QUANTITATIVE TWO-SCALE STABILIZATION ON THE POISSON SPACE

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 (Ann. Appl. Probab. 11 (2001) 1005-1041) and Penrose (Ann. Probab. 33 (2005) 1945-1991). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (Ann. Appl. Probab. 27 (2017) 1588-1645) 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.