APPROXIMATION OF SUMS OF CONDITIONALLY INDEPENDENT VARIABLES BY THE TRANSLATED POISSON DISTRIBUTION

Abstract. It is shown that the sum of a Poisson and an independent approximately normally distributed integer valued random variable can be well approximated in total variation by a translated Poisson distribution, and further that a mixed translated Poisson distribution is close to a mixed translated Poisson distribution with the same random shift but fixed variance. Using these two results, a general approach is then presented for the approximation of sums of integer valued random variables, having some conditional independence structure, by a translated Pois-son distribution. We illustrate the method by means of two examples. The proofs are mainly based on Stein’s method for distributional approximation. 1