Discrete Probability Models and Methods: Probability on Graphs and Trees, Markov Chains and Random Fields, Entropy and Coding(78 Probability Theory and Stochastic Modelling)

The emphasis in this book is placed on general models (Markov chains, random fields, random graphs), universal methods (the probabilistic method, the coupling method, the Stein-Chen method, martingale methods, the method of types) and versatile tools (Chernoff's bound, Hoeffding's inequality, Holley's inequality) whose domain of application extends far beyond the present text. Although the examples treated in the book relate to the possible applications, in the communication and computing sciences, in operations research and in physics, this book is in the first instance concerned with theory. The level of the book is that of a beginning graduate course. It is self-contained, the prerequisites consisting merely of basic calculus (series) and basic linear algebra (matrices). The reader is not assumed to be trained in probability since the first chapters give in considerable detail the background necessary to understand the rest of the book.

Table of Contents:
Introduction.- 1.Events and probability.- 2.Random variables.- 3.Bounds and inequalities.- 4.Almost-sure convergence.- 5.Coupling and the variation distance.- 6.The probabilistic method.- 7.Codes and trees.- 8.Markov chains.- 9.Branching trees.- 10.Markov fields on graphs.- 11.Random graphs.- 12.Recurrence of Markov chains.- 13.Random walks on graphs.- 14.Asymptotic behaviour of Markov chains.- 15.Monte Carlo sampling.- 16. Convergence rates.- Appendix.- Bibliography.

About the Author :
Pierre Brémaud obtained his Doctorate in Mathematics from the University of Paris VI and his PhD from the department of Electrical Engineering and Computer Science of the University of California at Berkeley. He is a major contributor to the theory of stochastic processes and their applications, and has authored or co-authored several reference or textbooks on the subject.

Review :
“This is a book that any discrete proababilist will want to have on the shelf. It is a comprehensive extension of the author's masterfully written text Markov Chains ... Surprisingly; the book contains an extensive amount of information theory. ... In my opinion the new book would be ideal for a year-long course on discrete probability.”  (Yevgeniy Kovchegov, Mathematical Reviews, May, 2018) “This is a very carefully and well-written book. The real pleasure comes from the contents but also from the excellent fonts and layout. Graduate university students and their teachers can benefit a lot of reading and using this book. There are more than good reasons to strongly recommend the book to anybody studying, teaching and/or researching in probability and its applications.” (Jordan M. Stoyanov, zbMATH 1386.60003, 2018) “This book is an excellent piece of writing. It has the strictness of a mathematical book whose traditional purpose is to state and prove theorems, and also has the features of a book on an engineering topic, where solved and unsolved exercises are provided. I appreciated the very carefully selected solved examples that are interwoven in each chapter. They provide an indispensable aid to digest the concepts and methods presented.” (Dimitrios Katsaros, Computing Reviews, February, 21, 2018) “This is a comprehensive volume on the application of discrete probability to combinatorics, information theory, and related fields. It is accessible for first-year graduate students. … Results are easy to find and reasonably easy to understand. … Summing Up: Recommended. Graduate students and faculty.” (M. Bona, Choice, Vol. 54 (12), August, 2017)