Probability theory

the long version

This book is intended for graduate students who have a good undergraduate introduction to probability theory, a reasonably sophisticated introduction to modern analysis, and 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. Although the book is primarily intended for students and practitioners of probability theory and analysis, it will also be a valuable reference for those in fields as diverse as physics, engineering, and economics.