Multivariate normal approximation in geometric probability

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 $d$-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 $\lambda \to \infty$; here we give an $O({\lambda }^{-1/(2d+\epsilon )})$ 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.

23 pages

References in corpus (5)

• Normal approximation under local dependence
• Gaussian limits for random measures in geometric probability
• Laws of large numbers in stochastic geometry with statistical applications
• Explicit laws of large numbers for random nearest-neighbour type graphs
• Limit theory for the random on-line nearest-neighbour graph

Cited by in corpus (2)

• Moderate deviations for stabilizing functionals in geometric probability
• Limit theorems for random spatial drainage networks