Quantitative two-scale stabilization on the Poisson space

TL;DR

This paper develops new inequalities to measure how close the distribution of certain Poisson space functionals is to a Gaussian distribution, with applications to geometric and spatial functionals, enhancing understanding of their Gaussian fluctuations.

Contribution

It introduces a two-scale stabilization approach on the Poisson space, extending recent CLT estimates and providing detailed applications to geometric functionals.

Findings

Established bounds for Poisson functionals' Gaussian approximation

Applied bounds to spatial CLTs for geometric and shot noise functionals

Extended CLT estimates to multidimensional settings

Abstract

We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one - that we evaluate and compare at two different scales - and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization - see Penrose and Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (2017) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbor graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.