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. [Article] [arXiv] [MR]



Abstract

Consider a measure μλ=xξxδx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in Rd, and ξx 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 μλ-measures (suitably scaled and centred) of disjoint sets in Rd are asymptotically independent normals as λ tends to infinity; here we give an O(λ1/(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.