Browse Books

Reviewer: Ramesh Garimella

Durrett provides an introduction to probability theory using a measure theoretic approach for advanced undergraduates and graduate students. This book is also an excellent resource. Several interesting and concrete examples are presented throughout the textbook, which will help novices obtain a better understanding of the fundamentals of probability theory. In this latest edition, the essential aspects of measure theory are presented as a part of the main text (in chapter 1) rather than in the appendix. Chapter 2 deals with weak and strong laws of large numbers. As an application of weak law, it is shown that a "high-dimensional cube is almost the boundary of a ball." Chapter 3 deals with the distribution of the sum of independent and identically distributed random variables. Several central limit theorems are presented. An interesting application for number theorists is the Erd?s-Kac central limit theorem on the number of prime divisors of large positive integers. Chapter 4 introduces the concept of stopping times, which is very useful later in the book. Chapter 5 gives basic facts about martingales, including sufficient conditions for martingales to converge. Chapter 6 discusses in detail the Markov chains on a countable state space by providing asymptotic behavior of n -step transition probabilities. Chapter 7 details ergodic processes, the ergodic theorem, and useful generalizations and applications. Chapter 8 deals with Brownian motion by developing several interesting aspects that include continuity properties, Markov property and 0-1 law, the zero set of Brownian motion, and properties of Brownian motion through its associated martingales. The book concludes by proving "laws of the iterated logarithm for Brownian motion and random walks with finite variance." Online Computing Reviews Service