Heads or Tails: An Introduction to Limit Theorems in Probability

Contents: Preface. Prerequisites and Overview. 1. Modeling a Probabilistic Experiment. 2. Random Variables. 3. Independence. 4. The Binomial Distribution. 5. The Weak Law of Large Numbers. 6. The Large Deviations Estimate. 7. The Central Limit Theorem. 8. The Moderate Deviations Estimate. 9. The Local Limit Theorem. 10. The Arcsine Law. 11. The Strong Law of Large Numbers. 12. The Law of the Iterated Logarithm. 13. Recurrence of Random Walks. 14. Epilogue. Biographies. Bibliography. Index.

Everyone knows some of the basics of probability, perhaps enough to play cards. Beyond the introductory ideas, there are many wonderful results that are unfamiliar to the layman, but which are well within our grasp to understand and appreciate. Some of the most remarkable results in probability are those that are related to limit theorems—statements about what happens when the trial is repeated many times. The most famous of these is the Law of Large Numbers, which mathematicians, engineers, economists and many others use every day.

In this book, Lesigne has made these limit theorems accessible by stating everything in terms of a game of tossing of a coin: heads or tails. In this way, the analysis becomes much clearer, helping establish the reader's intuition about probability. Moreover, very little generality is lost, as many situations can be modelled from combinations of coin tosses.

This book is suitable for anyone who would like to learn more about mathematical probability and has had a one-year undergraduate course in analysis.