In this post, we will see the book Limit Distributions For Sums Of Independent Random Variables – B. V. Gnedenko, A. N. Kolmogorov.

# About the book

This is a translation of the Russian book (1949). There are various points of contact with the treatises by P. Levy [76] and by H. Cramer [21], but much of the material in the book has been hitherto available only in periodical articles, many of which are in Russian. The systematic account presented here combines generality with simplicity, making some of the most important and difficult parts of the theory of probability easily accessible to the reader. Beyond a knowledge of the calculus on the level of, say, Hardy’s Pure Mathematics, the book is formally self-contained. However, a certain amount of mathematical maturity, perhaps a touch of single-minded perfectionism, is needed to penetrate the depth and appreciate the classic beauty of this definitive work.

The book was translated from Russian by K. L. Chung with Appendix by J. L. Doob. The book was published in 1954.

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You can get the book here.

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# Contents

PREFACE

# PART I. INTRODUCTION

## CHAPTER l. 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 R and in R^{1} 22

§ 7. Independence. Composition of distributions 26

§ 8. The Stieltjes integral 29

## CHAPTER 2. DISTRIBUTIONS IN R^{1} 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 theater 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

## 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 conditioning 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 L 149

§ 31. Conditions for convergence 152

§ 32. Unimodality of distributions of the Glass L 157

PART III. IDENTICALLY DISTRIBUTED SUMMANDS

## 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 a 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 monena {ог 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 245

APPENDIX II. NOTES ON § 32 252

BIBLIOGRAPHY 257

INDEX 262