Limit theorems of probability theory: sequences of independent random variables

Title page......Page 1
Preface......Page 3
Notation and abbreviations......Page 9
1.1 Random variables and their distributions......Page 11
1.2 Moments and quantiles......Page 15
1.3 Characteristic functions......Page 19
1.4 Convergence of distributions......Page 26
1.5 Concentration functions......Page 32
1.6 Infinitely divisible distributions......Page 38
1.7 Bibliographical notes......Page 46
1.8 Addenda......Page 47
2.1 Inequalities for the maximum of sums of independent random variables......Page 60
2.2 Exponential bounds......Page 64
2.3 Inequalities for moments of sums of independent random variables......Page 68
2.4 Inequalities for the concentration functions of sums of independent random variables......Page 73
2.6 Addenda......Page 87
3.1 The condition of infinite smallness......Page 98
3.2 Infinitely divisible distributions as limit laws......Page 101
3.3 Necessary and sufficient conditions for convergence to a given infinitely divisible distribution......Page 109
3.4 Limit distributions of class L and stable distributions......Page 111
3.5 Bibliographical notes......Page 117
3.6 Addenda......Page 118
4.1 The central limit theorem for a sequence of series of independent random variables......Page 122
4.2 Classical forms of the central limit theorem......Page 130
4.3 The weak law of large numbers for a sequence of series of independent random variables......Page 137
4.4 Classical forms of the weak law of large numbers......Page 141
4.5 Bibliographical notes......Page 144
4.6 Addenda......Page 145
5.1 Estimating the difference of distribution functions by the nearness of characteristic functions......Page 152
5.2 Esseen\'s inequality......Page 157
5.3 Generalizations of Esseen\'s inequality......Page 160
5.4 Upper and lower estimates having the same order......Page 167
5.5 Non-uniform estimates......Page 173
5.6 Asymptotic expansions in the central limit theorem: formal construction of the expansions......Page 179
5.7 Asymptotic expansions in the central limit theorem: the i.i.d. case......Page 182
5.8 Limit theorems for large deviations......Page 186
5.9 Bibliographical notes......Page 194
5.10 Addenda......Page 196
6.1 The Borel-Cantelli lemma......Page 209
6.2 Convergence of series of independent random variables......Page 214
6.3 The strong law of large numbers......Page 218
6.4 The strong law of large numbers: the i.i.d. case......Page 222
6.5 The strong law of large numbers: the necessary and sufficient conditions......Page 226
6.6 Estimates of the growth of sums of independent random variables in terms of sums of their moments......Page 229
6.7 Bibliographical notes......Page 236
6.8 Addenda......Page 237
7.1 Kolmogorov\'s theorem......Page 249
7.2 The Hartman-Wintner theorem......Page 258
7.5 Addenda......Page 265
Bibliography......Page 275
Author index......Page 297
Subject index......Page 301