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

TL;DR

This paper derives central limit theorems for U-statistics of Poisson point processes using Malliavin calculus, providing explicit bounds and applications to geometric processes like hyperplanes and graphs.

Contribution

It introduces a novel approach to CLTs for Poisson U-statistics via Malliavin calculus and chaos expansion, with explicit error bounds and practical applications.

Findings

Established CLTs with explicit Wasserstein bounds for Poisson U-statistics.

Derived formulas for variance using Wiener-Itô chaos expansion.

Applied results to Poisson hyperplane intersections and random geometric graphs.

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-It\^{o} 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.