Probability and Its Applications. New York, NY: Springer. ix, 269 p. DM 88.00; öS 642.40; sFr 77.50 (1996).
This is the first book devoted solely to the subject of Bonferroni-type inequalities. It is a useful guide in this exciting theory and the reader can find there a large variety of classical and new problems, results, methods for proof and applications.
The book contains ten chapters. The first four chapters are devoted to actual methods for proof of Bonferroni-type inequalities: the method of indicators, the method of polynomials, the geometrie method and the linear programming method. The method of indicators is a method of proving a probabilistic inequality involving probabilities of Boolean functions by the equivalent non-probabilistic inequality involving the corresponding indicator functions. It is based on a celebrated Rényi’s theorem (1958). Chapter I is devoted to this method and also includes inequalities known in the literature as graph-sieves. The method of polynomials (Chapter II) is justified by the observation that the validity of certain Bonferroni-type inequalities is equivalent to the validity of a corresponding collection of polynomial inequalities. This is the best method suited for finding new inequalities. The geometric method (Chapter III) is based on the geometric properties of the range of the binomial moments that is a convex polytope. It provides sharp Bonferroni-type bounds which are not linear over the whole space. Optimal Bonferroni-type inequalities can alternatively be viewed as solutions of maximization (minimization) problems of the linear programming. In Chapter IV the basic results of this theory are presented and an algorithm is used to derive Bonferroni-type inequalities.
Chapter V is devoted to linear bounds on the multivariate distribution of the number of occurrences in several sequences of events. In fact, the multivariate Bonferroni-type inequalities are at a much less developed stage than the univariate cases are. Yet a large number of interesting results obtained in recent years are unified in this chapter. Various applications to combinatorics, number theory, statistics and extreme value theory are considered in Chapters VI through IX. A special attention is payed to the applications of Bonferroni-type inequalities in extreme value theory. Limit theorems for maxima of r.v.’s are given in different models, namely the classical model of i.i.d. r.v.’s, the exchangeable model, the model where the r.v.’s are independent but not identically distributed, the graph dependent model. Miscellaneous topics are discussed in the last Chapter X, such as probability of occurrence for infinite sequences of events, quadratic inequalities, Borel-Cantelli lemmas proved by Bonferroni-type bounds. At the end the authors list some modern fields of applications developed in the recent years.
This book may be of interest to a wide readership since no previous knowledge of the subject matters is required. The necessary information is contained in an introduction section heading each chapter.