# Multivariate normal approximation in geometric probability

Research output: Contribution to journalArticle (Academic Journal)peer-review

7 Citations (Scopus)

## 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.
Translated title of the contribution Multivariate normal approximation in geometric probability English 293 - 326 34 Journal of Statistical Theory and Practice 2 (2) Published - Jun 2008

### Bibliographical note

Publisher: University of North Carolina Press
Other: http://journalstp.gracescientific.com/Volume2Number2.aspx