Theory of Probability and Random Processes: (Universitext)

The book consists of two parts. The first part concerns the classical probability theory. It contains a detailed analysis of Markov chains, Limit theorems, their relation to Renormalization Group theory. It also describes several applications of probability theory, which are not discussed in most other text-books. The second part includes the theory of stationary random processes, martingales, stochastic integrals and stochastic differential equations. One section is devoted to the theory of Gibbs random fields. The book contains essential material for an undergraduate or a graduate course in Probability theory and the theory of Random Processes. It may also serve as a reference for scientists who use modern probability theory in their research.

Table of Contents:
Probability Theory.- Random Variables and Their Distributions.- Sequences of Independent Trials.- Lebesgue Integral and Mathematical Expectation.- Conditional Probabilities and Independence.- Markov Chains with a Finite Number of States.- Random Walks on the Lattice ?d.- Laws of Large Numbers.- Weak Convergence of Measures.- Characteristic Functions.- Limit Theorems.- Several Interesting Problems.- Random Processes.- Basic Concepts.- Conditional Expectations and Martingales.- Markov Processes with a Finite State Space.- Wide-Sense Stationary Random Processes.- Strictly Stationary Random Processes.- Generalized Random Processes.- Brownian Motion.- Markov Processes and Markov Families.- Stochastic Integral and the Ito Formula.- Stochastic Differential Equations.- Gibbs Random Fields.

About the Author :
YAKOV SINAI has been a professor at Princeton University since 1993. He was educated at Moscow State University, and was a professor there till 1993. Since 1971 he has also held the position of senior researcher at the Landau Institute of Theoretical Physics. He is known for fundamental work on dynamical systems, probability theory, mathematical physics, and statistical mechanics. He has been awarded, among other honors, the Boltzmann Medal (in 1986) and Wolf Prize in Mathematics (in 1997). He is a member of Russian and American Academies of Sciences. LEONID KORALOV is an assistant professor at the University of Maryland. From 2000 till 2006 he was an assistant professor at Princeton University, prior to which he worked at the Institute for Advanced Study in Princeton. He did his undergraduate work at Moscow State University, and got his PhD from SUNY at Stony Brook in 1998. He works on problems in the areas of homogenization, diffusion processes, and partial differential equations.

Review :
From the reviews of the second edition: "The book is based on a series of lectures taught by the authors at Princeton University and the University of Maryland. The material of the book can be used to support a two-semester course in probability and stochastic processes or, alternatively, two independent one-semester courses in probability and stochastic processes, respectively. … will be found useful by advanced undergraduate and graduate students and by professionals who wish to learn the basic concepts of modern probability theory and stochastic processes." (Vladimir P. Kurenok, Mathematical Reviews, Issue 2008 k) “The text is well written and the concepts and results motivated and explained. Most of the chapters include a section with exercises of varying difficulty. The material of the book has been used by the authors to teach one-year lecture courses at Princeton University and the University of Maryland to advanced undergraduate and graduate students. Summarising, the book is enjoyable and provides a concise well-motivated presentation of the material covered, suitable for lecture courses at an advanced level.” (Evelyn Buckwar, Zentralblatt MATH, Vol. 1181, 2010)