Discrete Probability Models and Methods: Probability on Graphs and Trees, Markov Chains and Random Fields, Entropy and Coding

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Springer, Jan 31, 2017 - Mathematics - 559 pages

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.


 

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Contents

1 Events and Probability
1
2 Random Variables
20
3 Bounds and Inequalities
65
4 Almost Sure Convergence
78
5 The probabilistic Method
93
6 Markov Chain Models
117
7 Recurrence of Markov Chains
145
8 Random Walks on Graphs
184
14 Universal Source Coding
356
15 Asymptotic Behaviour of Markov Chains
373
16 The Coupling Method
397
17 Martingale Methods
416
18 Discrete Renewal Theory
441
19 Monte Carlo
457
20 Convergence Rates
475
21 Exact Sampling
509

9 Markov Fields on Graphs
215
10 Random Graphs
255
11 Coding Trees
287
12 Shannons Capacity Theorem
318
13 The Method of Types
341
Appendix A
535
Bibliography
545
Index
555
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About the author (2017)

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.