CENTRAL LIMIT THEOREMS FOR U-STATISTICS OF POISSON POINT PROCESSES

Abstract

A U-statistic of a Poisson point process is defined as the sum Sigma f(x(1),...,x(k)) over all (possibly infinitely many) k-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-Ito 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.