## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 4 (1975), 726-731.

### A General Poisson Approximation Theorem

#### Abstract

A sum of nonnegative integer-valued random variables may be treated as a Poisson variable if the summands have sufficiently high probabilities of taking 0 value and sufficiently weak mutual dependence. This paper presents simple exact upper bounds for the error of such an approximation. An application is made to obtain a new extension for dependent events of the divergent part of the Borel-Cantelli lemma. The bounds are illustrated for the case of Markov-dependent Bernoulli trials. The method of the paper is to reduce the general problem to the special case of independent 0-1 summands and then make use of known bounds for this special case.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 4 (1975), 726-731.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996313

**Digital Object Identifier**

doi:10.1214/aop/1176996313

**Mathematical Reviews number (MathSciNet)**

MR380946

**Zentralblatt MATH identifier**

0321.60018

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Poisson approximation dependent summands Borel-Cantelli lemma Markov-dependent Bernoulli trials

#### Citation

Serfling, R. J. A General Poisson Approximation Theorem. Ann. Probab. 3 (1975), no. 4, 726--731. doi:10.1214/aop/1176996313. https://projecteuclid.org/euclid.aop/1176996313