Central limit theorems for U-statistics of Poisson point processes

A U-statistic of a Poisson point process is defined as the sum ∑ f(x₁,…,xₖ) 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.