Approximation of sums of conditionally independent variables by the translated Poisson distribution

It is shown that the distribution of the sum of a Poisson random variable 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 Poisson distribution. We illustrate the method by means of two examples. The proofs are mainly based on Stein's method for distributional approximation