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Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.
This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book covers a wide variety of topics, many of which are not usually found in introductory textbooks.
Basic Measure Theory.- Independence.- Generating Functions.- The Integral.- Moments and Laws of Large Numbers.- Convergence Theorems.- Lp-Spaces and Radon-Nikodym Theorem.- Conditional Expectations.- Martingales.- Optional Sampling Theorems.- Martingale Convergence Theorems and their Applications.- Backwards Martingales and Exchangeability.- Convergence of Measures.- Probability Measures on Product Spaces.- Characteristics Functions and Central Limit Theorem.- Infinitely Divisible Distributions.- Markov Chains.- Convergence of Markov Chains.- Markov Chains and Electrical Networks.- Ergodic Theory.- Brownian Motion.- Law of the Iterated Logarithm.- Large Deviations.- The Poisson Point Process.- The Ito Integral.- Stochastic Differential Equations.- References.- Notation Index.- Name Index.- Subject Index.
From the reviews: "The book is indeed comprehensive, consisting of 26 chapters on different topics. ! can be well used as a reference book on a wide range of topics. The target audience is researchers and graduate students ! . Numerous advanced topics are included, so that the book is more inclusive ! . There is more than enough material for a two-semester course here. ! the book will primarily be used as a reference book. For that purpose, it is a rich and relatively inexpensive choice." (Miklos Bona, MathDL, January, 2008) "This book of over 600 pages gives a self-contained presentation of modern probability theory. It is based on courses on advanced probability given by the author. ! Most of the proofs are well detailed. ! This book will be helpful for graduate students in mathematics ! and for researchers in mathematics or theoretical physics." (Sophie Lemaire, Mathematical Reviews, Issue 2009 f)