57 research outputs found
Limit theory for the Gilbert graph
For a given homogeneous Poisson point process in 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 with kernel f:\X^k \to \R^d over a Poisson process
is defined in \cite{ReiSch11} as under appropriate integrability
assumptions on . 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
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