Counterexamples in probability / Jordan M. Stoyanov

Bibliogr. p. 293-309. Index

This book presents over 250 counterexamples involving a variety of topics in probability theory. It is intended both as a source of supplementary material for undergraduate and graduate courses in probability theory and stochastic processes, and as a monograph of interest, in its own right, for researchers.

This book contains twenty-five sections organized into four chapters. Each section begins with a review of the definitions and basic theorems pertaining to the title of the section, and then presents a number of relevant counterexamples. The first chapter is a short one which looks at basic properties of probability spaces and independence of events. The second chapter focusses on random variables, and includes counterexamples involving distribution functions, characteristic functions, infinite divisibility, stable distributions, independence of random variables and moments (including conditional expectations). Chapter 3 is devoted to limit theorems for sequences of random variables; interrelationships amongst the various modes of convergence are discussed, and examples involving the laws of large numbers, the central limit theorem and the three-series theorem are presented.

The final chapter examines stochastic processes, including stationary, Markov, Poisson and Wiener processes. Also included are counterexamples involving discrete- and continuous-time martingales and generalizations thereof. The book concludes with a set of supplementary remarks which give the source of each counterexample presented in the book. ... (Zentralblatt)