Multivariate normal approximation in geometric probability

Abstract

Consider a measure $μ_λ= \sum_x ξ_x δ_x$ where the sum is over points $x$ of a Poisson point process of intensity $λ$ on a bounded region in $d$-space, and $ξ_x$ is a functional determined by the Poisson points near to $x$, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the $μ_λ$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $λ\to \infty$; here we give an $O(λ^{-1/(2d + ε)})$ bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.