Limit Theorems of Probability Theory: Sequences of Independent Random Variables

VALENTIN V. PETROV
St Petersburg University
St Petersburg
Russia
CLARENDON PRESS OXFORD
1995
Oxford University Press, Walton Street, Oxford 0X2 6DP
Oxford New York



Contents
Notation and abbreviations x
1. Some basic concepts and theorems of probabiiity theory 1
.1 Random variables and their distributions 1
.2 Moments and quantiles 5
.3 Characteristic functions 9
.4 Convergence of distributions 16
.5 Concentration functions 22
.6 Infinitely divisible distributions 28
1.7 Bibliographical notes 36
1.8 Addenda 37
2. Probabiiity inequaiities for sums of independent random
variables 50
2.1 Inequalities for the maximum of sums of independent
random variables 50
2.2 Exponential bounds 54
2.3 Inequalities for moments of sums of independent random
variables 58
2.4 Inequalities for the concentration functions of sums of
independent random variables 63
2.5 Bibliographical notes 77
2.6 Addenda 77
3. Weak limit theorems: convergence to infinitely divisible
distributions 88
3.1 The condition of infinite smallness 88
3.2 Infinitely divisible distributions as limit laws 91
3.3 Necessary and sufficient conditions for convergence to a
given infinitely divisible distribution 99
3.4 Limit distributions of class L and stable distributions 101
3.5 Bibliographical notes 107
3.6 Addenda 108
4. Weak limit theorems: the central limit theorem and the
weak law of large numbers 112
4.1 The central limit theorem for a sequence of series of
independent random variables 112
4.2 Classical forms of the central limit theorem 120
4.3 The weak law of large numbers for a sequence of series of
independent random variables 127
4.4 Classical forms of the weak law of large numbers 131
4.5 Bibliographical notes 134
4.6 Addenda 135
5. Rates of convergence in the central limit theorem 142
5.1 Estimating the difference of distribution functions by the
nearness of characteristic functions 142
5.2 Esseen's inequality 147
5.3 Generalizations of Esseen's inequality 150
5.4 Upper and lower estimates having the same order 157
5.5 Non-uniform estimates 163
5.6 Asymptotic expansions in the central limit theorem: formal
construction of the expansions 169
5.7 Asymptotic expansions in the central limit theorem: the i.i.d.
case 172
5.8 Limit theorems for large deviations 176
5.9 Bibliographical notes 184
5.10 Addenda 186
6. Strong limit theorems: the strong law of large numbers 199
6.1 The Borel-Cantelli lemma 199
6.2 Convergence of series of independent random variables 204
6.3 The strong law of large numbers 208
6.4 The strong law of large numbers: the i.i.d. case 212
6.5 The strong law of large numbers: the necessary and
sufficient conditions 216
6.6 Estimates of the growth of sums of independent random
variables in terms of sums of their moments 219
6.7 Bibliographical notes 226
6.8 Addenda 227
7. Strong limit theorems: the law of the iterated logarithm 239
7.1 Kolmogorov's theorem 239
7.2 The Hartman-Wintner theorem 248
7.3 The generalized law of the iterated logarithm 250
7.4 Bibliographical notes ? 255
7.5 Addenda 255
Bibliography
Author index
Subject index