The core of this book is a one-year course in probability theory and the theory of random processes, taught at Princeton University to undergraduate and graduate students. The book provides a comprehensive, self-contained exposition of classical probability theory and the theory of random processes, and dwells on a number of modern topics not addressed in most textbooks. There is detailed discussion of Lebesgue integration, Markov chains, random walks, laws of large numbers, limit theorems, and their relation to Renormalization Group theory. It also includes the theory of stationary random processes, martingales, generalized random processes, and Brownian motion. One section is devoted to the theory of Gibbs random fields. The material is essential to many undergraduate and graduate courses. The book can also serve as a reference for scientists using modern probability theory in their research.

Comprehensive, self-contained exposition of classical probability theory and the theory of random processes Dwells on a number of modern topics, not addressed in most textbooks Author Ya. G. Sinai is one of the world's leading probabilists and mathematical physicists Includes supplementary material: sn.pub/extras

Autorentext

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.

Inhalt
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.