A non-uniform bound for translated Poisson approximation
Andrew D Barbour, Universitat Zurich
Kwok Pui Choi, National University of Singapore
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 $law(W)$ 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 $law(W)$ to be sufficiently
smooth; this requirement is incorporated into the bounds by way of a
parameter $al$, which measures the average overlap
between $law(X_i)$ and $law(X_i+1), , 1 le i le n.$
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