Rates of multivariate normal approximation for statistics in geometric probability

TL;DR

This paper establishes explicit rates of multivariate normal approximation for a broad class of statistics derived from marked Poisson processes, using stabilization methods and second order Poincaré inequalities, with applications in geometric probability, random graphs, and topological data analysis.

Contribution

It introduces new rates of convergence for multivariate normal approximation of stabilizing functionals of Poisson processes, applicable in various geometric and topological contexts.

Findings

Rates of convergence are of order s^{-1/d} for the covariance.

Results apply to statistics in random graphs and topological data analysis.

Rates are generally unimprovable and governed by covariance convergence.

Abstract

We employ stabilization methods and second order Poincar\'e inequalities to establish rates of multivariate normal convergence for a large class of vectors $(H_s^{(1)},...,H_s^{(m)})$, $s \geq 1$, of statistics of marked Poisson processes on $\mathbb{R}^d$, $d \geq 2$, as the intensity parameter $s$ tends to infinity. Our results are applicable whenever the constituent functionals $H_s^{(i)}$, $i\in\{1,...,m\}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the $d_2$-, $d_3$-, and $d_{convex}$-distances. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are in general unimprovable and are governed by the rate of convergence of $s^{-1} {\rm Cov}( H_s^{(i)}, H_s^{(j)})$, $i,j\in\{1,...,m\}$, to the limiting covariance, shown to be of order $s^{-1/d}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.