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arXiv · 1104.1039

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

Matthias Reitzner·Matthias Schulte
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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-Itô 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.

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BibTeXRIS

Matthias Reitzner, Matthias Schulte. 2013-12-12. Central limit theorems for $U$-statistics of Poisson point processes. https://doi.org/10.1214/12-aop817

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