Probability approximations via the Poisson clumping heuristic. (English) Zbl 0679.60013

Applied Mathematical Sciences, 77. New York etc.: Springer-Verlag. xv, 269 p. DM 88.00 (1989).
The Poisson clumping heuristic is a technique for obtaining estimates of probabilities associated with extrema. One begins by observing that “exceptional” values of a stochastic process tend to occur in “clumps”, which themselves occur more or less according to a Poisson process. This observation was already proved fruitful in classical extreme value theory, where the process of exceedances plays a prominent role. The chance that the most extreme value is not exceptional can then be written approximately as the probability that no clump of exceptional values occurs, which is just a Poisson probability involving the rate \(\lambda\) of the clump process.
Now, if clumps are rare, in the sense that they rarely overlap, the proportion p of values which are exceptional is approximately related to \(\lambda\) by the formula \(p=\lambda E C\), where E C denotes the expected size of the clump. Thus, if p can be calculated (which is often easy) and if E C can be estimated (which is often harder), \(\lambda\) can be determined, and from \(\lambda\) the required probabilities can be deduced. There are various methods for computing E C, among which the harmonic mean formula and the ergodic exit method are prominent.
The heuristic is illustrated by a cornucopia of examples, drawn from widely diverse areas of probability theory, ranging from sample path properties of diffusions through geometrical coverage problems to questions concerning combinatorial extrema over very large, but finite sets. Some examples are easy, others very hard: some have been well investigated in the literature, others are quite new. Throughout, the author takes the standpoint that the aim is to compute first and ask questions afterwards - rigour is explicitly played down, even if the arguments used are nonetheless suspiciously carefully motivated. The benefit derived is in the wealth of material which can then be compressed into 270 pages.
The book is delightful, stimulating, challenging, overwhelming: so many good problems, so little time... not just something for everyone, more a feast. Sure to be a classic, and deservedly so.
Reviewer: A.Barbour


60C05 Combinatorial probability
60D05 Geometric probability and stochastic geometry
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)