A modern introduction to probability theory
This textbok is designed for graduate students in probability theory. It merges measure theory with probability theory, and deal with "random objects" as well as "random variables". There is a chapter of introductory material for advanced topics, as well as over 1000 exercises.
Print Book, English, ©1996
Birkhäuser, Boston, ©1996
9780817638078, 9783764338077, 0817638075, 3764338075
List of Tables * Preface * Part I: Probability Spaces, Random Variables, and Expectations * Probability Spaces * Random Variables * Distribution Functions * Expectations: Theory * Expectations: Applications * Calculating Probabilities and Measures * Measure Theory: Existence and Uniqueness * Integration Theory * Part 2: Independence and Sums * Stochastic Independence * Sums of Independent Random Variables * Random Walk * Theorems of A.S. Convergence * Characteristic Functions * Part 3: Convergence in Distribution * Convergence in Distribution on the Real Line * Distributional Limit Theorems for Partial Sums * Infinitely Divisible and Stable Distributions as Limits * Convergence in Distribution on Polish Spaces * The Invariance Principle and Brownian Motion * Part 4: Conditioning * Spaces of Random Variables * Conditional Probabilities * Construction of Random Sequences * Conditional Expectations * Part 5: Random Sequences * Martingales * Renewal Sequences * Time-homogeneous Markov Sequences * Exchangeable Sequences * Stationary Sequences * Part 6: Stochastic Processes * Point Processes * Diffusions and Stochastic Calculus * Applications of Stochastic Calculus * Part 7: Appendices * Appendix A. Notation and Usage of Terms * Appendix B. Metric Spaces * Appendix C. Topological Spaces * Appendix D. Riemann–Stieltjes Integration * Appendix E. Taylor Approximations, C-Valued Logarithms * Appendix F. Bibliography * Appendix G. Comments and Credits * Index