PREFACE 1
PART Ⅰ.INTRODUCTION 13
CHAPTER 1.PROBABILITY DISTRIBUTIONS.RANDOM VARIABLES AND MATHEMATICAL EXPECTATIONS 13
1.Preliminary remarks 13
2.Measures 16
3.Perfect measures 18
4.The Lebesgue integral 19
5.Mathematical foundations of the theory of probability 20
6.Probability distributions in R1 and in R? 22
7.Independence.Composition of distributions 26
8.The Stieltjes integral 29
CHAPTER 2.DISTRIBUTIONS IN R1 AND THEIR CHARACTERISTIC FUNCTIONS 32
9.Weak convergence of distributions 32
10.Types of distributions 39
11.The definition and the simplest properties of the characteristic function 44
12.The inversion formula and the uniqueness theorem 48
13.Continuity of the correspondence between distribution and characteristic functions 52
14.Some special theorems about characteristic functions 55
15.Moments and semi-invariants 61
CHAPTER 3.INFINITELY DIVISIBLE DISTRIBUTIONS 67
16.Statement of the problem.Random functions with independent increments 67
17.Definition and basic properties 71
18.The canonical representation 76
19.Conditions for convergence of infinitely divisible distributions 87
PART Ⅱ.GENERAL LIMIT THEOREMS 94
CHAPTER 4.GENERAL LIMIT THEOREMS FOR SUMS OF INDEPENDENT SUMMANDS 94
20.Statement of the problem.Sums of infinitely divisible summands 94
21.Limit distributions with finite variances 97
22.Law of large numbers 105
23.Two auxiliary theorems 109
24.The general form of the limit theorems.The accompanying infinitely divisible laws 112
25.Necessary and sufficient conditions for convergence 116
CHAPTER 5.CONVERGENCE TO NORMAL,POISSON,AND UNITARY DISTRIBUTIONS 125
26.Conditions for convergence to normal and Poisson laws 125
27.The law of large numbers 133
28.Relative stability 139
CHAPTER 6.LIMIT THEOREMS FOR CUMULATIVE SUMS 145
29.Distributions of the class L 145
30.Canonical representation of distributions of the class L 149
31.Conditions for convergence 152
32.Unimodality of distributions of the class L 157
PART Ⅲ.IDENTICALLY DISTRIBUTED SUMMANDS 162
CHAPTER 7.FUNDAMENTAL LIMIT THEOREMS 162
33.Statement of the problem.Stable laws 162
34.Canonical representation of stable laws 164
35.Domains of attraction for stable laws 171
36.Properties of stable laws 182
37.Domains of partial attraction 183
CHAPTER 8.IMPROVEMENT OF THEOREMS ABOUT THE CONVERGENCE TO THE NORMAL LAW 191
38.Statement of the problem 191
39.Two auxiliary theorems 196
40.Estimation of the remainder term in Lyapunov's Theorem 201
41.An auxiliary theorem 204
42.Improvement of Lyapunov's Theorem for nonlattice distribution 208
43.Deviation from the limit law in the case of a lattice distribution 212
44.The extremal character of the Bernoulli case 217
45.Improvement of Lyapunov's Theorem with higher moments for the continuous case 220
46.Limit theorem for densities 222
47.Improvement of the limit theorem for densities 228
CHAPTER 9.LOCAL LIMIT THEOREMS FOR LATTICE DISTRIBUTIONS 231
48.Statement of the problem 231
49.A local theorem for the normal limit distribution 232
50.A local limit theorem for non-normal stable limit distributions 235
51.Improvement of the limit theorem in the case of convergence to the normal distribution 240
APPENDIX I.NOTES ON CHAPTER 1 245
APPENDIX II.NOTES ON 32 252
BIBLIOGRAPHY 257
INDEX 262