U-statistics in stochastic geometry

TL;DR

This survey explores the role of U-statistics in stochastic geometry, highlighting their properties, moment formulas, and limit theorems, with a focus on their applications in geometric functionals and the use of Malliavin calculus.

Contribution

It provides a comprehensive overview of U-statistics in stochastic geometry, including their fundamental properties, moment formulas, and recent limit theorems, emphasizing the use of Malliavin calculus.

Findings

Finite Wiener-Ito chaos expansion of U-statistics

Derivation of variance estimates and covariance approximations

Presentation of limit theorems for geometric functionals

Abstract

This survey will appear as a chapter of the forthcoming book [19]. A U-statistic of order $k$ with kernel $f:\X^k \to \R^d$ over a Poisson process is defined in \cite{ReiSch11} as$$ \sum\_{x\_1, \dots , x\_k \in \eta^k\_{\neq}} f(x\_1, \dots, x\_k) $$ 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, see for instance \cite{ LacPec13, LPST,ReiSch11}. 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, the 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.