Multivariate normal approximation in geometric probability

Consider a measure μλ = Σx ξx δx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, 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( 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.