Normal approximation on Poisson spaces: Mehler's formula, second order Poincar, inequalities and stabilization

• 成果类型: Article
• 署名作者: Last, Guenter; Peccati, Giovanni; Schulte, Matthias
• 署名单位: Helmholtz Association; Karlsruhe Institute of Technology; University of Luxembourg
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0643-7
• 发表日期: 2016
• 页码: 667-723
• 关键词: CENTRAL-LIMIT-THEOREM; minimal spanning-trees; gaussian fluctuations; geometric probability; U-statistics; functionals; tessellations; percolation

摘要

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincar, inequality, as well as on the use of Malliavin operators, of Stein's method, and of an (integrated) Mehler's formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be optimal, thus significantly improving previous findings in the literature. As a necessary step in our analysis, we also derive new lower bounds for variances of Poisson functionals.