Sums of Independent Random Variables

359 Pages·1975·20.533 MB·English

by  Valentin V. Petrov (auth.)

Ergebnisse der Mathematik
und ihrer Grenzgebiete
Band 82
Herausgegeben von P. R. Halmos . P. J. Hilton
R. Remmert· B. Szokefalvi-Nagy
Unter Mitwirkung von L. V. Ahlfors . R. Baer
F. L. Bauer· A. Dold . J. L. Doob . S. Eilenberg
K. W. Gruenberg· M. Kneser· G. H. Muller
M. M. Postnikov . B. Segre . E. Sperner
Geschiiftsftihrender Herausgeber: P. J. Hilton
V. V. Petrov
Sums of Independent
Random Variables
Translated from the Russian by
A.A. Brown
Springer-Verlag
Berlin Heidelberg New York 1975
Valentin V. Petrov
Leningrad University, J~eningrad, U.S.S.R.
Translator:
Arthur A. Brown
vVashington, D.C., U.S.A.
Title of the Russian Original Edition:
Summy Nezavisimyh
Slncamyh Velicin
Publisher: "Nauka", Moscow 1972
An edition of this book is published by thc Akadernie-Verlag, Berlin,
for distl'ib'ltion in socialist countries
AlVIS Subject Classifications (1970):
Primary 60-XX, 60 Fxx, 60 Gxx
Secondary 62-XX
ISBN -13:978-3-642-65811-2 e-ISBN-13:978-3-642-65809-9
DOl: 10.1007/978-3-642-65809-9
Library of Congress Cataloging iu PuulicatIon Data. Petrov, Valel~tin Vlaclimirovich.
Sums of independent random variables. (Brgebnisse del' nluthematik und ihrel' Grenz
gebiete; Bd. 82). Translation of Smnmy nezavisimykh slndmlnykh velichin. Biblio
graphy: p. Includes index. 1. Stochastic processes. 2. Summability theory. 3. Dis!!'i
bution (Probability theory) 1. Title. II. Series. QA274.P4813. 519.2 75-5766.
This work is subject to mpyright. All rights are reserved. whether the whole or part
of the TI1aterial is concerned. specifically those of translation, reprinting, re-usc of
illustrations, broadcasting, reproduction by photocopying machine or Bimilar TI1Cans,
and storage in data bank~. eIHler § 54 of the German Copyright La,v ,vhcre copies are
n1ade for other than private use, a fee is payable to the publisher, the amount of the
fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin.· Heidelberg 1975.
Softeover reprint of the hardcover 1st edition H)75
V. V. Petrov
Sums of Independent
Random Variables
Translated from the Russian by
A.A. Brown
Springer-Verlag
New York Heidelberg Berlin 1975
Valentin V. Petrov
Leningrad University, J~eningrad, U.S.S.R.
Translator:
Arthur A. Brown
vVashington, D.C., U.S.A.
Title of the Russian Original Edition:
Summy Nezavisimyh
Slncamyh Velicin
Publisher: "Nauka", Moscow 1972
An edition of this book is published by thc Akadernie-Verlag, Berlin,
for distl'ib'ltion in socialist countries
AlVIS Subject Classifications (1970):
Primary 60-XX, 60 Fxx, 60 Gxx
Secondary 62-XX
ISBN -13:978-3-642-65811-2 e-ISBN-13:978-3-642-65809-9
DOl: 10.1007/978-3-642-65809-9
Library of Congress Cataloging iu PuulicatIon Data. Petrov, Valel~tin Vlaclimirovich.
Sums of independent random variables. (Brgebnisse del' nluthematik und ihrel' Grenz
gebiete; Bd. 82). Translation of Smnmy nezavisimykh slndmlnykh velichin. Biblio
graphy: p. Includes index. 1. Stochastic processes. 2. Summability theory. 3. Dis!!'i
bution (Probability theory) 1. Title. II. Series. QA274.P4813. 519.2 75-5766.
This work is subject to mpyright. All rights are reserved. whether the whole or part
of the TI1aterial is concerned. specifically those of translation, reprinting, re-usc of
illustrations, broadcasting, reproduction by photocopying machine or Bimilar TI1Cans,
and storage in data bank~. eIHler § 54 of the German Copyright La,v ,vhcre copies are
n1ade for other than private use, a fee is payable to the publisher, the amount of the
fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin.· Heidelberg 1975.
Softeover reprint of the hardcover 1st edition H)75
Preface
The classic "Limit Dislribntions fOT slt1ns of Independent Ramdorn Vari
ables" by B. V. Gnedenko and A. N. Kolmogorov was published in 1949.
Since then the theory of summation of independent variables has devel
oped rapidly. Today a summing-up of the studies in this area, and their
results, would require many volumes. The monograph by I. A. Ibragi
mov and Yu. V. I~innik, "Independent and Stationarily Connected
VaTiables", which appeared in 1965, contains an exposition of the contem
porary state of the theory of the summation of independent identically
distributed random variables. The present book borders on that of
Ibragimov and Linnik, sharing only a few common areas. Its main
focus is on sums of independent but not necessarily identically distri
buted random variables. It nevertheless includes a number of the most
recent results relating to sums of independent and identically distributed
variables. Together with limit theorems, it presents many probahilistic
inequalities for sums of an arbitrary number of independent variables.
The last two chapters deal with the laws of large numbers and the law
of the iterated logarithm. These questions were not treated in Ibragimov
and Linnik; Gnedenko and KolmogoTOv deals only with theorems on the
weak law of large numbers.
Thus this book may be taken as complementary to the book by
Ibragimov and Linnik. I do not, however, assume that the reader is
familiar with the latter, nor with the monograph by Gnedenko and
Kolmogorov, which has long since become a bibliographical rarity.
I therefore include a number of classical results of the theory of the
summation of independent random variables. The greater part of the
text is nevertheless given over to material which has not appeared in
monographs on probability theory either here or abroad.
The book olllits from its scope such broad subjects as multidimensional
limit theorems, boundary problems for sums of independent random
variables, limit theorems for sums of a random number of independent
terms and some others.
Each chapter is accompanied by a supplement in which results that
border on the basic text are stated. The text itself does not depend on
VI Preface
the supplements. The bibliography is not complete; it includes only
material cited in the text.
It is assumed that the reader is familiar with the fundamentals of
probability theory, to the extent of the material contained in Chapters
1-8 of "KUTS Teon:i VCToyatnostci" by B. V. Gnedenko (A Course in
Probability Theory).1 A summary of the essential results is given in
Chapter 1.
I hope that this book will be useful to specialists in probability theory
and to students concerned with the theory of summation of independent
random variables.
I express my deepest gratitude to Professors A. A. Borovkov, V. A.
Egorov, 1. A. Ibragimov, L. V. Osipov, and Yu. V. Prohorov, who
have read the manuscript and given me many valuable suggestions.
I take advantage of the occasion, to express my warmest thanks to
Professor Yu. V. Linnik and Professor.T. Neyman for their consideration
and support. I owe to them the inspiration for the writing of this book.
Leningrad, 1972 V. V. Petrov
1 Gnedenko, B. V.: The Theory of Probability, tr. B. D. Seckler, New York:
Chelsea 1962.
Notation
The numbering of theorems and formulae begins anew in each chapter.
When the chapter number is omitted in the parentheses enclosing a
reference the reference itself is within the chapter where it occurs.
The Halmos square 0 indicates the end of a proof. The abbreviations
used are: d.f. for distribution function, c.£. for characteristic function,
a.c. for almost certainly, and i.o. for infinitely often.
The expression sup {(x) means sup f(x).
x -oo<x<co
The symbols 8, 81, 82, ••• denote quantities whose absolute value
does not exceed unity.
Unless otherwise specified, the symbols A, AI, A2, ••• denote abso
lute positive constants, and 0, 01, O2, ••• C, C1, ••• denote positive
constants. One and the same letter, used in different, even though
neighboring, portions of the text may stand for different values.
Unless otherwise specified, all limits are taken as n ~;> CXJ.
The relation an >< bn means that 0 < lim inf an ;;;:: lim sup an < CXJ.
an bn b"
The relation an 0.) bn means that - -'>- 1.
Finally, b"
Contents
Chapter I. Probability Distributions and Characteristic Functions. 1
§ 1. Random variables and probability distributions . 1
§ 2. Characteristic {unctions . . . . . . . . . . . . . . . .. 7
§ 3. Inversion formulae . . . . . . . . . . . . . . . . . . . 12
§ 4. The convergence of sequences of distributions and characteristic
functions 14
§ 5. Supplement. . . . . . . . . . . . . . . . . . . . . . . 18
Chapter II. Infinitely Divisible Distributions 25
§ L Definition and elementary properties of infinitely divisible
distributions . . . . . . . . . . . . . . . . . . 25
§ 2. Canonical representation of infinitely divisible characteristic
functions ..... 26
§ 3. An auxiliary theorem 32
§ 4. Supplement. . . . . 36
Chapter III. Some Inequalities for the Distribution of Sums of
Independent Random Variables 38
§ 1. Concentration functions . . . 38
§ 2. Inequalities for the concentrat.ion functions of sums of inde
pendent random variables. . . . . . . . . 43
§ 3. Inequalities for the distrihution of the maximum of sums of
independent random variahles . . . . . . , . . . 49
§ 4. Exponential estimates for t.he dL,tributions of sums of, indepen
dent random variables 52
§ 5. Supplement. . . 56
Chapter IV. Theorems on Convergence to Infinitely Divisible
Distributions . . . . 63
§ 1. Infinitely divisible distributions as limits of the distributions of
sums of independent. random variables. . . , . . . . . . . 63
Contents IX
§ 2. Conditions for convergence to a given infinitely divisible
distribution . . . . . . . . . . . . . . . 75
§ 3. Limit distributions of class L and stable distributions 82
§ 4. The central limit theorem 91
§ 5. Supplement. . . . . . . . . . . . . . . . . . . 102
Chapter V. Estimates of the Distance Between the Distribution of
a Sum of Independent Random Variables and the Normal
Distribution . . . . . . . . . . . . . . . . . . . . . . 104
§ 1. Estimating the nearness of functions of bounded variation by
the nearness of their Fourier-Stieltjes transforms 104
§ 2. The Esseen and Berry-Esseen inequalities 109
§ 3. Generalizations of Esseen's inequality 112
§ 4. Non-uniform estimates. 120
§ 5. Supplement. . . . . . . . . . . . 126
Chapter VI. Asymptotic Expansions in the Central Limit Theorem 134
§ 1. Formal construction of the expansions. . . . . . . . . . . 134
§ 2. Auxiliary propositions . . . . . . . . . . . . . . . . . . 139
§ 3. Asymptotic expansions of the distribution function of a sum of
independent identically distributed random variables 158
§ 4. Asymptotic expansions of the distribution function of a sum of
independent non-identically distributed random variables, and
of the derivatives of this function 172
§ 5. Supplement. . . . . . . . . . . . . . . . . . . . . . . 181
Chapter VII. Local Limit Theorems 187
§ 1. Local limit theorems for lattice distributions 187
§ 2. Local limit theorems for densities . . . . . 198
§ 3. Asymptotic expansions in local limit theorems 204
§ 4. Supplement. . . . . . . . . . . . . . . 213
Chapter VIII. Probabilities of Large Deviations. 217
§ 1. Introduction . . . . . . . . . . . . . . 217
§ 2. Asymptotic relations connected with Cramer's series. 218
§ 3. Necessary and sufficient conditions for normal convergence in
power zones 231
§ 4. Supplement. . . . . . . . . . 248
Chapter IX. Laws of Large Numbers . . . . . . . . . . 256
§ 1. The weak law of large numbers. . . . . . . . . . . 256
§ 2. Convergence of series of independent random variables 263