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Probability Theory: An Advanced Course (Universitext)

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In recent years many excellent graduate textbooks in probability have become available. Although these books provide the reader with the basic analytic tools of measure theoretic probability, they usually fall short of presenting probability in more general (and abstract) spaces. As far as I know, there are not many good books on the subject: Billingsley "Convergence of Probability Measures" comes to mind, and Jacod and Shiryaev's "Limit Theorems for Stochastic Processes" (both excellent, and very expensive). The goal of this booklet is to fill this void and to provide a researcher with some more advanced analytical tools. The book is a "selection of topics" in the author's words, and I think it has no pretense of completeness. What it presents, it presents very well, with short yet rigorous proofs (as far as I can tell: I studied only the first two chapters, and the last one). The reader should know probability at least at the level of Billingsley's "Probability and Measure". A basic command of topological spaces and Banach spaces is recommended. One criticism: having a PhD student in mind, the author could have spent a few more pages on examples, but this doesn't detract much from a book that is already a helpful reference and an example of good style.
Finally, here are the titles of the chapters, with a short description when needed: 1. Introduction [sets the stage for random variables in Polish spaces] 2. Spaces of probability measures 3. Conditioning and martingales 4. Basic limit theorems [SLLN, CLT, LIL, Large Deviations] 5. Markov chains 6. Foundations of continuous time processes.