Compound Poisson approximation: a user’s guide

Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecular sequence analysis. In this paper, we review the ways in which Stein's method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration.We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein's method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods. The Annals of Applied Probability 2001, Vol. 11, No. 3, 964–1002 COMPOUND POISSON APPROXIMATION: A USER’S GUIDE By A. D. Barbour1 2 and O. Chryssaphinou2 Universität Zürich and University of Athens Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecular sequence analysis. In this paper, we review the ways in which Stein’s method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration. We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein’s method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods. 1. Motivation. Many probability models [Aldous 1989] involve rare, isolated and weakly dependent clumps of interesting occurrences. A typical example is that of clusters of extreme events, such as earthquakes of magnitude exceeding 4 0; when one event occurs, several more may follow in quick succession, before normality returns. Clusters of events can then be expected to take place almost “at random,” according to a Poisson process, in which case the number of clusters occurring in a given time interval would have a distribution close to a Poisson distribution Po λ with some mean λ, and the sizes of the individual clumps might well also be assumed to be approximately independent and identically distributed with some distribution . If these assumptions were precisely true, the total number W of occurrences would then have a compound Poisson distribution CP λ , the distribution of the sum of a random Po λ -distributed number of independent random variables, each with distribution : more formally, CP λ is defined by