U-Statistics in Stochastic Geometry

Abstract

A U-statistic of order k with kernel $f: \mathbb{X}^{k} \rightarrow \mathbb{R}^{d}$ over a Poisson process η is defined as

$$\displaystyle{\sum _{(x_{1},\ldots,x_{k})}f(x_{1},\ldots,x_{k}),}$$

where the summation is over k-tuples of distinct points of η, under appropriate integrability assumptions on f. U-statistics play an important role in stochastic geometry since many interesting functionals can be written as U-statistics, like intrinsic volumes of intersection processes, characteristics of random geometric graphs, volumes of random simplices, and many others. It turns out that the Wiener–Ito chaos expansion of a U-statistic is finite and thus Malliavin calculus is a particularly suitable method. Variance estimates, approximation of the covariance structure, and limit theorems which have been out of reach for many years can be derived. In this chapter we state the fundamental properties of U-statistics and investigate moment formulae. The main object of the chapter is to introduce the available limit theorems.