In:Colloquia Math.
Soc.
Janos.
Boylai. Limit theorems of Proba-
bility Theory,
149-165. Amsterdam, North-Holland.
/57
La, G.S.(l986). A strong upper bound in an improved approximation of the
empirical k-spacings process and strong 1imits for the oscillation modulus.
Techn. report 47, L.S.T.A., Université Paris 6.
/67
Loève, M.(1963). Probability Theory. Van Nostrand Comp. Inc. Princeton.,
3rd ed.
/77
Mason, D.M., Shorack, G. and Wel1ner, A.(1983). Strong limit theorems for
oscillation moduli of the uniform empirica1 prOCéSS. Z. Wahrsch. verw.\\
\\
1
Gebiete, 65, 83-87.
/87
Pyke, R.(1965). Spacings. J. Roy. Statist. Soc. Ser. B, 27, 359-449.
/97
Shorack, G.R.(1972). Convergence of quanti1e and spacing process with
application. Ann. Math. Statist., 43, 1400-1411.
/107 Stute, W.(1982). The oscillation behavior of empirical process. Ann.
1
Probab., 10, 86-107.
1
1
1
1
'"", r ,
,
'"", r ,
l'

---

1
-186-
1
GAUSSIAN APPROXIMATIONS AND RELATED
QUESTIONS FOR THE SPACINGS PROCESS.
Gane Samb Lô (*)
Abstract.
AlI the available results on the approximation of the k-spacings
process to Gaussian processes have used only one approach, the Shorack-Pyke one .
.!.
It is shown here that this approach cannot yield a rate better than (N/loglogN)-4
1
(logN)2. Strong and weak bounds for that rate are specified both where k is fixed
and where k~. A Glivenko-Cantelli theorem is given while Stute's result for
the increments of the empirical process based on i. i. d. r. v .' s is extended to the
spacingsprocess. One of the Mason-Wellner-Shorack cases is also obtained.
Key words. Spacings, empirical
process, oscillation modulus, strong and weak
approximation, or der statistics, gamma distribution and function, law of the
iterated logarithm.
Research address. L.S.T.A., Université Paris VI. T.45/55, E.3. 4, Place Jussieu,
:
F-75230, Paris Cédex 05. France.
\\:
r
~ .
\\:
r
"

---

J
,.,
':-1137-
1- Introduction.
The non-overlapping uniform k-spacingsare defined by
D~,n = Uik,n - U(i-l)k,n' 1 ~i~ [n~lj==N,
where
O=U
~Ul
<...~U
<U
1=1 are the order statistics of the sequence
O,n-
,IF
- n,IF n+ ,n
Ul"",U
of independent rand04 variables (r.v. 's) uniformly distributed on
n
(0,1) ; [x] denotes the integer ipart of x. The study of these r. v. 's have
i
received a great amount of atteption in recent years (see L~7,L~7,L!Q7and Lï~7).
~articularly the related empirieal process
l
8
2
N(X) = N {FN(x) - Hk(x)}, O~x<+oo,
k
- t
t e x
where FN(x) == #={i, l;Si;SN, NkD~,n-~ x}/N and Hk(x) =f~
(k-l)!
@,
r(x)dt
flX'L '
,
X being the indicator function\\of A, plays a fundamental role in many area~
A
in statistics (see L~7). AlI of its aspects are being described by various
authors ..
(i) For the convergence of statistics based on spacings, it is helpful to have
a Glivenko-Cantelli Theorem for F (.). Such results for the overlapping case
N
are available in /37.
(ii) The limiting law of the spacings statistics may follow from suitable
approximations of ~ to Gaussian processes. It is clear that the better the
rates of those approximations are the less restrictive the conditions on the
underlying r.v. 's are. Such approximations also yield Kolroogorov-Smirnov tests.
(iii) Finally, the oscillation modulus of ~ have been studied in /77 establis-
hing the weak identity of the oscillation moduli of SN and that 'of the
empirical ~rocess based on a sequence of i.i.d. r.v. 's.
1
Our airo is to give strong versions of the weak
charaeterization of the ose il-
1 lation moduli we have already given in 177. As to the approximation of SN
ta Gaussian processes, we will show that the rate given in /17 is, in fact,
1
1
1
1

---

,
,
-188-
J
"
,
"
J
. "
. a strong one. Our best achievement is that this rate is the best attainab1e for
the approach used unti1 now and we pro vide the corresponding bounds. Withrespect
to /17 and /27, we do not let k fixed. We allow it to go to infinity. Fina1ly
we give the Gltvenko-Cantel1i theorem for F
with almost the same condition as
N
~n /37 for the overlapping case.
A- The Gaussian approximation.
Approximations of SN to Gaussian processes are availab1e since /Ï27. The best
rates among those a1ready given are due to /17 and to /27. Among other results,
L~7 proved the following theorem and coro11ary.
Theorem 1. There exist a probability space carrying a sequence U , U , '"
of
l
2
\\
i~dependent r.v.'s uniform1y distributed on (0,1) and a sequence of Gaussiàn
processes {WN(x), O~x<t=}, 14=0:1, 2, .,. satisfying
-1
(S) Ji- N>l, E(WN(x)WN(y))=min(Hk(x), Hk(y))-Hk(x)Hk(y)-k
xyHk(x)Hk(y)
such that
whenever k is fixed. Here Hk(x)=dHk(x)/dx.
1 Remark 1. From now on, we will say "according to the wording of Theorem 1" at
the place of "Ther~ exist a probability space ... such that".
1 Definition. A Gaussian process whose covariance function is given by (S) will
1 be called a Shorack process of parameter k or a k-Shorack process.
Corollary 1. According to the wording of Theorem l, we have
1
1.
_1.
1.
N4 (logN)2 (log1ogN)-4 sUPO~x<t= IS (x)-W
N
N(x)I=Op(l), as N~.
1
o
3/4 _1.
.l-
This means that aN=(logN)
N 4 is a strong rate of convergence while a =(logN)2
1
N
,
1
1
(210g10gN)4N-4 is a weak one. In fact f17 has shown
1
1

---

l
,
J
. 4,
J
'4,
-189-
1
Theorem 2. There exist an other sequence of processes B , N = 1,2, ... and a
N
1
sequence, of k-Shorack processes W , N = 1,2, ... such that, for k fixed,
N
(i) B~ ~ B , V N>l
N
1
'
1
(ii) sup
IBN (x) - W (x) la,;,s·O(a ), as N++=.
N
N
O~<-roo
1
1
AlI these results ~re based on representations of spacings by e~ponential r.v. 's.
1,
Namely, when n+l = kN,
k
(11)
d { ,j=ik
{D.
,1~i.s.N}
l,n
=
CL. (. 1)kE .) 1 SI'
J= 1-
J
n+
=:{Yi/Sn+1, l~i~},
. where El' E , ... is a sequence of indepenàent exponential rv' s with mean one and whose
2
partial SUffiS are SA, n~l. If llN=on=Sn+/ Nk itfol1ows that
d
l
l
(12) {BN(x), O~x<-roo} = {N 2 (~N(UNx) - Hk(x)) + 0(N 2 )}
*
= {AN(x) + RN(x), O~x<~} -. {BN(x), O~x<+oo},
where ~N(.) (resp. A (·)) is the empirical distribution function (resp. empirical
N
process) based on YI'.'" YN. The ci ted resul ts are deri ved from simul tanuous
approximations of AN and RN'
First we assert that the best rate attainable through this approach is that of
117 even when k~.
Theorem 3. According to the wording of Theorem 2, for any k satisfying
1
-0
(1) J 00>0, V 0<0<00' kN
-> 0 as N+~,
wehave
JK(k)=(kk+~ e-k/k!)~, (k fixed)
-1
lim sup aN
sup
*
*
l B (x) - W (x)la,;,s.
N
N
1
l
K = ( 2 TI ) - t,
( k + +(0) .
N++=
0.'S-x<oo
O
1
Our secund result is an improvement of Theorem 1 of 127.
l
1
t

---

1
,
,
,
J
' "
-19'0-':' J '"
1
1
Theorem 4.
According to the wording nf Theorem 1. we have for any k such that
-ho
'
4 0 1
for sorne 0 , o<oo<i. kN
+0 as; N+t=,
0
-1
{K(k)
(k fixed) ,
lim sup aN
sup
ISN(x;)-WN(x) 1 ~ K
•
a.S .•
N4+<o
O~x<+""
0
(k +
-f-OO),
Proof of Theorern 4.
d
*
From (12), we have SN = SN
for aIl N~l. Furthermore,
* . !
-~
SN(x) = i\\N(x) + N:I (Hk(lJ x ) - H (x)) - (i\\N(lJNx) -AN(x)}+O(N
)
N
k
=:i\\N(x) + RNl(x) + R
(x) + R
(x).
N2
N3
We shall proceed by steps, approx~mating each of the RNi's.
Lemma 1.
Let N = (l+P)~' p>O, p= 1,2, ... ,où and
p
N=N
Cl
l
p
C = U +
(sup
IR
(x) - N2 xHk(x) (uN-1)I>sBNK(k):4}.
Nl
Np N= N
O<x<-f-OO
p
=
Then if k/N~ as N+-f-OO, l P(C
)<-f-OO.
N
p
p
PROüF.
Apply the mean value theorem twice and get
(13)
( )
~(
)
' ( )
t(
.22
- l ) x H, , ( )
ANI = RN1 x - N lJN- l xHk x = N U
k x
N
N '
Where o<jxN/xl<max (l.lJ ). First, it may be easily seen that
N
xHk(x)
kt+ke-k
2
( 14)
K(k)
,
sup
l
k!
O~x<-f-OO k:l
(15)
lim
sup
lx Hk(x)/k~1 = K;.
k-+-f= O<x<-t=
and
(16)
OkM = sup
sup
Ix2Hk(x)/kl <+00.
~ 1
Q:Sx<+oo
Recall that for aIl E>O.
(17)
l P(max (1.lJN»l+d ~ l p( IlJNI>l+d< -t=
N
N
by the strong law of large nurnbers (SLLN) and
N=N
-1
(8)
Ip ( U p+l
P
N=Np
by th~l~w of!~~.iteraten.logarithm
(loglog-law~. We show in the Appendix

---

1
-191-
1
how to adapt the c1assica1 SLLN and log1og-law to these cases.
1 Now by (13), (14) and (15)
N=N
1-1
p+
1 (19) P(C t~ ~
P(max(l,
Np
N=NP
I I I
l
1 with c=EK(k)2/4M(l+E), e = (loglogN)4 N4(logN)2 (21og1ogNk)-2. But
N
2
loglogNk = (logN)(l+o(l)), K(k) is bounded and thus ce >(l+E)
for large N.
N
1 Thus we can app1y (17) and (18) to (19) and this completes the proof.
1
Lemma 2.
Let
e;>O and
1
N=N
-1
O
= { li p+1
(sup
IR (x)! > (1+E/4)aN K(k)}, p=1,2, ... Then for any
N
N2
p
N=N
~<.p:x>
p
r
k=k(N) su ch that k/N-tO as N-+-tOO,
~ P(D
) < +=.
N
,
p
p
PROOF.
The mean value theorem implies
(120)
IHk(~Nx) - Hk(x)1 ~ I~N-1IK(k)2 max(l, WN)k!.
Proceeding similar1y to (19), we get
N=N
-1
1/3
(121 )
PCD
) ~
~ p+1
P(max(1, W »(l+E/4)
)
N
N
P
N=Np
N=N
1-1
p+
+ P (
U
{
sup
N=N
!Hk(x)-Hk(y) l<c
p
N
: = R
+ R
N21
N22
(122)
\\:
r
, .
,,'
\\:' r
, .

---

-192-
1
where b
= (2logl08 N)2
2
2/3
K(k)
(1+E/4)
. Let Y (.) be the empirical process
N
N
N
based on Dl""
UN and P
be the secund terro of the right member of the ine-
Np
quality (122). Thus (11) implies
N=N
1-1
(123)
P
~ P( U p+
{
sup
NP
N=N
O~u~l-bN
p
- -
_11.
1'3
where we have used the fact that (2b logb
)2/
N
N
aN k(k) + (l+E)
1
as k/N+O, as
N+~. Finally, from line 14, p.95 and Line 23, p.98 in L!~7, we get LPN ~.
p
p
This and (120), (121), (122), and (123) together imply Lemma 2.
/
1
/'
Lemma 3. (Komlos, Major, Tusnady, 1975). There exist a probability space carrying
a sequence YI' Y2' . .. as defined in (11) and a sequence of
Brownian bridges
1
B
(s), ~s~l, N = 1,2, ...
such that
N
h
P(
sup
1 AN(x)
- B~ (Hk(x»
1 > Alog1N+x
);;; Be-
,
~x<+oo
N2
for aIl sequence (k=k(N»N~l and for aIl x, where N ,A,B, and À are absolute
1
positive constants.
1
1
PROO~. This doesn't need ta be proved. It is directly derived from Ib7 and
Corollary 4.4.4 of 147.
1
Proof of Theorem 3 (continued)~
1
1
On the probability space of Lemma 3, Lemmas 1 and 2 combined with the fact ~~N-2
imply that
1
N=N
-1
(124)
L P ( U p+l
P
N=Np
1
where
**
SN
(x) = AN(x)
x H~ (x) (uN-1),
Hence, the proof will be
complete if we approximate
** in the right way. But by Lemma 3, for any E>O,
SN
1
for large N,
-l-E
1
(125)
P(
N
,
1
1
"
r
"
r
1

---

1
-193-
1
where Al is sorne absolute constant. From Lemma 3.1 of /27,
1
I l s
C Nk
-1 J+""'{
1
}
-1 I+"'"
1
)
(126) N2 (l.J -l)=N2 k
n+
+k
0
J\\N(x)-BN(Hk(x))
dx+k
0
BN(Hk(x)
dx.
N
Nk
1
f.--o
1:
h
f
Let tN=N'
,0<O<u . On the one hand, one
as
or large N,
O
l
l
t
1
N
1
1
E(2loglogN)4(logN)2 )
P(
IIo {J\\N(x)-BN(x)}dxl>EaN/12)~P( sup
IAN(x)-BN(Hk~»I>
1
O
12 N4 -
O~<tro
1
~
1 ~P( sup IAN(x)-BN(Hk(x))]>AllogN/N2).
O~x<+"",
This and (15) together imply
1
1
t N
.
1
1
(127) P( sup
IxHk(x)k-
JO {~(t)-BN(Hk(t))}
dtl>EaNK(k)/l2)~N- -E,
1
O~x<+oo
for N large enough. On the ather hand, as N~oo,
r
t
(128) P( sup
II+oo{J\\N(t)-B~(Hk(t))}dtl>N-~) ~N~ exp(_N -Ù/4).
O~x<+oo t N
1
Ta see that apply Markov's inequality with EI;x'IJ\\N(x)-B~(Hk(x))ldx~)~4k-le-X/2
N
N
(k-l) /2
-1
k
t-o
1
x
'/(k-l)!dx~4k
t
exp(-t /2). Since k=o(N
0), as N-++oo,
(128) follows.
N
N
FinallY for large N,
l
S
-s
l
l
l
l
l
P( sup
IxHk(x)/k21
Nk
nt
I>Ea K(k)/16) ~ P(S >N2k2 )=1_H (N2k2 ).
~x<-fJX'
(Nk)2
NI
-
k
k
,
Integrating by parts we have: k/~~ =>l-H (x)~2xk-le-x / (k--:l)!. Then if k/N~~ for
k
large N, we get by Sterling's formula,
l
l
l
l
l
(129) I-Hk(k2N2)~canst. exp(-k2 N2 (1+(k/N)2l og (k/N)).
Thus,
u1timately as N-+~ whenever k/N+Q as N+~. Put tagether (125),
(126), (127),
(128) and (130) to~ get
(131)
L P(
**
sup~8N (x)-W**(x) !>e:a K(k)/4) < tro,
N
N
N
O~x<+"'"
, .
,,<. ( , "

---

-194-
**
1
-1 ,.
ftoo
1(
(»
4)
where W (x)=BN(Hk(x) )-xk
Hk(x) 0 t dB
H
N
N k t . , x~. And combine (12
with
(131) to have
N=N
Cl
*
**
(132)
L P( U p+
{sup
IBN(x)-W
(x) \\>(l+E:)aNK(k») < +"".
N
p
N=N
O<x <-t=
p
This together with Lemma 4.4.4. of 147 completes the proof.
Proof of Theorem 3.
As in the proof of Theorem 4, the spacings are always defined on the probability
space of Lemma 3. We shall study each of the RNi's once again. First we put to-
gether (13), (14), (15) and (16) to get
1
1
(133) sup
!RN1(x)-N2(on-1)xHk(X)1= O(N-2 loglogN), a.s., as N++oo.
~x<-j-ro
Now Lemma 2 says nothing else but
(134) Hm sup
sup
\\R
(x)/a I ~K(k) or K ' a.s.,
N2
N
O
N+t=
~x<+oo
whenever k is fixed or k++oo while k/N+O as N++oo. And the proof will be completed
1
through our fundamental lemma which is the following.
Lemma 4. Under the assumptions of Theorem 3, we have
1
\\ 1-1
lim sup
sup
aNRN2(x)I~K(k) or KO' a.s.,
N+-/<x>
~x<-tœ
according whether k is fixed or k++oo
and satisfies (L).
1
-1
k
-x
Proof of Lemma 4. Let l\\J(x)=«k-1)!)
x e , x~O. By the mean value theorem,
1
1 By Sterling's formula we can find a constant T>O such that
(135)
sur
k~1l\\J(x)--W(k) I~.lhk-l, for a11 ~ 1.
1
1x-k ~h~l
-
Now,
1 (136) AN(x)=Hk(O x)-Hk(x)=(o -1),,,(x) (x lx), O<x lx<max(1,o ).
n
n
\\jJ
n
n
=n
=
n
1
1
"'. (
, .
_ / l '
1

---

-195-
.
.1
Ix-kj~hg =:>A (x)=(1+0(1))k2 (On-1 ){ K(k)+O({h+(h+k)ll-0 l}/k)}, a.s.
N
n
1~
Let h=h(N)+O as N+~. Then by the loglog-law, there exists n {_ n anda sequence
(Nj(w)) extract from (N) (let n
and k
be the corresponding subsequences) satis-
j
j
fying
1
1
AN (x)=«2loglogn .)/N )2K(k .)2(1+0(1))= :(1+o(1))d
.
J
J
J
N .
J
J
uniformly in x, k.-h.~x~k.-h., where h.=h(N.) as
N++oo. Thus we have uniformly
J
J- -
J
J
J
J
in x e [k.-h., k .+h"J = I
,
k
J
J
J
J
(138)
IR
.2(x) l~ IY .(H
.(1+o(1))-Y
.(H .(x)) 1=: IR:.
N
N
k .(x)+d N
N
k
2 (X) 1.
J
J
J
J
J
J
J
We now prove that
-
1
*
(139)3no (_ n , p(n )=l, Jfw eno' ~im inf sup {IR .2(x)/b(d .)
o
N
N
1 }~1,
J++OO
xeIk .
J
J
J
-1 1-
where b(s)=(2s10g10gs
)
, O<s<l.
Proof of
(139).
Let
C
(p)= sup
sup
lyN(s)-yN(s+v)!/b(d ), ~1.
N
N
l
O~y~dN/P ~s~l-v
By Theorem 02 of LI17.
1
(140) -vp~1, jnp(=st, p(np)=l, Jfwél , lim sup C
(p)(W)<p-2.
N
P
N+-too
1
\\
p--too
1
2
Let st 2= n n Il st
.Obviously p(n )=1. And
p=l
p
(141) Jfwél 2 , C
2(w)=
sup
Y
(H
(x)+d
(1+o(1))-YN.(H
))·
N
k
N
k .(x)+d N .)=o(b(dN
Nj
O~x<+= j
j
j
J
J
J
J
This together with the following
(~42) JfxeI
.(~
k ' R: .2(x) =Y N
.(x)+d
.)-Y
.(\\(x))
N
N
.
j
J
J
J
J
J
+Y
.(~.(x)+dN.)'
N .(Hk .(x)+dN .(l+o(l))-YN
as }++n,
J
J
J
J
J
J
implies that
*
(143)
sup
RN .2(x) ~ sup
Y
(~ (x)+d )-Y (H (x))+o(b(d ))
N
N
N
k
xe \\
.
J
xci: k.
j
j
j
j
j
Nj
J
J
~: CN.3(h(N .))+o(b(d )).
N
J
J
j
\\:' r
\\:. r
1
~ " .'

---

-196-
2
-!
Nowput Jk=Hk(I ) and remark that the lenght of J
iSP(J )=2K(k) hk
(1+0(1».
k
k
k
2
-! -1/4p
For any p~l, choose h=h(N, p) =h
(with h.
=h(N., p) such that 2K(k) h k
d
=
p
J,P
J
P
N
1+0(1), as N~oo. Thus h+O as N++oo when (L) holds. A1so mN=max{i, i~O, Hk(k-h )
p
+id
E JJ + +00 as N~oo. Therefore we may use the lines of the proof of Lemma 2.9
N
of 1137 to conclude that for any p~l,
P(DN)=P(
1~~ {YN(C~+l)-YN(C~)} Ib( dN) ~(l-l/p)1)
--N
that is
1
(144) :V-p~l, =Irt', P(rl')=l, Jftœœ, liminf CN3(hp)/b(dN)~(l-1/p)2.
-
p p p
N++oo
2
p=+oo
.
Letting rlO=rt n n rl', we get P(rla)=l and for aIl WErl '
O
p=l
p
(145) 1im inf
sup
IR: 2(x) I/b(dN) ~ l.
j++oo
xEI k .
j
J
\\~e have used
in (138)
that representation for
commodity reasons as
it has appeared in the p~oof. The sa me may be done,
step by s~ep,
1
1
fo110wing Stute's results
(see {I~7) to get the version of (145)
for R
.
itself.
This remark completes the proof of
(~39).
N 2
J
Proof of Lemma 4 (continued). Remark that
lim sup
N-++oo
~lim inf
sup
r+oo
O~x<+oo
This combined with (139) and with the fact that b(dN)=K(k)a
(1+0(1»
as N++oo
N
proves Lemma 4.
'1;.. r
'1;.
r
, .'

---

-197-
Conclusion. It is clear by Theorern 3 that the approach used until now cannot
yield a rate better than aN. The problern is now: what new approach would be used
,
1
to reach, i f possible, the very best rate, that of /67 which is N-2 log N.
B- The Glivenko-Cantelli Theorem.
For the overlapping case, /37 obtained a Glivenko-Cantelli theorem when the
1 a
step satisfies kN- + + 0 as N++oo for sorne O<a<l. As to the overlapping case
only fixed steps have been handled in L~7. We give the general result in
Theorern 5. Let k~l be fixed or k++oo while k/N+O as N++oo. Then
lirn
sup
IFN(x)-Hk(x)l=o, a.s.
N++oo
O~<+oo
on the probabilty space where the spacings are defined.
Proof of Theorem 5. We have
First, it follows from Lemma 2 that for aIl E>O,
N=N
Cl
l
N-2 sup
1 R
(x)
\\ ~ P( N~NP+
N2
1 >E/ 4 )<+00.
O~x<+oo
p
Next, P(
sup
IRN4(x)I>E/4)~P( Il-UN' k~K(k)2>E/4). And direct calculations
O~x<+oo
irnply that for aIl À>l, we have
for large N. Thus by (18)
N=N
1-1
l P(
U p+
sup
IR
(x)I>E/4)< +00
N4
p
N=N
O<x<+oo
p
=
whenever k/N-+O as N+;-oo. Finally,
1
1
1
P(
sup
I~N(x)-Hk(x)I>N-4)
P(
sup
N-2 Iy (s)!>N-4 )
N
O~x<+oo
O~s<l
"=, r
1
1 . '

---

-198-
~~"
~.~:
(.
by the fact that Y (·) has stationary increments. Using now a representation of
N
Y
by a poisson process and an approximation of a Poisson distribution by a Gaus-
N
sian one (see LI]7 , Lemmas 2.7 and 2.9) to get for large N tha t
3/2
_1.
5/4
1/8
J ~ const. N
P(N(O, l »N 4 const. )~const. N
exp(-N
).
N
Thus LJ < +CXl. And the proof of Theorem 5 is now complete.
N N
c- The oscillation moduli.
The socilation modulus of a function R(s), O~s<l, is defined by K(d, R) ==
sup
sup
IR(s+h)-R(s)l. O<d<l. That of the empirical process pertaining to
O~d O~<1-h
i.i.d. r.v.'s has been studied for several choices of d in /97and /137. It is
remarkable that the weak versions of aIl those results are inherited by the re-
duced spacings process aN(s)==SN(H~l(s», O~s<l, (see /77). For the strong case,
we obtain these two results.
Theorem 6. 1. The Stute's case.
If (dN)~l is a sequence of non-~ncreasing positive reals such that
(SI)
1
(S4)
thenfor k~l fixed or k=k(N)~~ as N++co and satisfying
1
1
we have
lim sup K(d , aN)!qN=l
a.S.
N
N+..p:o
1
II. AcMason-Wellner-Shorack case.
1
L
(
-c
et aN=a 10gN)
,a>O, c>O. Then under the same assumptions on k used in
1
Part l, we have limsup K(d , aN)/qN~(l+c)2, a.s.
N
N-++oo
1
,-. r
1
1
1 . '

---

-199-
Proofof
Part l
of
Theorem 6.
We have by Lemmas 1 and 2,
(30) .lfN~l, {CtN(s), O<s<l} g, {l\\N(H~1(s»+RNS(s)+RN6(s),
O~s<l}=:{(~N(s), O~s<l},
1.
-1
-1
!
<l>
with R
(s)=N 2 (llN- 1)I\\ (s)H\\(H
(s»=:N. (~-1) (5), O~q,
NS
k
and
N=N
1-1
(31)
LP(
U p+
sup
\\R
(s) \\>(l+E)aNK(k»< +00.
N6
p
N=N
O~s<l
p
By (31) and (S4), we have
N=N
-1
(32)
L P( U p+1
P
N=N p
2
By Lemma A4 in LZ7, K(d , <l»=(l+o(l»~ as N-++oo for aIl k satisfying (SS). Thus,
N
by the 10glog-law,
N=N
1-1
LP(
U p+
P
N=N p
whenever
-1
(33) 1im k
d
loglog(l/d ) 10glog Nk = O.
N
N
N+t-oo
is satisfied. This obvious1y fo110ws from (Sl-2-3-4-S). By the resu1ts of /Ï37
1
1
as reca11ed in (123), for E>O,
N=N
-1
(34)
LP( li p+1
P
N=N p
when (Sl-2-3) ho1d. Since E is arbitrary and since (SI) and (S3) imp1y (33), we get
-1
(35)
lim sup
qN
K(d
, ~) ~1, a.s.
N
N++oo
1
Ta get the other inequali ty, define for 0 «:1 «:2 <+00, Q<d< l, for any function R(s) ,
0~<1,
1
(36) K'( d, R)=
sup
IR(u)-R(t)I/lu-t , O~u, t~l.
Cl cl<u-t<c d
2
1
Let RN(.) =R (.)+R (.) and rN(.)=l\\N(H~l(.». Now remark that for aIl E >O, there
NS
N6
1
.
«
' )
-1)1
-1 !
eXlsts E >O such that for a=
1-E
logd
and b=(E
1ogd
)
,a+b=( (l-E +E
+
~
2
1
N
2
N
1
2
1.
-1 1.
-1 1.
2(E (1-E »2)logd
)2=«1-E )logd
)2 with E >O, E ,E +O as E +O. Thus,
2
1
N
3
N
3
3
2
1
1
\\:;. r
, .'
1

---

-200-
(37)
P(K'(dtr aN)~a)~p({K'(~, ~~a} n{K'(d , RN»bl)
N
+P({K'(d N, aN)~}n{K'(dN' RN)~})
By (31) and (32),
N==N
-1
(38)
L
U p+1
P(
p
N==N p
for aIl E >0. Thus by (30), (36), (37) and (38) and Lemma 2.9 of ~~7 and sorne
2
straightforward considerations, we get lim inf K'(d , ~)~1, a.s., under (Sl-2-
N
N++oo
(39)
lim inf
K(d , aN) ~ hm inf K' (d , ~)~1, a.s.
N
N
N++oo
N++oo
(35) and (39) together complete the proof of Part l of Theorem 5.
Proof of Part II of Theorem 5.
Here (54) and (S3) are satisfied. It suffices thus to write again the pi~of of
the part one where one should use the probability inequality (13) of Lfj). It must be noted
that \\Part III of Theorem l in L27 holds for the g~neral case where 3 =a( logN) -c ,
N
\\
O<a, O<c.
APPENDIX. PROOFS OF STATEMENTS (17)
AND (l8).
a) Proof of Statement (17). Tchebychev's inequality yields a>l and S>l such that
2
a
'S
2
PC S 2/n > 1+E)~A2n-
and PC 1Sn-Sm(n) 1>nE/2)~3 n-, . as n+too, where men )=max{j •
n
j2~n, j=l, 2, ... }. Thus
1
(1) P(!]JNI>l+E)<P( \\UN-S
I/(n+1)\\>E/2)+P(S
1/n+1 ~ l+E/2)
=
n+
n+
1
since (n+1)~k as N++oo. Futhermore, by Tchebychev's inequality,
1
1
"
r
". ( 1 \\,'
M

---

1
-201-
1
1
4
3
P(SNk/ Nk _(Nk)2 > E/8) < 64 N-3 k-3 /E2, P(S/(Nk)2 - (Nk)2 > E/8) ~ 64 N- k- /E2
1
and P( iIJN-Sn+/(n+1) I~E/2) ~P(SNk/Nk > E/4) + P(Sk/(Nk)2 > E/4). Hence since Nk-++=,
2
N k-++oo as N++oo,
1
(2)
L P( IWN-Sn+/n+11 > E/2)<+oo.
N
Thus (1) and (2) together imp1y (17).
b) Proof of (18). We have
Sn+1-Nk
Sn+CSNk
SNk-Nk
-~---------'--l -
--~~"::';';;';;-"""1 + -~------'-l - .
(2Nk 10glog Nk)2
(2Nk 10g10g Nk)2
(2Nk 10glog Nk)2
First, since 0~(n+1)-Nk~k,
l
1
1
l
l
2
2
P(SN > E/2) ~ P(Sk>E:(2Nk 10g10g Nk) 2 /2) ~ 1-H (k2 N ) ~ const. exp( -tk2 N )
k
as k/N+O, N++oo (see Statement (129)). Thus
(3)
LP(SN > E/ 2) < +ro.
N
Now, let p=p(N)=inf{j, N>N.}
and q(N)=inf{j, k(N»N.}, N .=~l+P)jJ, j=l, 2,
J
J
.J
Then N
l~N~N ,N
IN
l~NK~N N , 10g10gN N =(loglogN )(1+0(1)), as N/k-++ro,
p- - -
p
p-
q- -
-
p q
p q
p
N+fOO, N
l/N -+ 1+0, as N-++ro. Thus (see /87, p.259-262),
p+
p
N=N
1-1
P(
U p+
{SN;;:; 1+E:/2})~A4P(SN N >1+0(E, P)(2Np10g10gNp)-!)~A5P-(1+O(E'P))
N=N
p q
p
as p+-f-OO, for P sma11 enough, O(E,p»O. The same ho1ds for -SN' Thus
N=N
-1
1
( 4 )
LP( U p+
( 1 SN 1> 1+E: / 2) <-tro .
p
N=N P
Fina11y (3) and (4) together imply (18).
"
r
,
"

---

-202-
REFERENCES.
II?
Aly, E.A.A.(1985). Strong approximation of empirica1 and quanti1e processes
of uniform and exponential spacings. Preprint.
127
Aly, E.A.A., Beirlant, J. and Horv~th, L.(1984). Strong and weak approxima-
tion of the k-spaeings proeesses. Z. Wahrsch. verw. Gebiete, 66, 461-488.
137
Beirlant, J. and Van Zuijlen, M.C.A.(1985). The empirical distribution
function and strong laws of order statistics of uniform spacings. J. Multiv.
Analysis, 16, 380-397.
147
Cs'~rg~, M. and Révèsz, P. (1981). St rong Appr oximat ions in Pr a ba bili t Y
and Statistics. Academie Press. New-York.
151 Deheuvels, P.(1984). Spacings and applications. Techn. Report, 13, L.S.T.A.,
Université Paris VI.
167
"
/
Kom1os, J., Major, P. and Tusnady, G.(1975). An approximation of the partial
suros of inde pendent rv's and the df. Z. Warhsch. verw. Gebiete, 32, 111-131.
177 Lô, G.S.(1987). On the increments of the k-spacings process for fixed or
moving steps. Submi tted and now in revision.
187
Loève, M.(l963). Probability Theory. Van Nostrand Camp. Inc. Princeton,
3rd Ed.
197 Masan, D.M., Shoraeck, G. and Wellner, J.A.(1983). Strong theorems for the
oscillation modu1i of the the uniform empirical process. Z. Wahrsch. Gebiete,
65, 83-97.
lï57 Pyke, R.(1965). Spac~ngs. J. Roy. Statist. Ser.B, 27, 359-449.
1
1II7 Pyke, R.(1972). Spacings revisited. Proc. Sixth. Berkeley Sympos. Math.
Statist. Probab., 417-427.
'-,<-
IÏ27 Shorack, G.(1972). Convergence of quantile and spacings processes with
applications. Ann. Math. Statist., 43, 1400-1411.
{!~7 Stute, W.(1982). The oscillation behavior of empirical process.
Ann. Probab., la, l, 86-107.
1
~
~
."
.". (
,.,
, '
~

---

"
, ' 1
1
-
,
,
J
4,
J
'4,
1
\\
QUATRIEME PARTIE.
STATISTIQUES D'ORDRE DANS LES ESPACES LATTICES

---

-203-
MAJORATION, MINORATION ET CONVERGENCE
FORTE DES QUANTILES CENTRALES ET EXTREMES DE VARIABLES
ALEATOIRES A VALEURS DANS DES ESPACES DE TYPE
CeS).
Gane Samb LO
UER de Mathématique et Informatique - Université de Saint-Louis-Sénégal
&
LSTA- Université Pierre et Marie Curie- Paris.
Résumé:
Nous introduisons les définitions de basè des statistiques d'ordre dans
les espaces de Banach lattices. Dans cette première publication, nous donnons
des 'bornes exactes et presque-sûres pour les quantiles et les extrêmes de toute
variable aléatoire à valeurs dans CeS) où S est un espace métrique compact.
Pour de telles variables aléatoires, nous
caractérisons celles d'entre elles
dont les quanLiles et/ou les extrêmes convergent.
Les résultats sont appliqués
au processus de Wiener et au pont Brownien.
Mots clés:
Statistique d'ordre, espaces métriques compacts et séparables,
quantiles, extrêmes.
Adresse postale: Gane Samb LO, UER de Mathématique et d'Informatique.
1
.
Université de Saint-Louis. Sénégal.
'1::. r
1
: .. .'

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-204-
I-INTRODUCTION.
a- Les structures.
Soit Xl'
... , X
une suite de copies indépendantes (s.c.i.) d'une variable
n
aléatoire (v.a.) réelle X. La théorie traitant des statistiques d'ordre
Xl
~ ...~
reçoit une attention touJ'ours plus considérable depuis une dizaine
,rr-
- n, n
-
dt années. Elle couvre l'étude des lois limites des extrêmes (voir De Haan [3J ou
Galambos
[SJ), celle des quantiles, celle de l'estimation des paramètres de
lois stables (voir LO
[6J), etc ... Plus recemment, les quantiles et les extrê-
mes ont été etudiés dans le cas multidimensionnel (voir Deheuvels
[4J
ou
voir Weiss [9J ).
Cet article traite des statistiques d'ordre_pour une v.a. définie sur un
/
espace de probabilité (n,E,p) à valeurs dans un espace de Banach lattice E.
Définition 1. (E,+,., Il. Il ,~) est un espace de Banach lattice sur 1R si
(E,+,., Il. Il) est un espace de Banach sur IR muni d'une relation d'ordre notée
~ et vérifiant les propriétés suivantes:
(al) ~ (x,y,z) e EXEXE,
x~ =>x+z~+z,
(a2) ~ (À,x) e IRxE, (1.';;;'0 et x~O)
>Àx~O,
(a3) ~ (x, z ) e EXE, o.~x~ => Il x " ~ Il y "
(a4) ~ (x,y) e ExE, sup(x,y) et inf(x,y) existent dans E.
Dans de
tels
espaces, les fonctions xl-->sup(x,y) et xl-----'?-ïnf(x,y) sont
uniformément continues (voir Schaeffer [8J ) . Ceci justifie la proposition
suivante:
Proposition 1. Si Xl' ... , X est une s.c.i. d'une v.a. X à valeurs dans un
n
espace de Banach lattice réel E, alors
1
x
==
sup
inf
X.
et
X
inf
sup
x. ,
k,n
l
k,n
l
K e [).n
K e [).n
i
e K
i
e K
n-k+l
k
- où [).~ désigne l'ensemble des parties de {l, ... ,n} ayant k éléments - sont aussi· =
des v.a. à valeurs dans E avec
(l.1)
x
- Xk,n
~ X
,
=
(-X)n-k+l,n '
k,n -
k,n
Ceci conduit à la définition des statistiques d'ordre dans E:
Définition 2. Soit Xl' X ' ... une s.c.i. d'une v.a. X à valeurs dans un espace
2
de Banach lattice E, les variables aléatoires Xl'
~ ... 2,X' (resP,X
~...~X
)
, n -
-
n, n
l , n-
- n, n
sont appelées les statistiques d'ordre à droite (resp. à gauche) de Xl, ... ,X .
n
'"
r
,
1
~ •• .'
.'

---

-205-
Soit E un espace de Banach lattice réel défini par un ordre canonique,
c-à-d, E={ (x ), aeA} et x=(x)
A < Y=(Y)
A si et seulement xa=;;;Y'" pour tout
a
a ae =
a ae
u.
aeA. Si en plus le "sup" et le "inf" sont obtenus point par point, c-à-d,
sup(x,y)=(sup(xa'Ya)aeA) et inf(x,y)=(inf(xa'Ya)aeA), alors les statistiques
d'ordre coincident à gauche et à droite et nous écrivons X
=X
=X
,1~~n.
k ,n
k ,n
k ,n
Cette égalité peut faire défaut si Sup et inf ne sont pas obtenus point 'par
point. Par exemple, cette égalité n'a pas lieu si E est l'espace des mesures
additives, régulières et dénombrales de Borel bien que l'ordre soit canonique.
Dans cet article, nous nous limiterons au cas où E=C(S) où (S,d) est un
espace métrique compact. Nous exigerons la séparabilité pour l'étude des extrêmes.
La norme utilisée est celle de la convergence ufliforme
Ilxil = sUPteSlx(t) 1,
xeC(S). L'ordre est canonique et obtenu point par point: x~
si et seulement
si x(t)~(t) pour tout teS. Ainsi pour toute V.a. à valeurs dans C(S) X=(X(t))teS'
X(t) est une v.a. réelle pour tout teS et (X(t))k
=(X
)(t)=:X
(t) pour
,n
k ,n
k ,n
tout teS, pour tout k, l~k~n.
Il est clair maintenant que l'objectif principal de c:t article est d'initier
la connexion entre le domaine très actif des probabilités dans les espaces de
Banach et celui (non moins actif) de l'étude des quantiles et des extrêmes dans
n
IR .
Avant d'exposer nos principaux résultats, nous allons reformuler quelques
résultats connus dans IR afin de les utiliser de manière convenable.
b- Retour au cas réel.
Les résultats exposés dans les deux premi~rs théorèmes sont bien connus.
Théorème 1.1: Soit Xl' X , ... une s.c.i. d'une v.a., X réelle avecP(X~)=F(x),
2
xeIR, A=suf{x;F(x)<1} et B=inf{x, F(x»O}. Alors, pour tout k fixé, l~~n,
X
+·A
et X
+ B, presque-sûrement
(p.s.) quandnt+oo.
k-n, n
k ,n
Théorème 1.2:·Soit Xl' X , ... une s.c.i. d'une v.a. réelle X avec P(X~x)~F(x),
2
xe œ. Alors, nous avons presque-sûrement,
-1
-1
-1
(i) Pour tout a, O<a<l, F
(a-O)=F
(a)~lim inf X~ !J
~ lim sup X~:l =F (a +0),
na ,n -
t+
nC!J ,n
nt+oo
n
00
-1
1
(ii) Si F- (.) est continue en a, O<a<l, alors Xr;;:l
+F
(a), p.s.
JEC!J ,n
1
où F- (u)=inf{s, F(s)~u}, O~u<l, est l'inverse genéralisée de F(.) et
-1
-1
-1
-1
1
:1
F(a-O)=lim u+O F
(a-u)=F
(a), F
(a+O)=lim ufO F- (d~u). Enfin [x~ désigne
la partie entière de x.
'<c. ( .
, '

---

-206-
Nous complétons le deuxième théorème comme suit.
Théorème 1.3: Sous les hypothèses et notations du Théorème 1.3, nous avons
l'équivalence entre les propositions suivantes pour tout a, O<a<l.
(i) X,-:l
converge vers un nombre non-aléatoire en probabilité ou presque-
L.!1C!J ,n
sûrement quand n tend vers + ~
1
(ii) F- (_) est continue en a, et X,-:l
tend vers F-l(a) presque-sûrement
L.!1C!J , n
quand n tend vers +00;
Nous ommettons la preuve de ce théorème qu'on peut obtenir en combinant
le Théorème de Glivenko-Cantelli et la loi du Zéro-Un.
11- Les Quantiles dans C(S).
A- Bornes et convergence des quantiles centrales.
Commençons par le
cas général où E est un espace de Banach lattice avec une
unité e, e e E, c'est-à-dire (c-à-d), pour tout xe E, Ilxl~l signifie -e;;"x;;"e.
Théorème 2.1. Soit Xl' X ' ... une s.c.i. d'une v.a. à valeurs dans E. SuppQSons
2
que E possède une unité e. Alors pour tout a, O<a<l,
pour tout E>O, il existe
un entier N(w,E) presque-sûrement défini (en wer2) tel que
V- n>N(w,E), - fF-l(l-a+O)+de-SX,-:l
(w)-SX,-:l
(w)-sfF-l(a+o)+de
-
L.!1C!J ,n
-
L.!1C!J , n
-
où F(x)=P( IIxil ;;"x), x e IR.
Preuve. Soit Zi= IIxill , i=1,2, ... On a -Zie;;,,\\~Zie pour tout i~l. Donc
-eZ
. 1
;;"X.
;;"X.
;;"Z.
e pour tout i, l;;"i;;"n et pour tout n. L'application du
n-l+ ,n
l,n
l,n
l,n
Théorème 1.2 achève la preuve.
Ces bornes sont valables dans C(S) puisque la fonction x\\_·->1 est bien
l'unité de C(S). Mais elles sont grossières. En nous restreignant à C(S), nous
obtenons les meilleures bornes possibles.
Théorème 2.2: Soit Xl' X , ... une s.c.i. d'une v.a. X=(X(t»teS à valeurs dans
2
C(S), où S est un espace'métrique compact. Soit Ft(x)=P(X(t)-S.x), xe IR, la fonc-
tion de répartition de la coordonnée X(t), teS. Alors, pour tout a, O<a<l, pour
tout u, O<a-4u<a+4u<1, il existe un entier nO(a,u,w) presque-sûrement défini
en w et tel que
1
.'

---

-207-
-1
-1
(i) :v.n>oO' .1fseS, -61J+F
(a-41J)~ Xr=::l
(s)-S.61J+F
(a+4IJ+O),
s
l.llC!J ,n
-
s
(ii) Par ailleurs, pour tout Ë, O<Ë<min(a/S,(l-a)/S),
l
l
sup
max( IF- (a-4Ë) l, IF- «H4(+O))<+co.
seS
s
s
Remarquons que (i) est tout à fait comparable au résultat (i) du Théorème
1.2. Il s'agit bien d'une généralisation du Théorème 1.2 dans C(S) muni de la
topologie de la convergence uniforme. Quant à (ii), il améliore de très loin le
résultat du Théorème 2.1.
DEMONSTRATION.
La démonstration est assez longue. Elle se fera à travers
une série de lemmes
et d'étapes. D'abord notons que ceS) est séparable-car S est compac,t. Nous aurons
donc (voir Prop. 1.2.1 dans Padgett et Taylor [7J):
Proposition 2.1. Soit X une v.a. à valeurs dans E, où (E,d) est un espace métrique
séparable. Alors pour tout À>O, il existe une v.a. dont l'ensemble des valeurs
est dénombrable dans E et telle que d(X(w), Y(w) )<À, pour tout w en.
Désormais nous supposons que
Il X+A-Y II~ Ë , où A sera précisé plus tard,
Y(W)=L,colx.IJ.(w), avec 1J.(w)IJ.(w)=O, 1J.(w)=O ou 1 pour tout i, j e t i#j,
1=
1
1
1
J
l
T
i=T
Y (w)=L. 1 x.IJ.(w), pour tout entier 1>0. Notons que les IJ. sont mesurables.
1=
1
1
l
T
Finalement, posons G (x)=P(Y(s)..'S.x) et G . (x)=P(Y (s)..'S.x), seS, xeIR. Nous aurons
s
-
T ,s
-
besoin de savoir comment sont liées les fonctions F , G et G
à travers
.
s
s
T ,s
trois lemmes .
.,
Lemme 2.1. 1 Avec les notations précédentes nous avons:
1
1
(i) sup
sup
1 F:'-
(a)+A_G- (a) !..'S.Ë,
0<a<1 seS
s
s -
(ii)
sup
-1
-1
sup·
1
F
(a+O)+A-G
(a+O) -S.E.
1
s
s -
0<a<1
seS
Preuve du lemme 2.1.
Il X+A-Y II_.<_E -> Jf (s, x)eSx IR, F (x-A-Ë)..'S.G (x)-S.F (x-A+E)
s
-
s
-
s
Posons x=A+F-l(a) , alors a-S.F (x-A)..'S.G (X+E) et donc
s
-
s · · · s
-1
-1
(2.1)
:If seS, :If a, O<a<l, G
(a)~Gs (Gs(X+E))~X+E,
s
1
-1
où nous avons utilisé les résultats: F(F- (u))G;U et F
(F(x.))~x pour tout
u, O<u<l, ~our tout xe IR, dès que F est une fonction de répartition continue à
droite. Nous aurons finalement:
"'. ( , ".,,'

---

-208-
(2.2)
Jf a, O<a<l, Jf seS,
En combinant la symmétrie des r~les de F
et de G , nous prouvons (i).
s
s
Pour obtenir (ii), il suffit de remplacer a par a+u et· de passer à la limite
quand ui-O.
T
Lemme 2.2. (i), Pour tout 00, i l existe T=T(E:) tel que P(
lIy_y II ~ E:)~E:
et tel que:
(ii) Jf seS, Jf xe IR, -E:+G
(x-E:)~G (x)~GT
(x+E:) ,
T,s
- s
-
,s
-1
-1
-1
(iii) Jf a, O<a<l, Jf seS, -E:+G
(a-E:)~GT,s(a)~Gs (a+E:)+E;,
s
(iv) les rôles de G et G
sont symmétriques
dans (i) et (ii).
s
T,s
Preuve du lemme 2.3.
T
(i) découle de la convergence ponctuelle de Y vers Y quand T-+-t=. (ii) et (iii)
s'obtiennent de la même manière que les résultats du Lemme 2.1.
Lemme 2.3. Avec les notations
du Lemme 2.2, nous avons
(ii )
sup
sup
O<a<l
d(s,t)<OT
(iii) Pour tout a, O<a<l,
Preuve du Lemme 2.3. Nous avons
T
T
1[...-'\\ i=T 1
1
Il y
(s)(w)-y (t)(~ ~i=l
xi(s)-xi(t)
2
pour tout ( w, s , t) e r2xS . Et la continuité uniforme de tl--~ I~=Tl x.(t) impli-
l=
l
que (i) duquel dérivent (ii) et (ii~) en procédant selon la méthode utilisée
\\
. :
dans la preuve du lemme 2.1.
~r, Nous allons reprendre la Démonstration du Théorème étape par étape.
\\
Soit O<a<l, O<E:<W, 0<a-4u<a+4w<1. A partir des Lemmes 1.1-2-3, choisissons
une v.a. à valeurs dénombrales Y et un entier T tels que
IIx+A-YII~E: et
T=min{t~l, Jf p~t, P( IIyP-Yli = IIRPII~E: )~E:}, où W>IQ(-A) et FC/(U»-A. Le lecteur
se convaincra facilement qu'un tel A existe toujours quelle que soit F '
O En
globalité, l'ensemble des choix de U, A et Y sera appelé et noté CH(E:). Les
T
v.a. X., Y., y~, R~, i=l, ... ,n seront des s.c.i. de X,y,yT: et R .
l
l
l
l
~ ..
"
r
,
.'
"
r
, .'

---

-209-
T
Etape 1:
Bornes de Yr::l
.
l..D<!J ,n
Il découle du Lemme 2.3 que:
(2.3a) Jf i, 1~~n,
IlyI,nIIQ(T) ~E, où o(T) = 0T et
T
où Y.
,i=I, ... ,n sont les statisques d'ordre des v.a. définies par
l,n
Y.
}
T
(w) si Y. (w) e {x l'
... , xT
Y.(w)= { l
l
l
0 sinon
Puisque S est compact, il existe un nombre fini d'ouverts B(u , 0T)={x,d(x,ui)<OT};
i
Ui=r
i=l, ... , r tel que S=
i=1 B(u , 0T)' Il existe, par le Théorème 1.2 un entier
i
n (a,T,E,W) presque-sûrement défini en w, tel que pour tout n>n ,
1
1
-1
T
-1
(2.3b) Jf i, l~i~r, -E+G
(a)~Yr::::l
(u.)~E+GT
(a+O).
- -
T , u .
-
L!1<!.J, n
l
-
, U .
l
-
l
En combinant le Lemme 2.3 aux propositions (2.3a) et (2.3b), on obtient
-1
T
-1
(2.4) Jf seS, -3E+G
(a )~Y r::::l
(s )~3E+GT
(a+O) .
T J S
-
L!1<!J, n
--
, s
Nous avons montré en fait un résultat plus général qui est le suivant:
((QI)): Pour tout a, O<a< 1, pour toute v. a. Yà
valeurs dénombrables dans C(S) et
pour tout entier T défini selon le. Lemme 2~, alors nous pouvons définir pr~sque
sûrement un entier N(a,Y,E,w) tel que (2.4) soit vraie pour tout n>N.
T
Etape 2. Approximation de Y.
par Y.
l,n
l,n
Nous devons démontrer la proposition fondamantale de cet article qui est:
-1
((Q2)) Soit O<W. On peut toujours choisir A tel que ~>FO(-A) et Fa (w»-A.
Alors, il existe EO(W»O, tel que pour tout choix CH(E) avec O<E<EO(W), on a:
(2.5) Jf k, l~k~n,
Y~-U3n~)/~,n~\\,,~Y~+[I3n~)/~,n ' p.s.
,
quand n est assez large avec la convenhion
T
T
Y
-y
1
= - co, pour ~O.
-p,n
n+ +p
Démonstration de CCQ2))'
Soit O<~<l, O<W-P-C<W-P-tE<1, w>FO( -A) et FOl (]J»-A. Proc;dons à un choix CH(E).
Soit H , F , G ,G
les fonctions de répartitions des coordonnées de V, X, Y
s
s
s
T,s
T
et y .
Soit enfin +X=sup(X,O) (resp. -X=sup(-X,O) ) avec +F
(resp. -F ) étant
s
s
les fonctions de répartitions
des coordonnées de +X (resp.
-X ). Les variables
+
+
+ T
-
-
- T
V,
y,
Y et
V,
y, et
Y sont aussi définies de la
meme manlere. Remarquons
+
+
+ T
que les Lemmes 2.1-2-3 sont encore valables pour les v.a.
V, Y, Y
et pour les
-
-
- T
v.a.
V, y, Y , T et &r étant choisis pour Y.
. r
1
•
'1::. r
'" .
~ " .'

---

-210
+T
-T
+
)
Si nous montrons que le nombre
des R. (resp.
R. ) parmi les
Y. (resp.
Y.
l
l
J.
l
pour l~i~n, est presque-sûrement inférieur à ~(~+nTI pour de larges valeurs
de n, pour tout O<n<~ , le lemme de l'Appendice impliquerait:
Jf k, l~k~n, +yr- [E(~+nil ,~+Yk,~+y~+ [E(~+n2J , n' p.s.
(resp. Jf k, ~k~n, -Y~-[Ë(~+nil,~-Yk,n ~yr+[!l(~+nil,n' p.s.)
Or nous avons: Jf j, 1~~, +y .
--y
. 1
::: Y.
. Et finalement «Q2»
serait
J,n
n-J+ ,n
J,n
prouvée dès que O<n<~/2. Il suffirait donc de prouver que
P( lim sup t
(1 /n»~)=P(Q«(»=O où l =}F{i, l<i<n Y.=R~}
n+OO
n
n
= = ' 1
J.
Supposons gue p(n(E»>O. Donc pour weQ(E) , on aura infiniment souvent (i.s.)
sur n l'égalité
+y~(w)=-Y~(w)=o au moins ~(~~pil+l fois. Et puisque les v.a.
l
l
+yT
-yT
. .
et
sont posltlves, on aura:
+ T
) - T
:v- wel2«() ,
Y[il(~-piJ,n(W = Y[E(~-piJ,n = 0
Ls.
+ T
Maintenant appliquons le Théorème 1.2 à la v.a. Y (0). Il existe alors nO ~
avec P(QO)=1 tel que
+ -1
+ T
(
+ -1
lf we QO' -(+ GT,O(~-P-()~ y [E(~-P2J ,n O)(w)~(+ GT,0(~-p+(+O),
quand n est assez large. En combinant cela avec les lemmes 2.1-2-3, on obtient:
+ -1
+ T
+ -1
:v- we12 , -6E+ HO (~-P-E)~ Y[E(~_p2J,n(O)(w)~6E+ HO (~-P+EtO),
0
quand n est
.assez large. Finalement, nous aurons pour wen nn( E);l<f> ,
O
+ -1
+ -1
(A) -6E+ HO (~-p-E)~~6E+ HO (~-p+E+O).
Si maintenant P(Q(E»>O infiniment souvent quand Efû, A reste vraie à la limite
!
+ -1
+ -1
:
pour tout O<P<~
et ort obtient
HO (~-p)~~ HO (IJ-p) pour tout O<fi)<~ . ·La con-
séquence est que +HOl(~)=O. Or +HOl(~)=A+FOl(~), pùisque +HO(x)=P(sup(X(O)+A,O)
+ -1
-1
+ -1
~x). Et donc
HO (u)=A+F
(u) si u>FO(-A) et
HO (u)=O si O~u<FO(-A). Par hypo-
O
+ -1
-1
thèse donc
HO (~)=A+FO (~»O. Cette contradiction complète la preuve.
Etape 3: Bornes de Xr:: :l
.
_ _ _ _ _ _ _ _ _ _ LilC!J ,n
Soit O<a<l, O<~, O<a-4~<a+4~<1. En utilisant «Q2»
pour un choix CH(E) où E<E ' on a
O
( 2 6) YT
<Y
< yT
.
~(a-2~2J ,n = rn~ ,n= [E(a+2~2J ,n' p.s., as n-++oo.
\\c. (
\\c. (
~" .'

---

l'
-211-
1
En appliquant maintenant les Lemmes 2.1-2-3 à (2.6), on obtient
1
1
Jf seS, -5€+A+F-
(a-2IJ-2€) ~ y r= ;;l
(s) ~ SE: +A+F- (a+2)J+2 € +0)
s
-
1l1<!J, n
-
s
r
presque-sûrement quand n++CO • Mais,
IYwiJ,n(s)(W)-X~iJ,n(S)(W)-AI<€,pour tout
couple (s,w) de SxQ. Il s'en suit, presque-sûrement, quand n+~,
1
1
1
Ji- seS, -6IJ+F- (a-4IJ).s.Xr-,0
(s)-S.6IJ+F- (a+4IJ+0).
S
-
lJ:I~, n
-
S
Et ceci prouve la partie (i) du Théorème 2.2.
Démonstration de la partie (ii) du Théorème 2.2.
Soit O<€.
Procédons à un choix CH(€)
avec A~O. Par les Lemmes 2.1-2-3, on a
-2€+G~~
1
u(a+3€+O) ~ F: (et+4€+0) ~ 2€+GT~ u(a+5€+0)
et
_2€+G- 1 (a-5€)~F-1(a-4€)~2€+G-1 (a-3€).
T,u
-
u
-
T,u
Puisque IlyTII91= sU P1.2.i':::'T Il xi Il , tous les nombres G~\\(I;;), 0<[,;<1, sont finis et
bornés par M. Finalement (ii) devient évident si 0<a-5€<a+5€<1
.
..
Exemples. Considérons le processus de Wiener {W(t), O~t~l} défini par P(W(t)~x)
l
=<lJ(xt -"2) où <lJ(.) est la fonction de répartition de la loi normale centrée réduite.
Le Théorème 2.2. affirme que
l.
1
l.
1
(2.7)
-6€+t 2 <lJ - (a-4€).s. Wr- :l
(t) -S. 6€+t 2 <lJ- (a+4€)
-
l..!lC!J, n
-
presque-sûrement quand n est large. Et pout le Pont Brownien associé B(t)~W(t)
+tW(l), nous aurons presque-sûrement, quand n~co,
1.
1
l.
1
(2.8) -6Ë+(t(l-t»2 <lJ- (a-4€)~ B~~ ,net) ~6E+(t(l-t»2 <lJ- (a+4€)
Il est clair ici que l'on aura la convergence uniform~ oes quantiles, c-à-d,
1 -1
1
limrr++co sUPse(O,l) 1W~iJ ,n(s)-s <lJ
(a)::: 0, p.s'.
et
l.
1
limrr++co sUPse(O,l) IB[piJ ,n(s)-(sCl-s»J <lJ- (a)1 ::: 0, p.s.
Ce sont là deux exemples particuliers du Théorème suivant.
". r
"
r

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-212-
Preuve. L'implication iii)
>i) découle du Théorème 2.3,
i)---~iii) du Théorème 1.3
et ii)
>iii) du Théorème de Dini.
Remarque. Les Théorèmes 2.2, 2.3 et 2.4 et le Corollaire 2.1 sont vrais dans le
cas plus général
X
où k/n-+ct
quand n++oo, O<a<l.
k ,n
B- Bornes et convergence des guantiles à la gueue.
Intéressons nous à X
où k/n + 0 ou k/n + 1 quand n+f-'X'.Puisque (-X).
:::::-X
. 1
k ,n
l,n
n-l+,n
l~i~n, nous étudierons uniquement X
avec k/n+l.
k,n
Théorème 2.5. Soit k, l~k~n, fixé. Sous les hypothèses du Théorèmes 2.2., nous
aurons pour tout U>O, 1-4U>O,
1
Jf seS, -6u+F- 0-4u) ~X k
(s) ~X
(s),
s
-
n-,n
-
n,n
presque-sûrement quand n++oo.
Preuve. Procédons à un choix CH(E) ou E est suffisamment petit et appliquons [02»·
Nous obtenons, presque-sûrement, quand n++oo,
yT
< y
.
~(l-2u-k/niJ -l,n =
n-k, n
Alors «01)) et les Lemmes 2.1-2-3 font le reste de la preuve. Et en raisonnant
analoguement, on aura
Corollaire 2.2. Sous les hypothèses du Théorème 2.2., ·nous avons pour tout U>O,
presque-sûrement, quand n++oo.
\\:' (
\\:. r
1" "

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-213-
Théorème 2.3. Soit 0<a<1. Supposons que les hypothèses du Théorème 2.2 soient
vérifiées. S'il exisite 8
tel que pour tout S. 0<a-8 <8<a+8 <1. la fonction
0
0
0
1
sl-->F-1(a) est continue et si F- (.) est continue en a, pour tout seS, alors
s
s
1
lim n++
(a)l::: 0, p.s.
OO
sup seS
IXr::::l
(s)_F-
l.l.lC!J ,n
s
Démonstration. On a par continuité F~l(S+O)=F~l(S) pour 0<a-8 <8<a+8 <1.
0
0
En plus la famille {sl-->F~1(a+80!p), pern, p~O} est une famille monotone de
fonctions continues définies sur la compact S et convergente ponctuellement vers
-1
s 1--7F
(a). Alors, par application du Théorème de Dini, on a:
s
(2.9)
sup
S IF-1(a+80!p)-F-1(a) 1+0 quand pt+oo ..
se
s
s
Similairement, on a
-1
-1
1
(2.10) sup
S 1F
(a-8 !p)-F
(a) + 0 quand pt+oo.
se
s
0
s
En combinant (2.9), (2.10) et le Théorème 2.2, on obtient le Théorème.
Enfin, nous caractérisons les v.a. dans C(S) par rapport à la convergence de
leurs quantiles.
Théorème 2.4. Soit l un intervalle ouvert de (0,1) . Avec les hypothèses du Théorème
2.2, les propositions suivantes sont équivalentes.
(i) .lfaeI, Xr:: :l
converge dans C(S). presque-sûrement ou en probabilité vers un
L!lC!J ,n
élément non aléatpire de C(S).
-1 ( )
.
F- 1( . )
( ii ) .lf aeI,
la fonction s ->F
a
est contlnue et
est continue en a pour
1
s
S'
tout seS.
(ii) * aeI, la famille
{ui-->F-1(u), seS} est équi-continue en a, c-à-d,
s
I
1
Hm
d \\
sup
S IF- (a+u)_F- (a) 1::: O.
U-rv
se
s
s
Corollaire 2.1: Supposons vérifiées les hypothèses du Théorème 2.2. Si la famille
1
{ul--->F- (a), seS}
est équi-continue en a, O<a<l, alors Xr:::':l
converge pres-
s
L.!lC!J ,n
que-surement vers
-le
A
F
a) quand n-++oo.
s
,",. r
1
1.'
, :

---

1
-214-
Il s'en suit
1
-1
.lf seS,
-61-1+F
0-41-1):s.X
:s.A (s)
F-l(l), p.s., quand
~L
n'Troo
s
-
n-k -
1
s
1 Le Théorème de Dini achève la preuve
1 La caractérisation complète des extr@mes est la suivante.
Thé rème 2.6. Sous les hypothèses du Théorème 2.5
avec un espace S séparable,
1 les seules possibilités de convergence de X k où k, l~k~n, est fixé, sont
n- ,n
les suivantes.
1
1) si le support de
1 Isup(X,O)11
est fini, c-à-d,
1
M+(X)= inf {a , Ilsup(X,O) Il,:S,a, p.s.}<c.o,
(i) ou bien Al(s) est continue en tout seS et alors 1im
Ilx k -Alll=o,p.s.
n~
n- ,n
(ii) ou bien il existe ST en lequel Al ( .) n'est pas continu et alors i l n'existe
pas d'é1é~ent non aléatoire de C(S), disons B, tel que
P(lim
IIx k -BII=O»o.
n+-tro
n- ,n
2) Si M+(X)=+OO, alors 1im
X k
(s) = -tro, p.s.
n+-tro
n- ,n
Démonstration. (1) Si M+(X)<+oo. Il s'en suit que Al est borné. Il est vrai aussi
que
:v- seS , =:los' pen )=1, JfweO , lim
X
k
(s)=Al(s), p.s.,
l
s
s
n++<'"
n- ,n
1
ùÙ SI est dense dans S. Soi t 04 la réunion des Os' sePl' On a
P(04)=1 et !fw ert ,V- seS , X _
n (s) (w) + Al (s) quand n++oo.
4
l
n k ,
(i) Maintenant si Al est continue, la famille {Al(s)-X
k
(s)(w), n~l}
est,
n- ,n
pour chaque w de 04' une fàmi11e monotone de fonctions continues sur SI et conven-
gente ponctuellement vers la fonction nulle. Une version du Théorème de
Dini
fondée sur la densité de SI dans S prouve que
.lf we Sl4'
sup
S
IAl(s)-X
k
(s)(w)1 + 0 quand n++c.o.
se 1
n- ,n
Il s'en suit immédiatement que sup
S IAl(s)-X
k
(s)(w)1 + 0 quand n++oo pour
se
n- ,n
tout we °4 ,
\\:' r
, .'
\\:
r
1
1 . '

---

l,
-215-
1 Exemples. Nous aurons pour le processus de Wiener et son pont Brownien associé
1 (voir exemple 1):
l.
1
1.
1
.!fu, O~u~l, Wk,n(u)~6w+u2 d>- (4w)~Wn_k,n(u»-6w+u2d>-
(1-4W) ,
1
1
l
1
Bk,n(u)~6~+(u(l-u»2 d>- (4U)~ Bn_k,n(u) ~ -6~+(u(l-U»2 <t>-_(l-4~)
presque-sûrement quand n++oo.
Ici nous aurons les résultats suivants:
et
a.s.
Bk,n - - -O:?,
pour tout couple (u ' u ) eJO, 1[2. Les mêmes résultats sont obtenus ~our Wk,n
1
2
et W k
. Cependant, nous n'avons pu obtenir
des résultats aussi forts dans
n- ,n
le cas général. Avant de donner la caractérisation complète des extrêmes, donnons
quelques conséquences du Théorème 2.5.
Corollaire 2.3. Supposons que les hypothèses du Théorème 2.5 soient satisfaites
avec un espace S séparable. Si sl--->F~1(1-4u) est continue pour tout O~<Uo<l
-1
et si u!-->F
(u) est continue à gauche de 1 pour tout SES, alors
s
lim
liA -X
n++oo
1
n-k, n II~
- - °
-1
où A1(s)~Fs (1), seS.
Preuve. Soit SI un sous-ensemble dénombrable et dense dans S. En appliquant le
Théorème 1.1, on a
Mais P(
rl Q )=P(Q3)=1 et X k-AI est continue. Alors de simples considérations
SES1
s
n-
sur la densité de SI dans S montrent que
"" ( 1 1.'

---

l'
-216-
(ii) Suppose que X k
(.)(w) converge dans C(S) vers, disons B, pour we~5 avec
n- ,n
P(~5»0. Puisque P( lim +
X
k
(s)=A (s) )=P(~(s))=l pour tout seS, on a
n +00
n- ,n
1
P(~(s) n~5»0 pour tout seS, c-à-d,
J{- seS,
P( B(s)=A (s) »0,
1
or le membre de gauche ne peut prendre que deux valeurs: zéro ou un. Donc
J{- seS,
B(s)=A1(s). Et ceci contredit l'hypothèse de (ii) et ainsi se prouve la
partie (ii).
2) La preuve est triviale.
Exemple. Soit le processus de Wiener sur (0,1), alors pour tout aeC(O,l),
Il (inf(W,a))n_k n - al 1+0 quand n4+00.
,
CONCLUSION. Ce premier article ouvre un champ d'investigations très vaste qui aura
naturellement besoin de toute la richesse des statistiques d'ordre dans R et aussi
des derniers développement de la théorie des probabilités dans les espaces de
Banach.
'"
r 1
: .. :

---

-217-
Appendice. Nous prouvons ici le résultat suivant:
Lemme: Soit (Yi)~~ une suite de nombres réels positifs telle qu'il existe
deux autres suites (xi)l~i~n et (Zi)l~i~n pour les on a
J x. et z.=O
l
l
l~i,;,n.
Yi =L
ou
,
z. et x.=o
l
l
Notons
1 = h, 1~
y.=z.}
et
1 ={ i, l~i~n, y.=X, }.
1
-
- '
l
l
2
l
l
Si
~ (Il)=m, alors on a les inégalités suivantes
(A)
x
< Y
< x
pour tout k, 1,9<.Sn_< ,
k-m,n =
k,n =
k+m+1,n '
-
où, par convention, -x
=x
= + ex> •
O,n
n+1,n
Preuve. Si k-m~l ou k+m~n, (A) est évident. Soit maintenant k-m>l et k+m<n.
Remarquons que si 8={ J', J'~_> , ( :=ln ., l <n .<n) (y"
=x
)} est vide, cela implique-
:::J J
= J=
J,n
n.J
rait que m~n-k+1, c-à-d, k+m~n+l, ce qui est contradictoire avec ce qui précède .
..
Donc p=min 8
est bien défini. Definissons pour tout intervalle l de {l, ... , n}
le nombre Ker)= =11= {J", jeI,(-::J n ., l<n.<n) (y.
=z
)}. Posons K =K(
II, kJ),
::J J
= J=
J , n
n.
1
~
J
K =K( J k, pJ ), K =K( ] p, nJ ). Il est aisé de vérifier que x
est le
2
3
n p
(k-K +1)-ième minimum
de l'ensemble {xi' ieI
}. Et puisque les xi sont tou?
1
2
positifs et puisque les xi sont nuls pour ieI , on conclut que
l
(R)
(B) prouve l'inégalité droite de (A). Pour prouver l'inégalité gauche, definissons
q=max {J', J"<k,(::J n,;, l<n.<n) (y"
=x
)}, qui est bien defini sinon on aurait
=
::J J
= J=
J , n
n.J
m~k. En posant Tl'=K([}, q]), T2=K( ] q, 1] ), T3=K( ] k, JÎI), on véfifie aussi
que x
est le (k-T -T )-ième minimum de l'ensemble {Xi' ieI } et finalement
n
1
2
2
q
(C)
y
~ x
= x
x
~ x
.
k,n
n
(k-T -T )+(T +T
k -T ,n
k-m.n
q
I
2
1
2
3
Finalement (B) et (C) impliquent (A).
"o. (
\\
(
,
1
~ •• .'
:,,'

---

-218-
REFERENCES.
1-Billingsley, P.(l968). Convergence of Probability measures. Hiley, New-York.
2-David, H.(l970). Order Statis.tics. Wiley, New-York.
3- De Haan, L.(l970). On regular Variation and its application to the
Convergence of Sample Extreme. Mathematical Centre Tracts, n_o32,
Amsterdam.
4-Deheuvels, P.(1984). Probabilistic Aspects of Multivariate Extremes. In:
Statistical Extreme and Applications,
Proc.
NATO ADV.
STUD.
INST.
Vimeiro/Port. 1983., NATO ASI. Ser., Ser. C, 131, pp. 117-130.
5- Galambos, J. (1978).
The As ympto tic Theor y of Ex t reme Order S ta t i st i cs.
Wiley, New-York.
6-Lo, G.S.(1986). Asymptotic behavior of Hill's estimate and Applications, J. Appl.
Probab, 23, 992-936.
7-Padgett, W.L. and Taylor, R.L.(l973).
Laws of Large Numbers for, Normed
Spaces and Certain Fréchet 1 s
Spaces. Lecture Notes in Math., 360,
Springer-Verlag, Berlin.
8'-Schae~er, H. H.(l974). Banach Lattices and Positive Operators. Spripger
Verlag, Berlin.
9-Heiss, L.(1963). On the asymptotic joint normality of quantiles from a multiva-
riate distribution. Journ., Res. Nat. Bur. Stand., B, Math. and Math. Physics.,
Vol. 68B, n02.
~ ...'.
,
"
.,,'

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