Multivariate normal approximation in geometric probability

TL;DR

This paper establishes a quantitative multivariate normal approximation for measures derived from Poisson point processes with stabilization properties, providing explicit convergence rates and applications to geometric graph models.

Contribution

It offers a new rate of convergence bound for multivariate normal approximation in geometric probability models with stabilization.

Findings

Derived an explicit rate of convergence of O(λ^{-1/(2d + ε)})

Proved asymptotic independence of measures for disjoint sets

Applied results to nearest-neighbour graph on Poisson points

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.