Probability Theory

Contents: 1. Probability Spaces. 2. Independence. 3. Laws of Large Numbers. 4. Convergence in Distribution. 5. Characteristic Functions. 6. Central Limit Theorems. 7. Conditional Expectation and Conditional Probability. 8. Discrete Parameter Martingales. 9. Markov Chains and MCMC. 10. Stochastic Processes. 11. Limit Theorems for Dependent Processes. 12. The Bootstrap. 13. Branching Processes. A. Review of Measure Theory. A.1. Measures A.2 Integration. A.3. L p ­Spaces. A.3.1 Inequalities. A.3.2. L p ­Spaces. A.3.3. Duality. A.4 Differentiation. A.5. Product measures. B. List of Abbreviations and Symbols. B.1 Abbreviations. B.2 Symbols. References. Author Index. Subject Index.

"This book provides a comprehensive treatment of modern probability theory based on measure theory. It can be regarded as a companion volume to "Measure Theory", Vol. 36, TRIM Series. The topics covered include Kolmogorov's existence theorem for stochastic processes, the fundamental notion of stochastic independence, the laws of large numbers including ergodic theorems, weak convergence of probability measures on the real line and Euclidean spaces, characteristic functions and the Levy-Cramer continuity theorem, the central limit theorems, stable laws, conditional expectation, discrete time martingales, discrete time Markov chains, continuous time Markov chains, Brownian motion, mixing sequences, theory of resampling methods in statistics, and an introduction to branching processes. The appendix reviews basic results from measure theory.The book includes a large number of exercises at varying levels of difficulty that will be helpful to the instructor and students. This book should be valuable to M.Sc. and Ph.D. students and research workers in mathematics, statistics, econometrics, engineering, physical and biological sciences."