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

Röllin, A (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli, 11(6):1115-1128.

## Abstract

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.

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.

## Citations

18 citations in Web of Science®
20 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Stein's method; total variation metric; translated Poisson distribution English 2005 29 Nov 2010 16:26 05 Apr 2016 13:24 Bernoulli Society for Mathematical Statistics and Probability 1350-7265 https://doi.org/10.3150/bj/1137421642 http://www.ams.org/mathscinet-getitem?mr=2189083http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1102.60022
Permanent URL: https://doi.org/10.5167/uzh-21741
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