Poisson process approximation

Abstract

Abstract Another reason for wishing to do so comes from the study of the extremes of a sequence of random variables. In this context, it is interesting to know not only when exceptional values occur but also how big they are, and the natural framework for the description of such phenomena is the marked point process of exceedances. It is often useful to know that the exceedance process can be reasonably approximated by a Poisson process on an appropriate space, and finding bounds for the error in such an approximation is therefore of considerable interest. A further reason is provided by multivariate Poisson approximation, as for instance when approximating the joint distribution of the numbers of cycles of lengths 1, 2 and 3 in a random permutation. Since a finite collection of independent Poisson random variables can be thought of as a Poisson process on a finite set, multivariate approximation can be regarded as a special case of Poisson process approximation.