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Heads or Tails: An Introduction to Limit Theorems in Probability

Emmanuel Lesigne
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 28
[Reviewed by
Gudmund Iversen
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This is a delightful little book; little because it has only 150 pages, and delightful because in these pages the author conveys an impressive and well-written account of central ideas of limit theorems in probability. The author is a well-know French authority on probability theory.

In the first part the author provides the mathematical model for finite probabilistic experiments based on tosses of a coin, as promised in the title. The second part contains theorems about finite sequences, and the third part models infinite experiments. These parts provide us with the weak law of large numbers and the central limit theorem, as well as the arcsine law and the local limit theorem. Furthermore, we get discussions of various forms of the strong law of large numbers, the law of the iterated logarithm, and some results about the recurrence of random walks.

The reader should have had a course in analysis, but measure theory is not needed. The author sees the book aimed at undergraduates in mathematics, science and engineering. In addition, I think the book will also be very valuable for students seriously interested in statistics. It fits well into an undergraduate program in statistics, and students in graduate programs in statistics will get a solid introduction to limit theorems. Those students interested in the more mathematical parts of statistics will need to go on beyond the book and into measure theory, but a more applied statistician is well served with the material in this book. While the book does not show the importance of such limit theorems for estimation problems in statistics, that is a topic for another book.

There is a scattering of footnotes referring to the originators of many of the ideas in probability theory. For example, Christiaan Huygens is given credit for the expected value, introduced in 1657. Also, the book ends with short biographies of 33 leading writers on probability, arranged chronological from John Wallis (1616–1703) to Andrei Kolmogorov (1903–1987) (perhaps we would have expected a book published in 2005 to also include his year of death?). All this adds a nice, human touch to the mathematical results.

The notation is very clear and to the point. Proofs are well stated, and they should provide nice challenges for readers who want to follow them. It is refreshing to have a book that starts with such a simple experiment with two outcomes and takes us as far as it does into the world of probability theory.

Gudmund R. Iversen ( is professor emeritus of statistics at Swarthmore College.  Among his interests are statistics education and uses of Bayesian statistics.

* Prerequisites and overview
* Modeling a probabilistic experiment
* Random variables
* Independence
* The binomial distribution
* The weak law of large numbers
* The large deviations estimate
* The central limit theorem
* The moderate deviations estimate
* The local limit theorem
* The arcsine law
* The strong law of large numbers
* The law of the iterated logarithm
* Recurrence of random walks
* Epilogue
* Biographies
* Bibliography
* Index