Synopses & Reviews
'In this book probability theory is presented as an integral part of analysis. Thus, the selection of topics covered has been dictated less by so-called probabilistic considerations than by the author\'s personal fascination with the insight that probability theory brings to certain aspects of analysis. By comparison to most recent treatments of the subject, the orientation is classical. In Chapter I, independent random variables are summed and their averages shown to converge; Chapter II deals with Central Limit phenomena; the general theory of weak convergence is introduced in Chapter III, where it is applied to the study of infinitely divisible laws, the construction of processes with independent increments, and finally a proof of Donsker\'s Invariance Principle for Wiener\'s measure. Taking Wiener\'s measure as the basic model, Chapter IV introduces the reader to the elements of both the Gaussian and Markovian theory of measures on function space. All the material in Chapters I through IV deals with quantities that are derived more or less directly from independent random variables. For this reason, the introduction of conditional expectations is postponed until Chapter V, where it is immediately applied to the study of martingales. Chapter VI is a digression in which the connection between martingales and various aspects of classical analysis are established and exploited; and Chapter VII contains applications of martingales to the analysis of Wiener\'s measure and elementary diffusion processes. Finally, Chapter VIII is devoted to some of the profound and beautiful connections between Wiener\'s measure and classical potential theory.'
Review
'\"The various topics are nicely interwoven in what is obviously a well-thought-out exposition. The author\'s style makes key ideas easy to grasp.\" Michael Cranston, Journal of the American Statistical Association\"...it is 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...the present book would be a good answer to what every probabilist should learn. It is solidly mainstream probability from an analytical standpoint, and the methods are modern...We can expect it to take its place alongside the classics of probability theory.\" F.B. Knight, Mathematical Review\"It is a first-class work of reference, for technique as well as for results and difinitive theory; even an extended review as this does not do justice to its detail and density.\" Peter Whittle, The Mathematical Intelligencer\"...a first-class work of reference, for technique as well as for results and definitve theory; even as extended a review as this does not do justice to its detail and density.\" Peter Whittle, The Mathematical Intelligencer'
Synopsis
Revised edition of a first-year graduate course on probability theory.
Synopsis
This revised edition is suitable for a first-year graduate course on probability theory. It is intended for students with a good grasp of introductory, undergraduate probability and a reasonably sophisticated introduction to modern analysis who now want to learn what these two topics have to say about each other. By modern standards the topics treated here are classical and the techniques used far-ranging. No attempt has been made to present 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. The introduction of conditional expectation values is postponed until the second part of the book where it is applied to the study of martingales. This section also explores the connection between martingales and various aspects of classical analysis and the connections between Wiener's measure and classical potential theory.
Table of 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.