Discrete Probability Models and Methods

To paraphrase from the preface: The book is an introduction to a few of the vast and flourishing domains of applied probability. It is self-contained. The mathematical level is that of a beginning graduate student. The prerequisites are calculus and linear algebra. No prior training in probability is assumed.

True to his word, Brémaud begins with chapters on Events and Probabilities and Random Variables. From there we progress to Bounds and Inequalities, and suddenly find ourselves discussing Almost Sure Convergence and the Probabilistic Method including the Lovasz Local Lemma, which is not as fancy as it sounds.

Returning to more traditional topics there are two chapters on Markov chains that describe the basics when time and space are discrete. From here we go to Random Walks on Graphs which describes the connection between reversible Markov chains and electrical networks, and then to Markov Fields on Graphs.

Four chapters on coding theory follow (Shannon’s Theorem, Lempel-Ziv) then we are back to Markov chains, Coupling, Martingale Methods, Discrete Renewal Theory, Monte Carlo Methods, Rates of Convergence, and Exact Sampling. In most cases all one gets is a taste. Fro example, in the Random Graphs chapter we meet the Erdős-Rényi model and hear a little about percolation.

The extreme heterogeneity in what is covered does not make the book suitable for any one course, but by selecting from the 21 chapters that total more than 550 pages one could build a number of different courses from it. I found the sections that I read to be very clear and instructive. Several taught me new facts and viewpoints about subjects that I thought I knew well. My one complaint is that it weighs 2.75 pounds. That does not sound like much, but for those of us who like to lay on the sofa and read math books on a Saturday afternoon, it is a little awkward to handle. However, if you are willing to sit in a chair or at your desk and read this then you will spend many enjoyable hours learning useful things from this book.

Richard Durrett taught at UCLA and Cornell before he came to Duke in 2010. He is a member of the National Academy of Science, who for the last thirty years has used probability to study problems that arise from ecology, genetics, and cancer modeling.