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 is a reasonably sophisticated introduction to modern analysis for those who want to learn what these two topics have to say about each other. 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.

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.' F. B. Knight, Mathematical Reviews

'... a first class work of reference, for technique as well as for results and definitive theory ...' Peter Whittle, The Mathematical Intelligencer

'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 Ststistical Association

'As the above list of topics shows, this book is much more than an introduction to probability ... the wealth of material presented and the strong emphasis on the intimate links between probability and analysis make this book worth having.' M. Schewizer, Zentralblatt f�r Mathematik

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