Probability Theory: An Analytic View

by Daniel W. Stroock

• ISBN13: 9780521132503
• ISBN10: 0521132509
• Edition: 2nd
• Format: Paperback
• Copyright: 2010-12-31
• Publisher: Cambridge University Press

Summary

This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, where they are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motion to the analysis of partial differential equations and potential theory.

Table of Contents

Sums of independent random variables

The central limit theorem

Infinitely divisible laws

Levy processes

Conditioning and martingales

Some extensions and applications of martingale theory

Continuous parameter martingales

Gaussian measures on a Banach space

Convergence of measures on a Polish space

Wiener measure and partial differential equations

Some classical potential theory

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