This revised edition of Daniel W. Stroock's classic text is suitable for a first-year graduate course on probability theory. By modern standards the topics treated are classical and the techniques used far-ranging: Dr. Stroock does not approach the subject as a monolithic structure resting on a few basic principles. The first part of the book deals with independent random variables, Central Limit phenomena, the general theory of weak convergence and several of its applications, as well as elements of both the Gaussian and Markovian theory of measures on function space. Stroock covers conditional expectation values in the second half where he applies them to the study of martingales. He also explores the connection between martingales and various aspects of classical analysis and the connections between Wiener's measure and classical potential theory. Student prerequisites are a good grasp of introductory, undergraduate probability theory and a reasonably sophisticated knowledge of analysis.
Contents
1. Sums of independent random variables; 2. The central limit theorem; 3. Convergence of measures, infinite divisibility, and processes with independent increments; 4. A celebration of Wiener's measure; 5. Conditioning and martingales; 6. Some applications of martingale theory; 7. Continuous martingales and elementary diffiusion theory; 8. A little classical potential theory.
Review
"...uniformly well written and well spiced with comments to aid the intuition, so the readership should include a wide range, both of students and of professional probabilists.... We can expect it to take its place alongside the classics of probability theory." Mathematical Reviews
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