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Probability Theory: An Advanced Course (Universitext), by Vivek S. Borkar
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This book presents a selection of topics from probability theory. Essentially, the topics chosen are those that are likely to be the most useful to someone planning to pursue research in the modern theory of stochastic processes. The prospective reader is assumed to have good mathematical maturity. In particular, he should have prior exposure to basic probability theory at the level of, say, K.L. Chung's 'Elementary probability theory with stochastic processes' (Springer-Verlag, 1974) and real and functional analysis at the level of Royden's 'Real analysis' (Macmillan, 1968). The first chapter is a rapid overview of the basics. Each subsequent chapter deals with a separate topic in detail. There is clearly some selection involved and therefore many omissions, but that cannot be helped in a book of this size. The style is deliberately terse to enforce active learning. Thus several tidbits of deduction are left to the reader as labelled exercises in the main text of each chapter. In addition, there are supplementary exercises at the end. In the preface to his classic text on probability ('Probability', Addison Wesley, 1968), Leo Breiman speaks of the right and left hands of probability.
- Sales Rank: #3887753 in eBooks
- Published on: 1995-10-05
- Released on: 2013-10-04
- Format: Kindle eBook
Review
"It has a well tied and balanced selection of topics, presented in an elegant, concise but not standard fashion." -- ZENTRALBLATT MATH
From the Back Cover
This book is an advanced text on probability theory. By presupposing the background of a standard first course in real analysis and a 'soft' course in probability theory, it gives a compact treatment of several key topics in probability, selected on the basis of their importance in forming the foundations of the modern theory of stochastic processes.
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By Giuseppe A. Paleologo
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.
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