Kybernetika 50 no. 6, 896-913, 2014

Functionals of spatial point process having density with respect to the Poisson process

This article was granted Editor's award of the year 2014Editor's award 2014

Viktor Beneš and Markéta ZikmundováDOI: 10.14736/kyb-2014-6-0896

Abstract:

$U$-statistics of spatial point processes given by a density with respect to a Poisson process are investigated. In the first half of the paper general relations are derived for the moments of the functionals using kernels from the Wiener-Itô chaos expansion. In the second half we obtain more explicit results for a system of $U$-statistics of some parametric models in stochastic geometry. In the logarithmic form functionals are connected to Gibbs models. There is an inequality between moments of Poisson and non-Poisson functionals in this case, and we have a version of the central limit theorem in the Poisson case.

Keywords:

moments, difference of a functional, limit theorem, U-statistics

Classification:

60G55, 60D05

paper.pdf

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