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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

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