Full Description
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.
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.
