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.
Author(s): Jordan M. Stoyanov
Series: Dover Books on Mathematics
Edition: Third
Publisher: Dover Publications
Year: 2013
Language: English
Pages: 404
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
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