Central Limit Theorems for U-Statistics of Poisson Point Processes

Details Central Limit Theorems for U-Statistics of Poisson Point Processes Central Limit Theorems for U-Statistics of Poisson Point Processes

2011-04-06

arxiv.org/abs/1104.1039v2

Mathematics

Probability

Scientific paper

A U-statistic of a Poisson point process is defined as the sum $\sum f(x_1,..., x_k)$ over all $k$-tuples of distinct points of the point process. Central limit theorems for U-statistics of Poisson point processes are shown. The (finite) Wiener-It\^o chaos expansion of such a functional is computed and used to derive a formula for the variance. Bounds for the Wasserstein distance of such functionals to a Gaussian random variable are proved. In order to estimate this bound, it is necessary to compute products of multiple Wiener-It\^o integrals. As examples, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.

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