RATES OF MULTIVARIATE NORMAL APPROXIMATION FOR STATISTICS IN GEOMETRIC PROBABILITY

• 成果类型: Article
• 署名作者: Schulte, Matthias; Yukich, J. E.
• 署名单位: Hamburg University of Technology; Lehigh University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1822
• 发表日期: 2023
• 页码: 507-548
• 关键词: CENTRAL LIMIT-THEOREMS; gaussian limits; steins method; functionals; CONVERGENCE; graphs

摘要

We employ stabilization methods and second order Poincare inequalities to establish rates of multivariate normal convergence for a large class of vec-tors (H(1) s ,..., H(m) s ), s >= 1, of statistics of marked Poisson processes on Rd, d >= 2, as the intensity parameters tends to infinity. Our results are appli-cable whenever the functionals H(i) s , i is an element of {1, ... , m}, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the d2-, d3-, and dconvex-distances and are in general unim-provable. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are gov-erned by the rate of convergence of s-1 Cov(H(i) s , H(j) s ), i, j is an element of {1, ... , m}, to the limiting covariance, shown to be at most of order s-1/d. We use the general results to deduce rates of multivariate normal convergence for statis-tics arising in random graphs and topological data analysis as well as for mul-tivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.