QUANTITATIVE TWO-SCALE STABILIZATION ON THE POISSON SPACE

• 成果类型: Article
• 署名作者: Lachieze-Rey, Raphael; Peccati, Giovanni; Yang, Xiaochuan
• 署名单位: Universite Paris Cite; University of Luxembourg; Brunel University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1768
• 发表日期: 2022
• 页码: 3085-3145
• 关键词: minimal spanning-trees; central limit-theorems; random sequential packing; berry-esseen bounds; normal approximation; gaussian limits; excursion sets; stein method; functionals; percolation

摘要

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.