Counterexamples in Probability

Counterexamples (in the mathematical sense) are powerful tools of mathematical theory. This book covers counterexamples from probability theory and stochastic processes. This new expanded edition includes many examples and the latest research results. The author is regarded as one of the foremost experts in the field. Contains numbers examples.

Counterexamplesin Probability Second Edition Jordan M. Stoyanov Bulgarian Academy of Sciences and University of Sofia Following the success of the first edition, widely regarded as the classic reference work on the subject, Professor Stoyanov has expanded his work to include many new counterexamples and the latest research results. Nearly 300 counterexamples are included, selected for their interest and for the importance of the theory they illustrate. A summary of definitions and main results is provided at the beginning of each section, followed by counterexamples in order of content and difficulty. These counterexamples demonstrate the power and non-triviality of stochastics. They cover the main results used in undergraduate and graduate courses in probability and stochastic processes and provide new starting points for students, teachers and researchers. Lecturers and examiners will find these counterexamples a useful source of illustrations and ideas. Contents Preface to the Second Edition Preface to the First Edition Basic Notation and Abbreviations Part 1 Classes of Random Events and Probabilities

1. Classes of Random Events
2. Probabilities
3. Independence of Random Events
4. Diverse Properties of Random Events and their Probabilities

1. Distribution Functions of Random Variables
2. Expectations and Conditional Expectations
3. Independence of Random Variables
4. Characteristic and Generating Functions
5. Infinitely Divisible and Stable Distributions
6. Normal Distribution
7. The Moment Problem
8. Characterization Properties of Some Probability Distributions
9. Diverse Properties of Random Variables

1. Various Kinds of Convergence of Sequences of Random Variables
2. Laws of Large Numbers
3. Weak Convergence of Probability Measures and Distributions
4. Central Limit Theorem
5. Diverse Limit Theorems

1. Basic Notions on Stochastic Processes
2. Markov Processes
3. Stationary Processes and Some Related Topics
4. Discrete-Time Martingales
5. Continuous-Time Martingales
6. The Poisson Process and the Wiener Process
7. Diverse Properties of Stochastic Processes

About the author Jordan Stoyanov graduated from Moscow University in 1970. He is now a Senior Research Fellow at the Bulgarian Academy of Sciences and a Professor at the University of Sofia. His main research interests are in the field of stochastic processes and probability theory. Professor Stoyanov is a member of the Bulgarian Mathematical Society, the Bernoulli Society for Mathematical Statistics and Probability, and the Mathematical Association of America. He has been a visiting professor at many universities in Europe, Canada and the USA. His interest in collecting counterexamples dates from his student days.