Zero bias transformation and asymptotic expansions II : the Poisson case

TL;DR

This paper develops a recursive asymptotic expansion for expectations of functions of sums of independent integer-valued variables using a discrete zero bias transformation, extending previous Gaussian-based methods.

Contribution

It introduces a discrete zero bias transformation approach for Poisson sums and provides detailed remainder estimations for the asymptotic expansions.

Findings

Recursive asymptotic expansion for Poisson sums derived

Remainder estimations for the asymptotic series provided

Extension of Gaussian-based methods to discrete Poisson case

Abstract

We apply a discrete version of the methodology in \cite{gauss} to obtain a recursive asymptotic expansion for $\esp[h(W)]$ in terms of Poisson expectations, where $W$ is a sum of independent integer-valued random variables and $h$ is a polynomially growing function. We also discuss the remainder estimations.