Foundations of Modern Probability

Description

This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry. Apart from important improvements and revisions, some of the earlier chapters have been entirely rewritten. To help the reader, the material has been grouped together into ten major areas, each arguably indispensable to any serious graduate student and researcher, regardless of their specialization. Each chapter is largely self-contained and includes plenty of exercises, making the book ideal for self-study and for designing graduate-level courses and seminars in different areas and at different levels. Extensive notes and a detailed bibliography make it easy to go beyond the presented material if desired. Preface to the First Edition Preface to the Second Edition Preface to the Third Edition Acknowledgments Words of Wisdom and Folly Contents Introduction and Reading Guide I. Measure Theoretic Prerequisites Chapter 1 Sets and Functions, Measures and Integration Chapter 2 Measure Extension and Decomposition Chapter 3 Kernels, Disintegration, and Invariance II. Some Classical Probability Theory Chapter 4 Processes, Distributions, and Independence Chapter 5 Random Sequences, Series, and Averages Chapter 6 Gaussian and Poisson Convergence Chapter 7 Infinite Divisibility and General Null Arrays III. Conditioning and Martingales Chapter 8 Conditioning and Disintegration Chapter 9 Optional Times and Martingales Chapter 10 Predictability and Compensation IV. Markovian and Related Structures Chapter 11 Markov Properties and Discrete-Time Chains Chapter 12 Random Walks and Renewal Processes Chapter 13 Jump-Type Chains and Branching Processes V. Some Fundamental Processes Chapter 14 Gaussian Processes and Brownian Motion Chapter 15 Poisson and Related Processes Chapter 16 Independent-Increment and Lévy Processes Chapter 17 Feller Processes and Semi-groups VI. Stochastic Calculus and Applications Chapter 18 Itô Integration and Quadratic Variation Chapter 19 Continuous Martingales and Brownian Motion Chapter 20 Semi-Martingales and Stochastic Integration Chapter 21 Malliavin Calculus VII. Convergence and Approximation Chapter 22 Skorohod Embedding and Functional Convergence Chapter 23 Convergence in Distribution Chapter 24 Large Deviations VIII. Stationarity, Symmetry and Invariance Chapter 25 Stationary Processes and Ergodic Theory Chapter 26 Ergodic Properties of Markov Processes Chapter 27 Symmetric Distributions and Predictable Maps Chapter 28 Multi-variate Arrays and Symmetries IX. Random Sets and Measures Chapter 29 Local Time, Excursions, and Additive Functionals Chapter 30 Random Measures, Smoothing and Scattering Chapter 31 Palm and Gibbs Kernels, Local Approximation X. SDEs, Diffusions, and Potential Theory Chapter 32 Stochastic Equations and Martingale Problems Chapter 33 One-Dimensional SDEs and Diffusions Chapter 34 PDE Connections and Potential Theory Chapter 35 Stochastic Differential Geometry Appendices 1. Measurable maps 2. General topology 3. Linear spaces 4. Linear operators 5. Function and measure spaces 6. Classes and spaces of sets 7. Differential geometry Notes and References Bibliography Indices Authors Topics Symbols