Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using a concentration inequality for the convex distance

Elementary moves on triangulations

It is proved that a triangulation of a polyhedron can always be transformed into any other triangulation of the polyhedron by using only elementary moves. One consequence is that an additive function (valuation) defined only on simplices may always be extended to an additive function on all polyhedra.Comment: 10 pages; some corrections, extended proofs of Lemma 4 and Corollary

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.Comment: 22p