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Probability Theory: A Comprehensive Course

Achim Klenke
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
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The book is indeed comprehensive, consisting of 26 chapters on different topics. Only the first eight are required reading in order to understand the rest of the book; the other chapters are organized in fairly independent clusters as explained in the preface. Therefore, the book can be well used as a reference book on a wide range of topics. The target audience is researchers and graduate students, to which this reviewer would add that graduate students reading this book should be advanced graduate students.

Numerous advanced topics are included, so that the book is more inclusive than most introductory graduate textbooks on probability. A few of these topics are percolation, stochastic differential equations, several aspects of martingales, limit theorems for sums of random variables, and statistical physics. Given the wide scope of the book, this reviewer would have liked to see more on discrete probability, but this may be just personal bias.

Using the book as a textbook could be problematic. There are very few exercises, roughly 3–4 per section on average. None come with solutions, or even answers, though a few come with hints. Assigning homework could be a problem without these, and students would be likely to complain about the lack of exercises to test their understanding. The writing is rather terse, and I would have liked to see more connections to other parts of mathematics, such as combinatorics and graph theory.

If one does have the courage to teach from the book, then one can use the book at a variety of levels. There is more than enough material for a two-semester course here. Or one could teach a one semester course from the first eight chapters (ending with conditional expectations) and one or two selected independent blocks from the later clusters of chapters. Or, one could teach a special topics course or a reading course on a few of those clusters.

Still, I believe that the book will primarily be used as a reference book. For that purpose, it is a rich and relatively inexpensive choice.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.

Basic Measure Theory.- Independence.- Generating Functions.- The Integral.- Moments and Laws of Large Numbers.- Convergence Theorems.- Lp-Spaces and Radon-Nikodym Theorem.- Conditional Expectations.- Martingales.- Optional Sampling Theorems.- Martingale Convergence Theorems and their Applications.- Backwards Martingales and Exchangeability.- Convergence of Measures.- Probability Measures on Product Spaces.- Characteristics Functions and Central Limit Theorem.- Infinitely Divisible Distributions.- Markov Chains.- Convergence of Markov Chains.- Markov Chains and Electrical Networks.- Ergodic Theory.- Brownian Motion.- Law of the Iterated Logarithm.- Large Deviations.- The Poisson Point Process.- The Itô Integral.- Stochastic Differential Equations.- References.- Notation Index.- Name Index.- Subject Index.