Edgeworth expansion with error estimates for power law shot noise

Abstract

Consider a homogeneous Poisson process in ${\mathbb{R}}^d$, $d\ge 1$. Let $R_1<R_2<\dots$ be the distances of the points from the origin, and let $S=R_1^{-\gamma }+R_2^{-\gamma }+\dots$, where $\gamma >d$ is a parameter. Let ${\overline{S}}^{(r)}={\sum }_k{R}_k^{-\gamma }{1}_{R_k\ge r}$ be the contribution to $S$ outside radius $r$. For large enough $r$, and any $\overline{s}$ in the support of ${\overline{S}}^{(r)}$, consider the change of measure that shifts the mean to $\overline{s}$. We derive rigorous error estimates for the Edgeworth expansion of the transformed random variable. Our error terms are uniform in $\overline{s}$, and we give explicitly the dependence of the error on $r$ and the order $k$ of the expansion. As an application, we provide a scheme that approximates the conditional distribution of $R_1$ given $S=s$ to any desired accuracy, with error bounds that are uniform in $s$. Along the way, we prove a stochastic comparison between $(R_1,R_2,\dots )$ given $S=s$ and unconditioned radii $(R_1^{\prime },R_2^{\prime },\dots )$.

35 pages, 1 figure