Counterexamples in probability.

*(English)*Zbl 0629.60001
Wiley Series in Probability and Mathematical Statistics. Chichester etc.: John Wiley & Sons. XXIII, 313 p.; £38.50 (1987).

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.

With the success of well-known earlier books of counterexamples in analysis, graph theory and topology, it is not surprising to witness the appearance of such books in other fields. Indeed, the book under review was preceded by a related work by J. P. Romano and A. F. Siegel [Counterexamples in probability and statistics (1986; Zbl 0587.60001)]. Despite their similar titles, there appears – to this reviewer, at any rate – to be a surprisingly small overlap of material between the two books. To a large extent, this may be explained by the focus of each book: Romano and Siegel devote nearly half of their wok to statistical counterexamples, while Stoyanov looks exclusively at probability theory and includes topics (such as stochastic processes) not examined in the other book. In summary, the author has produced a well- organized and carefully-written book which will be a welcome addition to the bookshelves of teachers, students and researchers alike.

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.

With the success of well-known earlier books of counterexamples in analysis, graph theory and topology, it is not surprising to witness the appearance of such books in other fields. Indeed, the book under review was preceded by a related work by J. P. Romano and A. F. Siegel [Counterexamples in probability and statistics (1986; Zbl 0587.60001)]. Despite their similar titles, there appears – to this reviewer, at any rate – to be a surprisingly small overlap of material between the two books. To a large extent, this may be explained by the focus of each book: Romano and Siegel devote nearly half of their wok to statistical counterexamples, while Stoyanov looks exclusively at probability theory and includes topics (such as stochastic processes) not examined in the other book. In summary, the author has produced a well- organized and carefully-written book which will be a welcome addition to the bookshelves of teachers, students and researchers alike.

Reviewer: R.Tomkins

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60Exx | Distribution theory |

60Fxx | Limit theorems in probability theory |

60Gxx | Stochastic processes |

60Jxx | Markov processes |