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.