This lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for applications, this comprehensive treatment is a rigorous graduate text and reference. Operating under the philosophy that the best way to learn probability is to see it in action, the book contains extended examples that apply the theory to concrete applications.

This fifth edition introduces a new chapter on multidimensional Brownian motion and its relationship to partial differential equations (PDEs), an advanced topic finding new applications. Setting the foundation for this expansion, Chapter 7 now includes a proof of Itô’s formula. Key exercises previously left as proofs for the reader have been directly incorporated into the text as lemmas. The revised edition reinstates discussion about the central limit theorem for martingales and stationary sequences.