Rates of multivariate normal approximation for statistics in geometric probability

Abstract

We employ stabilization methods and second order Poincaré inequalities to establish rates of multivariate normal convergence for a large class of vectors $({\mathit{H}}_{\mathit{s}}^{(1)},\dots ,{\mathit{H}}_{\mathit{s}}^{(\mathit{m})})$, $\mathit{s}\ge 1$, of statistics of marked Poisson processes on ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 2$, as the intensity parameter s tends to infinity. Our results are applicable whenever the functionals ${\mathit{H}}_{\mathit{s}}^{(\mathit{i})}$, $\mathit{i}\in \{1,\dots ,\mathit{m}\}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the ${\mathit{d}}_2$-, ${\mathit{d}}_3$-, and ${\mathit{d}}_{\mathrm{convex}}$-distances and are in general unimprovable. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are governed by the rate of convergence of ${\mathit{s}}^{-1}Cov({\mathit{H}}_{\mathit{s}}^{(\mathit{i})},{\mathit{H}}_{\mathit{s}}^{(\mathit{j})})$, $\mathit{i},\mathit{j}\in \{1,\dots ,\mathit{m}\}$, to the limiting covariance, shown to be at most of order ${\mathit{s}}^{-1/\mathit{d}}$. 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.