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Central limit theorems for $U$-statistics of Poisson point processes

By Matthias Reitzner and Matthias Schulte

Abstract

A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_1,\ldots,x_k)$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-It\^{o} 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.Comment: Published in at http://dx.doi.org/10.1214/12-AOP817 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Topics: Mathematics - Probability
Year: 2013
DOI identifier: 10.1214/12-AOP817
OAI identifier: oai:arXiv.org:1104.1039
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