Counterexamples in Probability

Description

Most mathematical examples illustrate the truth of a statement; conversely, counterexamples demonstrate a statement's falsity if changing the conditions. Mathematicians have always prized counterexamples as intrinsically enjoyable objects of study as well as valuable tools for teaching, learning, and research. This third edition of the definitive book on counterexamples in probability and stochastic processes presents the author's revisions and corrections in addition to a substantial new appendix. Suitable as a supplementary source for advanced undergraduates and graduate courses in the field of probability and stochastic processes, this volume features a wide variety of topics that are challenging in both content and detail. The text consists of four chapters and twenty-five sections. Each section begins with short introductory notes of basic definitions and main results. Counterexamples related to the main results follow, along with motivation for questions and counterstatements that range in difficulty. A familiarity with basic notions and results in probability and stochastic processes is assumed, and a chapter of supplementary remarks provides a wealth of information about original sources as well as references for further studies. Cover Copyright Contents Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Basic Notation and Abbreviations Chapter 1 Classes of Random Events and Probabilities Section 1 Classes of Random Events Section 2 Probabilities Section 3 Independence of Random Events Section 4 Diverse Properties of Random Events and Their Probabilities Chapter 2 Random Variables and Basic Characteristics Section 5 Distribution Functions of Random Variables Section 6 Expectations and Conditional Expectations Section 7 Independence of Random Variables Section 8 Characteristic and Generating Functions Section 9 Infinitely Divisible and Stable Distributions Section 10 Normal Distribution Section 11 The Moment Problem Section 12 Characterization Properties of Some Probability Distributions Section 13 Diverse Properties of Random Variables Chapter 3 Limit Theorems Section 14 Various Kinds of Convergence of Sequences of Random Variables Section 15 Laws of Large Numbers Section 16 Weak Convergence of Probability Measures and Distributions Section 17 Central Limit Theorem Section 18 Diverse Limit Theorems Chapter 4 Stochastic Processes Section 19 Basic Notions on Stochastic Processes Section 20 Markov Processes Section 21 Stationary Processes and Some Related Topics Section 22 Discrete-time Martingales Section 23 Continuous-time Martingales Section 24 Poisson Process and Wiener Process Section 25 Diverse Properties of Stochastic Processes Supplementary Remarks References Appendix Index Back