2015/02/28 by Raphaël Lachèze-Rey, Lachèze-Rey, Raphaël, Matthias Reitzner +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #Topological and Geometric Data Analysis
paper · doi:10.48550/arxiv.1503.00110
openalex publication_date 2015/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This survey will appear as a chapter of the forthcoming book [19]. A U-statistic of order k with kernel f:\Xk → \Rd over a Poisson process is defined in \citeReiSch11 as∑_x_1, … , x_k ∈ ηk_≠ f(x_1, …, 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.