A Modern Approach to Probability Theory

Students and teachers of mathematics and related fields will find this book a comprehensive and modern approach to probability theory, providing the background and techniques to go from the beginning graduate level to the point of specialization in research areas of current interest. The book is designed for a two- or three-semester course, assuming only courses in undergraduate real analysis or rigorous advanced calculus, and some elementary linear algebra. A variety of applications-Bayesian statistics, financial mathematics, information theory, tomography, and signal processing-appear as threads to both enhance the understanding of the relevant mathematics and motivate students whose main interests are outside of pure areas.

Contenu

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