Ustatistics are universal objects of modern probabilistic summation theory. They appear in various statistical problems
112 63 32MB
English Pages 432 [435] Year 1996
Table of contents :
Dedication
Contents
Preface
Introduction
1 Basic definitions
2 Inequalities
3 Law of large numbers
4 Weak convergence
5 Functional limit theorems
6 Approximation estimates
7 Asymptotic expansions
8 Large deviations
9 Law of iterated logarithm
10 Dependent variables
Bibliographical supplements and comments
Bibliography
Index
UStatistics in Banach Spaces
USTATISTICS IN B A N A C H SPACES
Yu.V. Borovskikh
///VSP/// Utrecht, The Netherlands, 1996
VSP BV P.O. Box 346 3700 AH Zeist The Netherlands
© VSP BV 1996 First published in 1996 ISBN 9067642002
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
CIPDATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Borovskikh, Yuri V. Ustatistics in Banach spaces / Yuri V. Borovskikh. Utrecht: VSP With index, ref. ISBN 9067642002 bound NUGI815 Subject headings: Ustatistics / Banach spaces
Printed
in The Netherlands
by Koninklijke
Wöhrmann
BV,
Zutphen.
Managing Editors:
E.V. Cherkashin International Mathematical Center, Kiev, Ukraine O.A. Cherkashina Institute of Mathematics, Kiev, Ukraine
Language Editor:
J. Thomas University of Salford, Manchester, UK
To My grandson Anton
God, Thou great symmetry, Who put a biting lust in me From whence my sorrows spring, For all the frittered days That 1 have spent in shapeless ways Give me one perfect thing. Anna Wickham {"Envoi", 1921)
Contents Preface
xi
Introduction
1
1 Basic definitions
5
1.1
Onesample [/Bstatistics
1.2
Multisample l/Bstatistics
15
5
1.3
Von Mises' statistics
21
1.4
Banachvalued symmetric statistics
23
1.5
Permanent symmetric statistics
24
1.6
Multiple stochastic integrals
27
1.7
Bvalued polynomial chaos
29
2 Inequalities
3
31
2.1
Inequalities based on the Hoeffding formula
31
2.2
Martingale moment inequalities
37
2.3
Maximal inequalities
42
2.4
Contraction and symmetrization inequalities
44
2.5
Decoupling inequalities
48
2.6
Hypercontractive method in moment inequalities
64
2.7
Moment inequalities in Banach spaces of type p
71
Law of large numbers
73
3.1
Onesample L/Bstatistics
73
3.2
Multisample [7Bstatistics
79
3.3
Von Mises' statistics
84
3.4
Estimates of convergence rates
86
4 Weak convergence 4.1
95
Central limit theorem
95
4.2
Convergence to a chaos
102
4.3
Multisample [/Bstatistics
114 vii
Vili
5
6
7
8
Contents 4.4
Poisson approximation
124
4.5
Stable approximation
129
4.6
Approximation with increasing degrees
132
4.7
Symmetric statistics
146
4.8
[/statistics with varying kernels
152
4.9
Weighted (/statistics
160
Functional limit t h e o r e m s
165
5.1
Nondegenerate kernels
166
5.2
Degenerate kernels
170
5.3
Weak convergence to a chaos process
174
5.4
Weak convergence in the Poisson approximation scheme
182
5.5
Invariance principle for symmetric statistics
184
5.6
Functional limit theorems with varying kernels
187
5.7
Weak convergence of (7processes
191
Approximation estimates
201
6.1
General methods of estimation
201
6.2
Rate of normal approximation of (/Kstatistics
205
6.3
Estimates with increasing degree
215
6.4
Nonuniform estimates
224
6.5
Rate of chaos approximation
226
6.6
Normal approximation of [/Hstatistics
237
6.7
Multisample L/Hstatistics
277
6.8
Estimates in central limit theorem in
279
6.9
Rate of Poisson approximation
291
A s y m p t o t i c expansions
295
7.1
Expansions for nondegenerate i/Rstatistics
296
7.2
General method of expansions
311
7.3
Expansions with canonical kernels
316
7.4
Expansions with arbitrary kernels
323
Large deviations
327
8.1
Exponential inequalities
327
8.2
Moderate deviations
335
8.3
Power zones of normal convergence
337
8.4
Probabilities of large deviations for l/Hstatistics
339
Contents 9
ix
Law of iterated logarithm
349
9.1
i/ftstatistics
350
9.2
l/Hstatistics
351
9.3
Bounded LIL
352
9.4
Compact LIL
357
9.5
Functional LIL
360
9.6
Multisample [/Bstatistics
361
10 D e p e n d e n t variables
365
10.1 Symmetrically dependent random variables
365
10.2 Weakly dependent random variables
366
10.3 Bootstrap variables
369
10.4 Order statistics
375
Bibliographical s u p p l e m e n t s and c o m m e n t s
377
Bibliography
385
Index
419
Preface [/statistics are universal objects of modern probabilistic summation theory. T h e y appear in various statistical problems and have very important applications.
The
mathematical nature of this class of random variables has a functional character, and therefore, leads to the investigation of probabilistic distributions in infinitedimensional spaces. T h e situation when the kernel of a [/statistic takes values in a Banach space, turns out to be the most natural and interesting. In this case the construction of the probabilistic theory of such [/statistics is based on methods of probability theory in B a n a c h spaces. T h e mathematical beauty and universal attractiveness of [/statistics is clearly explained by the universal idea of s y m m e t r y lying in its definition. T h e idea of s y m m e t r y is dominant in modern probability theory in Banach spaces. T h a t is why [/statistical objects are natural in all aspects of probability theory in Banach spaces.
In concentrated form this
idea is revealed in the representation of every [/statistic as an integral functional with respect to a permanent random measure.
A s y m p t o t i c properties of random
permanents tell us that a [/statistic is a statistical image of an element of chaos, and a f a m i l y of [/statistics serves as a statistical model of chaos processes. In this book I present in a systematic form the probabilistic theory of [/statistics with values in B a n a c h spaces (i/Bstatistics). which has been developed up to the present time.
T h e exposition of the material in this book is based around the
following topics:  algebraic and martingale properties of [/statistics;  inequalities;  law of large numbers;  the central limit theorem;  weak convergence to a Gaussian chaos and multiple stochastic integrals;  invariance principle and functional limit theorems;  estimates of the rate of weak convergence;  asymptotic expansion of distributions;  large deviations;  law of iterated logarithm;  dependent variables; xi
xu
Preface
 relation between Banachvalued (/statistics and functionals from permanent random measures. All asymptotic properties of i/Bstatistics depend on Banach space B, rank r, kernel $ and moment limitations for
For r = 1 (nondegenerate kernel) the
asymptotic behaviour of (7Bstatistics can essentially be reduced to the asymptotic behaviour of sums of Bvalued random elements. If r > 2 (degenerate kernel), then the family of possible limit distributions for (7Bstatistics is extended by the distributions of functionals of infinitedimensional Gaussian vectors defined by kernel
distributions of multiple stochastic integrals
and functionals of them, distributions of Gaussian chaos and Poisson chaos and so on. I hope that this book will serve to introduce many people to the beautiful theory of (/statistics in Banach spaces. Many people have helped me in the writing of this book, either through discussions or by reading one of the versions of the manuscript. I am extremely grateful to V.S. Korolyuk for his continuous support in my efforts and his diligence in exploring stochastic laws of ¡7statistics during the last twenty years. My deep gratitude is expressed to O.A. Cherkashina (Adamenko) (Institute of Mathematics, Kiev) and E.V. Cherkashin (International Mathematical Center, Kiev) for their activity in publishing mathematical manuscripts, and especially for their editorial work on several versions of this book. I would also like to thank Joseph Thomas (British) for his English language editing of the text.
Komarovo, near St
Petersburg,
Russia
8th May, 1995
Yuri V. Borovskikh
Introduction In this book we consider (/statistics £/„ = nH
£
,Xlm)
with Banachvalued kernels $ : Xm —> B. The exploration of the stochastic laws of random elements of this type is accompanied by all the problems which are inherent in the investigations of stochastic properties of sums of independent Banachvalued random vectors (this corresponds to the case m = 1). In Chapter 1 we present preliminary information which includes the definition of canonical functions gc; rank r of kernel
1 < r < m; canonical Hoeffding
representation; multiple stochastic integrals and Banachvalued polynomial chaos. Naturally, our treatment of these definitions is relevant to (7J3statistics. Chapter 2 is devoted to inequalities. Here we have moment inequalities related to the geometry of Banach spaces and maximal inequalities following from martingale properties of l/Bstatistics. Decoupling inequalities, consisting in the comparison play a very important role in what follows.
between £ $ (   i / n   ) and Here [/ dec =
n
H
£
is a decoupling version of Un, and random variables {X{,...
,Xl}™ = 1 are indepen
dent copies of { X i , . . . , Xn}. Laws of large numbers for [/Bstatistics and for Banachvalued von Mises' statistics are considered in Chapter 3. The estimates of the rate of convergence in these theorems are obtained by martingale inequalities. Chapter 4 is devoted to the study of the convergence in distribution of UBstatistics and the different functionals of them.
Conditions of the theorems are
formulated in terms of canonical functions gc, c = r , . . . ,m, on rank r, 1 < r < m.
and are dependent
In the nondegenerate case (r = 1) in limit we have
asymptotic normality. If kernel $ is degenerate (r > 2), then Gaussian chaos arises if the considered Banach space is of type 2. Under additional moment conditions for canonical functions gc these statements hold true for multisample [/Bstatistics and symmetric statistics. We also consider weak convergence under the conditions 1
2
Introduction
of Poisson approximation, stable approximation and with increasing degrees. T h e results for [/statistics with varying kernels and for weighted [/statistics are adduced as well. Weak convergence of probability measures generated by [/statistics in functional spaces is considered in C h a p t e r 5. R a n d o m [/processes of a stochastic sequence { [ / n ( $ ) } can be defined in different ways. It is possible to consider a partial sum process similar to (i/[ n ( ]($), 0 < t < 1) as a sequence of processes indexed by p a r a m e t e r t when $ is fixed. Another interpretation of ( [ / „ ( $ ) , $ £ J 7 ) is a stochastic process indexed by family T consisting of measurable s y m m e t r i c kernels. In this chapter, for such [/processes we study t h e weak convergence in nondegenerate and in degenerate cases, and in Poisson approximation schemes. Martingale functional limit theorems are applied to [/statistics with varying kernels. In C h a p t e r 6, t h e rate of weak convergence is investigated, and various approximation estimates depending on a Banach space
a kernel $ and a distribution P
are obtained. In different situations we apply different m e t h o d s of analysis. For the estimation of t h e rate of Gaussian convergence we apply t h e characteristic function m e t h o d and t h e m e t h o d of compositions. In Poisson approximation cases we apply the C h e n  S t e i n m e t h o d .
Rank r of kernel $ has an effect upon t h e form of t h e
limit distribution. T h a t is why it is natural t h a t t h e rate of convergence depends on this rank r . We consider nonuniform estimates and estimates with increasing degree both in Gaussian approximation and in Poisson approximation schemes. T h e most interesting estimates are obtained for l/Hstatistics. C h a p t e r 7 is devoted to t h e problem of t h e refinement of t h e limit theorems of t h e weak convergence of probability distributions generated by [/Bstatistics. This problem is connected with a s y m p t o t i c expansion and clarification of t h e conditions for it to be true.
These conditions can depend on the m e t h o d of investigations.
T h e m e t h o d of characteristic functions d e m a n d s t h e well known C r a m e r condition and t h e conditions of t h e m o m e n t s of individual terms. In this problem, rank r of kernel $ has an essential effect. Depending on different values of r this problem has different solutions. But an asymptotic expansion can be written for every r and t h e coefficients of this expansion can be represented in t h e form of multiple stochastic integrals if kernel $ satisfies some additional conditions. In C h a p t e r 8 we consider t h e behaviour of probability P(nr'2\\Un6\\ in t h e region x = 0(nf3)
>x)
with some 0 < ¡3 < 1. First we obtain some exponential
inequalities for all x > 0, and then we study t h e large deviations of this probability when x = x(n)
—* oo as n —> oo. T h e results depend on rank r of kernel $ and
t h e properties of a Banach space B. For [/Hstatistics t h e estimations of t h e rate of convergence in t h e regions of large deviations are obtained.
Introduction
3
Chapter 9 is devoted to the law of iterated logarithm for (7Bstatistics.
For
[/statistics with values in Banach spaces of type 2, the validity of bounded LIL, compact LIL and functional LIL is proved. Corresponding theorems include conditions for canonical functions gc. Chapter 10 serves as an introduction to the theory of l/Bstatistics when the random variables Xi,...
,Xn
are dependent.
Theoretically, for this class of ran
dom 5  v a l u e d elements, we can consider the whole spectrum of asymptotic and nonasymptotic problems which was considered for independent random variables. T h e author confines himself to the variables for which he has a personal preference: symmetrically dependent random variables, weakly dependent random variables, bootstrap variables and order statistics. T h e bibliographical supplements and comments contain remarks on the scientific works which were used during the writing of this book, and those which influenced the author. Additionally in the bibliography, the principal papers where the exploration of the stochastic laws of [/statistics were actively conducted, can be found at the beginning with the appearance of the fundamental work of Hoeffding (1948a).
Chapter 1 Basic definitions 1.1
Onesample UBstatistics
Definition of UBstatistics. Let B be a real separable Banach space with a norm  •  and let B* be the dual to space B. Denote by x*(x) the value of functional x* £ B* at x € B. Let Xi,... ,Xn be independent random variables taking values in the measurable space (X, X) and all with identical distribution P. Consider a Bochner integrable symmetric function $ : Xm —• B of m variables given on Xm and taking values in B. We define a Ustatistic as follows (1.1.1)
It is clear that Un £ B. Using terminology of Borovskikh (1986) [/statistic (1.1.1) with a Bvalued kernel $ is called a UBstatistic. In particular, if B = R it is called a URstatistic and if B = H, where H is a real separable Hilbert space, it is called a UHstatistic. Let V = {P} be a class of probability distributions on ( X , X). By 6 : P —> 9(P), we denote a functional given on V and taking values in B where 9{P) = j . . . j $ ( x l 5 . . .
,xm)P{dXl)...P{dxm)
If £ < oo, then Un is an unbiased estimate of the 5valued element 6{P) — E9{XU... ,Xm). In particular, if in (1.1.1), m = 1, then Un = n _1 ($(A"i) + • • • + $(A"„)) is the sum of independent identically distributed (i.i.d.) Bvalued random variables. Therefore, [/statistic (1.1.1) can be treated as a functionalalgebraic generalization of usual mean. Without loss of generality, we suppose that $ is a symmetric function of its arguments, since otherwise we can use a symmetric kernel 5 m $ according to the 5
Basic
6
definitions
formula Sm$(xi,...
,xm) = — ,x,J m! where the summation is carried out over all m! permutations ( ¿ 1 , . . . ,im)
of the
numbers ( 1 , . . . , m). Then [/statistic (11.1) is defined as follows £
£/„=nH
(Xn,... ,Xlm),
(1.1.2)
where rJ" 1 ' = n(n — 1) • • • (n — m + 1). Note, that for all m < n, we have CT
l< ll7 i...^, m n.
Consequently, for
different values of parameter m we have 1)
if m = 1,
then i/n = rT1
2)
i f m = 2,
then Un = (§)
n1)
n)
if m = n — 1,
i f m = n,
then Un = n
+ • • • + $(Xn))
X3),
Ei : (Ai,...
, Xn), Xn+i,
Xn+2,...
}.
It is clear that 25n D 25 n + i for all n = 1 , 2 , . . . . For the stochastic sequence {Un, ®n). Lemma 1.1.3 holds. E x a m p l e 1 . 1 . 3 . Let X = R and Xni
< Xn2
corresponding to random variables Xi,...
Xn.
®n = a{w : Xni,... the cralgebra generated by Xni,... It is clear that 5Bn D Un
< ...
< Xnn
be ordered statistics
Denote by
,Xnn,Xn+i,Xn+2,
, Xnn, Xn+i,
• • •}
Xn+2,
•..
.
for every n > 1. Therefore =
E(*(Xil,...,Xim)
=
E(*(Xiit...
®n) ,Xlm)\Xnl,...
,Xnn)
for all 1 < ¿i < . . . < im < k and m < k < n.
H o e f f d i n g f o r m u l a . Let k = [n/m] be the integral part of the number n / m . We set tpn(xi,...
,xn)
=
k~1(^(x1,...
,xm)
+
+ $ ( z m + l , . . . , 22m) H Let and let
h $(z/fcmm+l, • • •
,Xkm)).
be the sum over all nl permutations ( ¿ i , . . . ,in) of the numbers ( 1 , . . . , n ) denote the sum over all
combinations ( ¿ i , . . . ,im)
constructed from
the numbers ( 1 , . . . , n). Then £ 1, j = 1 , . . . , c, and taking its values in Banach space B. A multisample f /
n = n ( n j ) \mi/ ]=\
UBstatistic
is defined as follows
a = i j u . . . ,ijmj,
J = 1,... ,c),
(1.2.1)
where n = ( n j , . . . , n c ) and the summation is carried out over all 1 < iji < . . . < ijm,
o.
32
Inequalities T h e integrability of Sn has been studied by a number of authors.
T h e most
general result in that direction is the following inequality obtained by T a l a g r a n d (1989): 5„,
1.
E\\Un0\\p 2, where one can choose a„ = ( l 8 \ / 2 ) V 
Note that if B — R, then the module of smoothness pit) is equal to max(0, t — 1). Therefore, p(t) = 0(t2),
t —• oo, and we can assume that the space R is 2uniformly
smooth. Thus, for the case B = R Corollary 2.2.2 also holds. Corollary 2.2.3. Assume that Banach space B is 2uniformly
smooth, r = m, and
< oo for some q > 1. Then
for 1 < q < 2, and / \ g+i E \ \ u
n
\ \ < < a ™ r )
n
2
^ 
2
^
E \ m
q
for q> 2.
Inequalities for multisample lIBstatistics. Consider the random element 5 fic j in (1.2.9). smooth and £jf1j'7 < oo for
T h e o r e m 2.2.3. Let Banach space B be puniformly q > 1. Then the inequality E\\S
h S
\r d, d > r where d = d\ + • • • + dc, r = r^ + • • • + rc and r is the rank vector in (1.2.7); aq is the constant in (2.2.1). Proof. We can write Snd=
in (1.2.9) as follows
'"' 9dix]t» 1 1. Then the inequality £ 1 1 « <
1, and let {a^}, k > 1, be a nonincreasing sequence of nonnegative numbers.
Then
P( max a t 5 f c c  > t) < r ' { < £   S n c   ' + c n For the reverse martingale (Un, ®„), n = r, r + 1 , . . . , the following assertions are valid, which are similar to Theorem 2.3.2 and Corollary 2.3.2. T h e o r e m 2 . 3 . 3 . Assume nondecreasing
sequence
that
< oo for q > I, and that {&&}, k > 1, is a
of positive numbers.
Then
P( m a x U k  9 \ \ > t) < n m.
T h e o r e m 2.3.4. The following inequalities are valid: P( sup \\Uk  0 > t) < ( " V i k>n
+ PcMUn
 0 (ln+£/n  A l l ) '  1 ) }
for any t > 0 and £(sup \\uk  0?) < k>n
lcqE\\Un
 6\\\
q>
1
where In+z = m a x ( l , l n 2 ) , 2 > 0;
Afe)'.
^ ( ^ r
P r o o f . By Lemma 1.2.2 for n > k
E{Un  0®s) = uke. Hence, we obtain
for any q > 1. Thus, the stochastic sequence verse submartingale with multivariate index k.
—
is an i?valued re
Therefore, the conclusion of the
present theorem comes from the known inequalities for such submartingales (see, e.g. Fazekas (1983, 1985)).
2.4
Contraction and symmetrization inequalities
Contraction and symmetrization inequalities provide a comparison between moments (and tails) of sums
Xi,
an 0,
P( max St > t) < 2P(5„ > t).
Contraction
and symmetrization
inequalities
45
The proof of Levy's inequalities are in many papers (see, for example, Araujo and Gine (1980), Theorem 2.6). L e m m a 2.4.2. Let ty: B
R be a convex function
Y and EY = 0 in the sense of Bochner's EV(X) In
on B. If X is independent
integration,
< EV{X
of
then
+ Y).
particular, £11X11" < E\\X + Y\\P,
p> 1.
P r o o f . For every x € B we have x = E(x + Y) and \t(x) = ty(E(x + Y))
[by Jensen's inequality]
< EV{x + Y). Then apply Fubini's theorem.
The following inequality is an obvious and simple consequence of Lemma 2.4.1 and Lemma 2.4.2. L e m m a 2.4.3. Let X\,... elements.
,Xn
be independent
zeromean
Let ^ ( z ) , x > 0, be a convex increasing function.
random
Banachvalued
Then
Proof. Let { X / } be an independent copy of {A";}. Then {Xi — X / } is a sequence of independent symmetric random elements. Further, [by Lemma 2.4.2] [by Lemma 2.4.1] [because of convexity] t=i n
< 2
i=1 t=l
n
46
Inequalities
L e m m a 2.4.4.
Let Xi,...
,Xn
be independent
with values in B. Then, for each sequence a\,... P
2P
(X>.*
and symmetric
random
variables
,an 6 R, and any t > 0,
(,^f) +
X}) + SiiWiXj)
X,) + + t3(X}:x)
[by formula (2.5.6) on the first term] = E*(sup4 
y
£
E^EHiAX^X])
E ^ ^ x i x , ) 
Y
Y
Exl^3(Xt,X^)
¿^«m*/,*;)!)
n m / p ) +
Hence, if condition (3.1.1) takes place, then n m P (   $   >nm/p)^
0
oo J P (   $   > t) dt.
Onesample
and
75
UBstatistics
oo nm(pl)/p
J
p(\\Q\\>t)dt>0
n">/p
as n —y oo. If we choose n = [t\plm for t > 1 where [t)dt
=
o o
n
for 1 < p < 2. Then
m ( 2  p ) / ( 2 p)
J
„m/p
¿P($ >
+
j
tP($>i) 0
valid:
as n  > oo .
P r o o f . T h e case p = 1 is contained in Theorem 3.1.1. We assume in the following that 1 < p < 2. Using formula (3.1.4), we write the following inequality E\\nm'pSn\\p
< 2p1£nm/p5n($1)p
+
2p1£nm^5n($1)p.
B y Corollary 2.7.1 i5n'»/P5n($1)r
ml
r
P
nm/p),
i.e. £nm/p5n($1)''0
as
R. It is natural to consider a twoparameter convergence in the strong law of large numbers for Unin2,
i.e. convergence as n\ A n? tends to infinity,
where ni A n2 is the minimum of n\ and n 2 . In (3.1.5) we assume that X i , X 2 , . . . are independent random variables with a common uniform distribution on (0,1). Clearly, EUnin2 where
— 0,
l I 6
" I I 0 0
®(x'y)dxdy
However, if m i n ( n i , n 2 ) —> oo, then the integrability of $ does not guarantee the validity of the strong law of large numbers. This fact is reflected in the following theorem. Denote
WMTSWFW where l n + z = l n ( m a x ( l , z)).
(3 I6)

Multisample T h e o r e m 3.1.4. Let
E\$(XI,X
2
)\
nij,
i , j

= 1 , . . . , c},
(3.2.2)
0 < A < 1. Denote by a vector q — (qi, ••• ,qc) with components 1 < qi < 2,
i = 1 , . . . , c.
Let f = ( r i , . . . , rc) be a vector of ranks and a vector d = ( d j , . . . , dc) from (1.2.7). Let r = ri H
b rc, d = di H
7 f J = d/(dr i.e.
E,c=iiri/
n

J=i
0
as
n
x
^
oo .
By Corollary 2.7.1 E\\Sni(g})\\2
< ft j^mAY^r'nfEWgA'lilM 3 =1