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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.
Brownian motion is a process of tremendous practical and theoretical significance. It originated (a) as a model of the phenomenon observed by Robert Brown in 1828 that “pollen grains suspended in water perform a continual swarming motion,” and (b) in Bachelier's (1900) work as a model of the stock market. These are just two of many systems that Brownian motion has been used to model. On the theoretical side, Brownian motion is a Gaussian Markov process with stationary independent increments. It lies in the intersection of three important classes of processes and is a fundamental example in each theory.
The first part of this chapter develops properties of Brownian motion. In Section 8.1, we define Brownian motion and investigate continuity properties of its paths. In Section 8.2, we prove the Markov property and a related 0-1 law. In Section 8.3, we define stopping times and prove the strong Markov property. In Section 8.4, we take a close look at the zero set of Brownian motion. In Section 8.5, we introduce some martingales associated with Brownian motion and use them to obtain information about its properties.
The second part of this chapter applies Brownian motion to some of the problems considered in Chapters 2 and 3. In Section 8.6, we embed random walks into Brownian motion to prove Donsker's theorem, a far reaching generalization of the central limit theorem.
In 1989 when the first edition of this book was completed, my sons David and Greg were 3 and 1, and the cover picture showed the Dow Jones at 2650. The past 20 years have brought many changes, but the song remains the same. The title of the book indicates that as we develop the theory, we will focus our attention on examples. Hoping that the book would be a useful reference for people who apply probability in their work, we have tried to emphasize the results that are important for applications, and have illustrated their use with roughly 200 examples. Probability is not a spectator sport, so the book contains almost 450 exercises to challenge readers and to deepen their understanding.
This fourth edition has two major changes (in addition to a new publisher):
(i) The book has been converted from TeX to LaTeX. The systematic use of labels should eventually eliminate problems with references to other points in the text. In addition, the picture environment and graphicx package has allowed the figures lost from the third edition to be reintroduced and a number of new ones to be added.
(ii) Four sections of the old appendix have been combined with the first three sections of Chapter 1 to make a new first chapter on measure theory, which should allow the book to be used by people who do not have this background without making the text tedious for those who have.