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!"