Multivariate normal approximation in geometric probability

Mathew D. Penrose and Andrew R. Wade

Journal of Statistical Theory and Practice, 2, no. 2, June 2008, 293–326. DOI: 10.1080/15598608.2008.10411876.

Abstract

Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $\mathbb{R}^d$, and $\xi_x$ 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 $\mu_\lambda$-measures (suitably scaled and centred) of disjoint sets in $\mathbb{R}^d$ are asymptotically independent normals as $\lambda$ tends to infinity; here we give an $O( \lambda^{-1/(2d + \varepsilon)})$ 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.