Foundations of Modern Probability

• * Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

"The second edition of this admirable book has grown by well over one hundred pages, including such new material as: multivariate and ratio ergodic theorems, shift coupling, Palm distributions, entropy and information, Harris recurrence, invariant measures, strong and weak ergodicity, Strassen's LIL and the basic large deviation results. Also, a lot of existing material has been rewritten and expanded. I repeat a statement from my review of the first edition: "From the table of contents it is difficult to believe behind all these topics a streamlined readable text is at all possible. It is: Convince yourself." Those who own the first edition should make some extra space for this second edition. Those who do not yet own a copy: buy one!" P.A.L. Embrechts in ISI Short Book Reviews, Vol. 22/3, December 2002From the reviews of the first edition: "...Kallenberg's present book would have to qualify as the assimilation of probability par excellence. It is a great edifice of material, clearly and ingeniously presented, without any non-mathematical distractions. Readers wishing to venture into it may do so with confidence that they are in very capable hands." F.B. Knight, Mathematical Reviews"... Indeed the monograph has the potential to become a (possibly even ``the'') major reference book on large parts of probability theory for the next decade or more." M. Scheutzow, Zentralblatt"The theory of probability has grown exponentially during the second half of the twentieth century and the idea of writing a single volume that could serve as a general reference for much of the modern theory seems almost foolhardy. Yet this is precisely what Professor Kallenberg has attempted in the volume under review and he has accomplished it brilliantly. ... It is astonishing that a single volume of just over five hundred pages could contain so much material presented with complete rigor and still be at least formally self-contained..." R.K. Getoor, MetricaFrom the reviews of the second edition: "The first edition of the author’s magnum opus drew universal acclaim when it appeared in 1997 for its scope, clarity and precision. For the second edition … Professor Kallenberg has added over a hundred pages in attempt to keep it up to date with the exponentially increasing body of knowledge which is probability theory. Moreover, certain sections … have been comprehensively rewritten in order to be self-sufficient. … This is an essential purchase for any serious probabilist." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (511), 2004)"The first edition of this most outstanding monograph has been praised in many reviews. The second edition is a revised and enlarged version of the first … . Of course, the praise for the first edition applies to the second edition as well. Nevertheless, it should be noted that the style in which this monograph is written is concise and particular. ... It has been said that the present monograph is a modern classic in probability theory, and this is true." (Klaus. D. Schmidt, Mathematical Reviews, Issue 2002 m)"The second edition of this admirable book has grown by well over one hundred pages … . Also, a lot of existing material has been rewritten and expanded. … ‘From the table of contents it is difficult to believe behind all these topics a streamlined readable text is at all possible. It is: Convince yourself.’ Those who own the first edition should make some extra space for this second edition. Those who do not yet own a copy: buy one!" (P. A. L. Embrechts, Short Book Reviews, Vol. 22 (3), 2002)"This second edition presents now even more material in the concise and elegant style of the former edition which by now has become a highly praised standard reference book for many areas of probability theory. … The chapters of the first edition have been carefully revised and corrected so that it has become an even more accurate and reliable reference work. Starting a book with a page entitled ‘Praise for the first edition’ is always daring, but in the case of Kallenberg’s treatise indeed justified." (Markus Reiß, Zentralblatt Math, Vol. 996 (21), 2002)"Foundations of Modern Probability is generally … useful at a graduate level. … It concision and abstractness makes it a useful reference." (Wordtrade, 2008)