Quantitative two-scale stabilization on the Poisson space

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 stabilizationsee 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 neighbour 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.

Variance asymptotics and central limit theory for geometric functionals of Poisson cylinder processes

“…Originally introduced in [30] and further developed in many subsequent works it has turned out to be a versatile device with a vast of potential applications. As concrete examples we mention the works [8,20,21,22,23,31,32,37] on various models for geometric random graphs, the paper [7] dealing with geometric random simplicial complexes, the application to the classical Boolean model [17], the works [8,16,25,31] dealing with Poisson hyperplane tessellations in Euclidean and non-Euclidean spaces, the applications in [22,36] to Poisson-Voronoi tessellations, the works on excursion sets of Poisson shot-noise processes [19,21] as well as the papers [4,5,22,40,41,42] considering different models for random polytopes. For an illustrative overview on the Malliavin-Stein method for functionals of Poisson processes we refer to the collection of surveys in [29].…”