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 contains a new chapter on multidimensional Brownian motion and its relationship to partial differential equations (PDEs), an advanced topic that is finding new applications. Setting the foundation for this expansion, Chapter 7 now features a proof of Itô’s formula. Key exercises that previously were simply proofs left to the reader have been directly inserted into the text as lemmas. The new edition re-instates discussion about the central limit theorem for martingales and stationary sequences.
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Probability: Theory and Examples
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Update 12/23/2017:
After finishing the semester, the book seemed to hold up well. It is still intact though the cover is now a bit bent. I did not travel to often with it and the bends in the soft cover mostly came from when I did do the traveling. I have put clear contact paper on a number of my soft covers which helps them stay sturdy, and I did not get the chance to put the contact paper on this book, so it likely would have held up even better with that added protection. As far as the content, I found the writing to be rather dry. However, reading the text was still useful and there are numerous practice problems, a few even with answers given. The course I took covered complex integration, series solutions of differential equations, integral equations, orthogonality, Green's functions, and group theory. I found Mary Boas' Mathematical Methods in the Physical Sciences to be helpful in my course, as well. It covers many of the same topics as this book and Boas' book often covered the topics more clearly, so if you have the extra cash to purchase Boas' book or if you can find a PDF copy of it, I would highly recommend that book in addition to this one.