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Limit Distributions For Sums of Independent Random VariablesPDF|Epub|txt|kindle电子书版本网盘下载

B.V.Gnedenko and A.N.Kolmogorov 著
出版社： Inc.
ISBN：
出版时间：1954
标注页数：264页
文件大小：33MB
文件页数：273页
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图书目录

PREFACE1

PART Ⅰ．INTRODUCTION13

CHAPTER 1．PROBABILITY DISTRIBUTIONS．RANDOM VARIABLES AND MATHEMATICAL EXPECTATIONS13

1．Preliminary remarks13

2．Measures16

3．Perfect measures18

4．The Lebesgue integral19

5．Mathematical foundations of the theory of probability20

6．Probability distributions in R1 and in R？22

7．Independence．Composition of distributions26

8．The Stieltjes integral29

CHAPTER 2．DISTRIBUTIONS IN R1 AND THEIR CHARACTERISTIC FUNCTIONS32

9．Weak convergence of distributions32

10．Types of distributions39

11．The definition and the simplest properties of the characteristic function44

12．The inversion formula and the uniqueness theorem48

13．Continuity of the correspondence between distribution and characteristic functions52

14．Some special theorems about characteristic functions55

15．Moments and semi-invariants61

CHAPTER 3．INFINITELY DIVISIBLE DISTRIBUTIONS67

16．Statement of the problem．Random functions with independent increments67

17．Definition and basic properties71

18．The canonical representation76

19．Conditions for convergence of infinitely divisible distributions87

PART Ⅱ．GENERAL LIMIT THEOREMS94

CHAPTER 4．GENERAL LIMIT THEOREMS FOR SUMS OF INDEPENDENT SUMMANDS94

20．Statement of the problem．Sums of infinitely divisible summands94

21．Limit distributions with finite variances97

22．Law of large numbers105

23．Two auxiliary theorems109

24．The general form of the limit theorems．The accompanying infinitely divisible laws112

25．Necessary and sufficient conditions for convergence116

CHAPTER 5．CONVERGENCE TO NORMAL,POISSON,AND UNITARY DISTRIBUTIONS125

26．Conditions for convergence to normal and Poisson laws125

27．The law of large numbers133

28．Relative stability139

CHAPTER 6．LIMIT THEOREMS FOR CUMULATIVE SUMS145

29．Distributions of the class L145

30．Canonical representation of distributions of the class L149

31．Conditions for convergence152

32．Unimodality of distributions of the class L157

PART Ⅲ．IDENTICALLY DISTRIBUTED SUMMANDS162

CHAPTER 7．FUNDAMENTAL LIMIT THEOREMS162

33．Statement of the problem．Stable laws162

34．Canonical representation of stable laws164

35．Domains of attraction for stable laws171

36．Properties of stable laws182

37．Domains of partial attraction183

CHAPTER 8．IMPROVEMENT OF THEOREMS ABOUT THE CONVERGENCE TO THE NORMAL LAW191

38．Statement of the problem191

39．Two auxiliary theorems196

40．Estimation of the remainder term in Lyapunov＇s Theorem201

41．An auxiliary theorem204

42．Improvement of Lyapunov＇s Theorem for nonlattice distribution208

43．Deviation from the limit law in the case of a lattice distribution212

44．The extremal character of the Bernoulli case217

45．Improvement of Lyapunov＇s Theorem with higher moments for the continuous case220

46．Limit theorem for densities222

47．Improvement of the limit theorem for densities228

CHAPTER 9．LOCAL LIMIT THEOREMS FOR LATTICE DISTRIBUTIONS231

48．Statement of the problem231

49．A local theorem for the normal limit distribution232

50．A local limit theorem for non-normal stable limit distributions235

51．Improvement of the limit theorem in the case of convergence to the normal distribution240

APPENDIX I．NOTES ON CHAPTER 1245

APPENDIX II．NOTES ON 32252

BIBLIOGRAPHY257

INDEX262