The Annals of Probability

Two Moments Suffice for Poisson Approximations: The Chen-Stein Method

Abstract

Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable result is due to Chen (1975). The method also provides an upper bound on the total variation distance to the Poisson distribution, and succeeds in cases where third and higher moments blow up. This paper presents Chen's results in a form that is easy to use and gives a multivariable extension, which gives an upper bound on the total variation distance between a sequence of dependent indicator functions and a Poisson process with the same intensity. A corollary of this is an upper bound on the total variation distance between a sequence of dependent indicator variables and the process having the same marginals but independent coordinates.

Article information

Source
Ann. Probab. Volume 17, Number 1 (1989), 9-25.

Dates
First available: 19 April 2007

http://projecteuclid.org/euclid.aop/1176991491

JSTOR

Digital Object Identifier
doi:10.1214/aop/1176991491

Mathematical Reviews number (MathSciNet)
MR972770

Zentralblatt MATH identifier
0675.60017

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

Citation

Arratia, R.; Goldstein, L.; Gordon, L. Two Moments Suffice for Poisson Approximations: The Chen-Stein Method. The Annals of Probability 17 (1989), no. 1, 9--25. doi:10.1214/aop/1176991491. http://projecteuclid.org/euclid.aop/1176991491.