CENTRAL LIMIT THEOREMS FOR U-STATISTICS OF POISSON POINT PROCESSES

• 成果类型: Article
• 署名作者: Reitzner, Matthias; Schulte, Matthias
• 署名单位: University Osnabruck; Helmholtz Association; Karlsruhe Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP817
• 发表日期: 2013
• 页码: 3879-3909
• 关键词: gaussian fluctuations; wiener; functionals; cumulants

摘要

A U-statistic of a Poisson point process is defined as the sum Sigma f(x(1),...,x(k)) over all (possibly infinitely many) k-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-Ito chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for U-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.