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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