A non-uniform bound for translated Poisson approximation

Abstract

Let $X_1, \ldots , X_n$ be independent, integer valued random variables, with $p^{\text{th}}$ moments, $p >2$, and let $W$ denote their sum. We prove bounds analogous to the classical non-uniform estimates of the error in the central limit theorem, but now, for approximation of ${\cal L}(W)$ by a translated Poisson distribution. The advantage is that the error bounds, which are often of order no worse than in the classical case, measure the accuracy in terms of total variation distance. In order to have good approximation in this sense, it is necessary for ${\cal L}(W)$ to be sufficiently smooth; this requirement is incorporated into the bounds by way of a parameter $\alpha$, which measures the average overlap between ${\cal L}(X_i)$ and ${\cal L}(X_i+1), 1 \le i \le n$.