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In this book, limit theorems are made accessibly by stating everything in terms of a game of tossing a coin: heads or tails. This book is suitable for anyone who would like to learn more about mathematical probability and has a one-year undergraduate course in analysis.

Cover 1 Title 4 Copyright 5 Contents 6 Preface 8 Prerequisites and Overview 10 Chapter 1. Modeling a Probabilistic Experiment 12 §1.1. Elementary Experiments 12 §1.2. Sequences of Elementary Experiments 14 Chapter 2. Random Variables 16 Chapter 3. Independence 20 Chapter 4. The Binomial Distribution 24 Chapter 5. The Weak Law of Large Numbers 28 Chapter 6. The Large Deviations Estimate 32 Chapter 7. The Central Limit Theorem 38 §7.1. Statement of the Theorem 38 §7.2. Remarks 39 §7.3. Applications 42 §7.4. Proof of the Theorem 45 Chapter 8. The Moderate Deviations Estimate 54 Chapter 9. The Local Limit Theorem 62 Chapter 10. The Arcsine Law 68 §10.1. Introduction 68 §10.2. Statement of the Theorems 69 §10.3. The Reflection Principle 70 §10.4. Proof of the Arcsine Law 75 §10.5. Proof of the Law of Returns to the Origin 82 Chapter 11. The Strong Law of Large Numbers 86 §11.1. Almost Sure Events, Independent Events 87 §11.2. Borel's Strong Law of Large Numbers 91 §11.3. Random Sequences Taking Several Values 94 §11.4. Normal Numbers 95 §11.5. The Borel-Cantelli Lemmas 98 Chapter 12. The Law of the Iterated Logarithm 106 §12.1. Introduction 106 §12.2. Hausdorff's Estimate 108 §12.3. Hardy and Littlewood's Estimate 109 §12.4. Khinchin's Law of the Iterated Logarithm 109 Chapter 13. Recurrence of Random Walks 118 §13.1. Introduction and Definitions 118 §13.2. Nearest Neighbor Random Walks on Z 120 §13.3. General Results about Random Walks 121 §13.4. Recurrence of Random Walks on ZN 136 Chapter 14. Epilogue 140 §14.1. A Few More General Results 140 §14.2. Closing Remarks 144 Biographies 146 Bibliography 156 Index 158 A 158 B 158 C 158 D 158 E 158 G 158 H 158 I 158 K 7 L 158 M 159 N 159 O 159 P 159 R 159 S 159 U 7 V 159 W 159 Back Cover 162

Emmanuel Lesigne: Université François Rabelais, Tours, France

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