Heads or tails an introduction to limit theorems in probability Emmanuel LesignePDF电子书下载

Prerequisites and Overview 1

Chapter 1.Modeling a Probabilistic Experiment 3

1.1.Elementary Experiments 3

1.2.Sequences of Elementary Experiments 5

Chapter 2.Random Variables 7

Chapter 3.Independence 11

Chapter 4.The Binomial Distribution 15

Chapter 5.The Weak Law of Large Numbers 19

Chapter 6.The Large Deviations Estimate 23

Chapter 7.The Central Limit Theorem 29

7.1.Statement of the Theorem 29

7.2.Remarks 30

7.3.Applications 33

7.4.Proof of the Theorem 36

Chapter 8.The Moderate Deviations Estimate 45

Chapter 9.The Local Limit Theorem 53

Chapter 10.The Arcsine Law 59

10.1.Introduction 59

10.2.Statement of the Theorems 60

10.3.The Reflection Principle 61

10.4.Proof of the Arcsine Law 66

10.5.Proof of the Law of Returns to the Origin 73

Chapter 11.The Strong Law of Large Numbers 77

11.1.Almost Sure Events, Independent Events 78

11.2.Borel's Strong Law of Large Numbers 82

11.3.Random Sequences Taking Several Values 85

11.4.Normal Numbers 86

11.5.The Borel-Cantelli Lemmas 89

Chapter 12.The Law of the Iterated Logarithm 97

12.1.Introduction 97

12.2.Hausdorff's Estimate 99

12.3.Hardy and Littlewood's Estimate 100

12.4.Khinchin's Law of the Iterated Logarithm 100

Chapter 13.Recurrence of Random Walks 109

13.1.Introduction and Definitions 109

13.2.Nearest Neighbor Random Walks on ZZ 111

13.3.General Results about Random Walks 112

13.4.Recurrence of Random Walks onZN 121

Chapter 14.Epilogue 131

14.1.A Few More General Results 131

14.2.Closing Remarks 135

Biographies 137

Bibliography 147

Index 149