Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: U-statistics in stochastic geometry   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.

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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc