Edgeworth expansion with error estimates for power law shot noise

Consider a homogeneous Poisson process in $\mathbb{R}^d$, $d \ge 1$. Let $R_1 < R_2 < \dots$ 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 $\overline{S}^{(r)} = \sum_k R_k^{-\gamma} \, \mathbf{1}_{R_k \ge r}$ 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 $(R'_1, R'_2, \dots)$.