Poisson Approximation and the Chen-Stein Method

• Published in: Statistical science Vol. 5; no. 4; pp. 403 - 424
• Main Authors: Arratia, Richard, Goldstein, Larry, Gordon, Louis
• Format: Journal Article
• Language: English
• Published: Institute of Mathematical Statistics 01-11-1990; The Institute of Mathematical Statistics
• Subjects: Approximation; Birthdays; Coincidence; Index sets; invariance principle; Mathematical inequalities; Mathematical moments; Mathematical theorems; Poisson approximation; Poisson process; Random variables; Stein's method; Vertices
• ISSN: 0883-4237; 2168-8745
• DOI: 10.1214/ss/1177012015

The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given in terms of first and second moments alone. We present a background of the method and state some fundamental Poisson approximation theorems. The body of this paper is an illustration, through varied examples, of the wide applicability and utility of the Chen-Stein method. These examples include birthday coincidences, head runs in coin tosses, random graphs, maxima of normal variates and random permutations and mappings. We conclude with an application to molecular biology. The variety of examples presented here does not exhaust the range of possible applications of the Chen-Stein method.