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Understanding Markov Chains: Examples and Applications

Nicolas Privault
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
John D. Cook
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Understanding Markov Chains by Nicolas Privault is an attractive book. Like most math books, it was typeset using LaTeX, but it looks better than most math books. Perhaps the author uses LaTeX particularly well. The paper is slightly cream-colored and the figures are well done. Even the solutions to the exercises, where some authors are wont to skimp on presentation quality, are well done.

The usual Markov chain topics are here. Discrete chains are emphasized, though there is some material on continuous chains. The book says in the introduction that measure theory is “outside the scope of this text.” It does contain some measure theory, though measure-theoretic aspects of Markov chains are understated.

The subtitle of the book is “Examples and Applications,” though the book does not have an unusual number of examples or applications, depending on your idea of “definition” and “example.” The book spends a good amount of time on gambling processes and random walks, which may be considered part of the theory of Markov chains or an application of Markov chains, depending on your perspective. You won't find many applications to areas not closely related to Markov chains. For example, Markov chain Monte Carlo (MCMC) is covered in only one page, and there is no mention of using MCMC to evaluate integrals.

Much of the book is devoted to stating and proving theorems, and the proof are very clear. However, there is not much narrative holding the book together. There is some narrative arc, though I wish the book had included more text explaining how the subject of Markov chains adheres together.

John D. Cook is an independent consultant and blogs at The Endeavour.