Edgeworth expansion with error estimates for power law shot noise

Consider a homogeneous Poisson process in ${\mathbb{R}}^d$, $d\ge 1$. Let $R_1 < R_2 < \dots$bethedistancesofthepointsfromtheorigin,andlet$S = R_1^{-\gamma} + R_2^{-\gamma} + \dots$,where$\gamma > d$isaparameter\mathrm{.}Let$\overline{S}^{®} = \sum_k R_k^{-\gamma} , $\mathbf{1}$_{R_k \ge r}$ be the contribution to $S$ outside radius $r$. For large enough $r$, and any $\bar{s}$ in the support of ${\bar{S}}^{(r)}$, consider the change of measure that shifts the mean to $\bar{s}$. We derive rigorous error estimates for the Edgeworth expansion of the transformed random variable. Our error terms are uniform in $\bar{s}$, 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$S = s$andunconditionedradii$(R’_1, R’_2, \dots)$.