This monograph is based on a series of lectures given by the authors - three well-known specialists on extreme value theory - for a DMV seminar (a winterschool of the German Mathematical Association). The book gives a short introduction to topics on classical extreme value theory but mainly emphasizes recent results and new directions in the field, in particular the author's own research.
The title suggests that Poisson approximations to the binomial law are the basic tools. But Poisson approximation is mainly understood in a modern sense as approximation of point processes by Poisson random measures. This is what the authors call a `functional law of small numbers'. Point process techniques are the bread and butter of contemporary extreme value theory: see, for example, the standard monographs by Leadbetter, Lindgren and Rootzen (1983), Resnick (1987), and Reiss (1989). The reader should have some background on point processes (see Cox and Isham 1980, Daley and Vere-Jones 1988, Karr 1986, or Resnick 1987 for the necessary tools) and on classical extreme value theory. Thus the book aims at the graduate student, the mathematical researcher, and everybody interested in a recent account of extreme value theory.
Point process techniques are also used for `conditional curve estimation', which means estimating the regression curve P(X < x|Y = y). The authors show convincingly that point processes are the right tool for dealing with these objects in a similar way as extremes of a sequence of random variables; however, extremes are the book's main purpose. This includes the classical theory and rates of convergence in limit theorems for extremes (the authors make extensive use of the Hellinger distance as appropriate means), multivariate maxima, and extremes.
Almost half of the book is dedicated to extremes for non-identically distributed sequences (i.e., general independent, stationary, and Gaussian sequences). A survey of the statistical methods (e.g., for estimating the extremal index and the parameters of the generalized Pareto distribution) is given.
The book is accompanied by a diskette, XTREMES, by S. Hassmann, R.-D. Reiss, and M. Thomas, providing an entertaining and educational statistical software system (for IBM compatible PC's under MS-DOS). This software seems to be unique. It is easy to handle and can be recommended for classroom purposes to illustrate basic distributions, fundamental notions, and the quality of statistical procedures in extreme value theory. Using the software is well documented in a chapter and an appendix.
It is my impression that books on extreme value theory must be very technical, and this monograph is no exception. At least in some of the more specialized chapters the reader will be confronted with some very technical assumptions and proofs. But this does not reduce the book's general value as a compendium of modern extreme value theory with introductions, surveys and long reference lists to different topics.
In summary, `Laws of Small Numbers: Extremes and Rare Events' is a very recommendable book for those interested in regression, extremes, point processes, dependence in probability theory, and statistics.