Multivariate normal approximation in geometric probability

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 $d$-space, 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 $R^d$ are asymptotically independent normals as $\lambda \to \infty$;herewegivean$O(\lambda^{-1/(2d + \epsilon)})$ 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.