Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces

170 Pages·1989·6.73 MB·English

by  V. Paulauskas

Approximation Theory in the Central Limit Theorem
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.s.R.
Yu. I. MANIN, Steklov Institute ofMathemalics, Moscow, U.S.s.R.
N. N. MOISEEV, Computing Centre, Academy ofS ciences, Moscow, U.s.s.R.
S. P. NOVIKOV, Landau Institute ofT heoretical Physics, Moscow, U.s.S.R.
M. C. POLYVANOV, SteklovInstitute ofM athematics, Moscow, U.s.s.R.
Yu. A. ROZANOV, Steklov Institute ofM athematics, Moscow, U.S.S.R.
Volume 32
Approximation Theory
in the Central
Limit Theorem
Exact Results in Banach Spaces
by
v. Paulauskas
and
A. Rackauskas
Department of Mathematical Analysis,
Vilnius University, Vilnius, U.s.s.R.
Kluwer Academic Publishers
Dordrecht / Boston / London
Library of Congress Cataloging in Publication Data
I.
Paulauskas, V. (Vigantas I~novich)
Approximation theory in central limit theorems-
exact results in Banach spaces.
(Mathematics and its applications. Soviet series)
Translation of: Tochnost' a pproksirnat~ii v t'1entral' noi
peredel'not teoreme v banakhovykh prostranstvakh.
Bibliography: p.
Includes index.
1. Asymptotic distribution (Probability theory)
2. Banach spaces. 3. Central limit theorem. 4. Con
iD.
vergence. I. Rachkauskas, A. (Alfredas IUrgevich)
II. Title. III. Series: Mathematics and its applications
(Kluwer Academic Publishers). Soviet series.
QA273.6.P3813 1989 519.2 88-23101
ISBN-13: 978-94-011-7800-6 e-ISBN-13: 978-94-011-7798-6
DOl: 10.1007/978-94-011-7798-6
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus NiJboff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Translated by B. Svecevicius and V. Paulauskas
Original publication:
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TeopeMe B 6aHaxoBblx rrpoCTpaHcTBax
published by Mokslas, Vilnius, © 1987
All Rights Reserved
This English edition © 1989 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 1989
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To our reviving Homeland
LITHUANIA
Contents
Editor's Preface ix
Preface xi
Introduction xiii
list of Notations xvii
Chapter 1. General Questions of the Distribution Theory in Banach Spaces
1.1. Random Elements in Banach Spaces
1.2. Main Facts on Gaussian Measures in Banach Spaces 5
1.3. Weak Convergence of Probability Measures and Some Inequalities 7
Chapter 2. Some Questions of Non-linear Analysis 11
2.1. Differentiation in Normed Spaces 11
2.2. Smooth Approximation in a Banach Space 22
2.3. Supplements 41
Chapter 3. The Central limit Theorem in Banach Spaces 44
3.1. Operators of 'JYpe p 44
3.2. The Central Limit Theorem 52
3.3. Supplements 59
Chapter 4. Gaussian Measure of e-Strip of Some Sets 62
4.1. General Remarks 62
4.2. Distribution Density of the Norm of a Gaussian Element in ~ 63
4.3. Distribution Density of the Norm of a Gaussian Element in Co 67
4.4. Supplements 7S
viii CONTENTS
Chapter S. Estimates of Rate of Convergence in the Central Umit Theorem 80
5.1. Estimates of the Rate of Convergence on Sets 80
5.2. Estimates of the Rate of Convergence in the Central Limit Theorem in Some
Banach Spaces 11 0
5.3. Estimates of the Rate of Convergence in the Prokhorov Metric and the
Bounded Lipschitz Metric 122
5.4. Supplements 135
Bibliographical Notes 139
References 144
Index 154
SERIES EDITOR'S PREFACE
~Et mai .... , si j'avait su comment en revenir. One service mathematics has rendered the
je n'y serais point aIIe.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non-
The series is divergent: therefore we may be sense' .
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Lie algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together_ And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more_ Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
x SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depresaing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if dcctronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
1bere is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All thm: have applications in both
dectrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'rea!' results. Similarly, the first two topics named have
alIcady provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunatdy.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), ydiow (China), red (USSR), blue (Eastern Europe), and green (everything
dse), stilI applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series stilI aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one fidd of enquiIy have, and have had,
on the devdopment of another.
The laws of large numbers, or, better, central limit theorems for sums of identically distributed
random variables, are basic to (both applications and foundations of) probability and statistics.
Given the frequent and natural role of Hilbert and Banach spaces as the spaces of values for such
random variables, the question of how wdl these sums are approximated by Gaussian distributions
becomes of fundamental importance. The present volume presents an up-to-date unified treatment
of this topic for the case of Banach spaces.
The complementary topic of rates of convergence to the Gaussian limit law and the topic of
large deviations in a HilbeJ:t space is the subject of a forthcoming monograph by Prokhorov, Sazo
nov and Jurinsky (1990, translation in this series 1991). Together the two books should give the
reader a very good idea of what is known at present concerning these matters.
The shortest path between two uuths in the Never lend books, for no one ever returns
real domain palset through the complex them; the only book. I have in my library
cIomaiD. arc books that other folk have lent me.
J. Had.mard Anatole France
La physique De nous doone pas seuJ_t The fuDCtion of an expert is not to be more
!'occ:uion de raoudre des probIanes ••. eIIe right than other peop1c, but to be wrong for
IIOUa fait pRllClltir Ia soIutica. more sophisticated rasoDS.
H. PoiDcut David Butler
Bussum, June 1989 Michid Hazewinkd
Preface
The intention of this book is to analyse the accuracy in approximating the distribution of sums
of independent identically distributed random elements in a Banach space by Gaussian
distributions. During the past two decades quite a number of results, resembling the final
ones, have been obtained in this field. At the same time a close relation has been found
between the estimates of the rate of convergence in the central limit theorem and some other
branches of mathematics (in particular, non-linear analysis). We have tried to systematize the
known results by presenting them in a unified form. An 'operator language' is used for this
purpose.
Due to limited space, it was not possible to touch upon certain trends in the asymptotic
analysis of distributions of random element sums such as the rate of convergence to stable
laws, or asymptotic expansions in the central limit theorem, approaching infinitely divisible
laws in Banach spaces. In the future, the above questions, no doubt, may comprise a separate
book.
Even concerning rates of convergence to the Gaussian law, not all questions have been
discussed to the same extent. It is the hope to the authors, however, that the present book and
the forthcoming monograph of I.V. Prokhorov, V.V. Sazonov and v.v. Iurinskii on the
estimates of convergence rate and large deviations in a Hilbert space will provide the reader
with a present-day review of research in this field.
The book consists of five chapters. Each chapter is divided into sections. At the end of
Chapters 2 -5 'Supplements' are provided which present short formulations of the results not
included in the main text, but closely related to it.
The sign 0 denotes the end of a proof. The following abbreviations are used in the text: a.s.
= almost surely; i.i.d. = independent identically distributed; r.v. = random variable; CLT =
the central limit theorem, B-r.e. = random element in the space B. In each chapter
theorems, lemmas, propositions and formulas are marked by two numbers: (4.3), for instance,
denotes formula number 3 in Section 4 of the given chapter. If a different chapter is referred
to, three numbers are used, e.g. (5.4.3), the first number denoting the number of the chapter.
The authors' express their sincere thanks to their teachers, Prof. I. Kubilius and Prof. V.
Statulevicius, for constant attention to the present work.
We also thank Dr. v.Yu. Bentkus who read all the proofs with the utmost care and gave a
number of remarks that contributed much to the improvement of the book. Last, but not least,
we express our gratitude to allow colleagues who helped us prepare the book for publication.