In this paper we give a historical account of the development of Poisson approximation using Stein’s method and present some of the main results. We give two recent applications, one on maximal arithmetic progressions and the other on bootstrap percolation. We also discuss generalisations to compound Poisson approximation, Poisson process approximation and multivariate Poisson approximation, an...

An information-theoretic foundation for compound Poisson approximation limit theorems is presented, in analogy to the corresponding developments for the central limit theorem and for simple Poisson approximation. It is shown that the compound Poisson distributions satisfy a natural maximum entropy property within a natural class of distributions. Simple compound Poisson approximation bounds are...

Given an i.i.d. sequence of n letters from a finite alphabet, we consider the length of the longest run of any letter. In the equiprobable case, results for this run turn out to be closely related to the well-known results for the longest run of a given letter. For coin-tossing, tail probabilities are compared for both kinds of runs via Poisson approximation.

Compound Poisson processes are often useful as approximate models, when describing the occurrence of rare events. In this paper, we develop a method for showing how close such approximations are. Our approach is to use Stein's method directly, rather than by way of declumping and a marked Poisson process; this has conceptual advantages, but entails technical difficulties. Several applications a...

i=1 pi ≤ max 1≤i≤n pi, (6.1) where λ := EW = ∑n i=1 pi. Clearly, if the pi’s are not all small, there may be little content in (6.1). This is to be expected, since then EW = λ and VarW = λ − ∑n i=1 p 2 i need no longer be close to one another, whereas Poisson distributions have equal mean and variance. This makes it more natural to try to find a family of distributions for the approximation wit...