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.
Citation
Matthias Reitzner. Matthias Schulte. "Central limit theorems for $U$-statistics of Poisson point processes." Ann. Probab. 41 (6) 3879 - 3909, November 2013. https://doi.org/10.1214/12-AOP817
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