Geodesic
Functionals of spatial point processes having a density with respect to the Poisson process
Kybernetika, Tome 50 (2014) no. 6, pp. 896-913
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$U$-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.
$U$-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.
DOI : 10.14736/kyb-2014-6-0896
Classification : 60D05, 60F05, 60G55
Keywords: difference of a functional; limit theorem; moments; U-statistics 
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Beneš, Viktor; Zikmundová, Markéta. Functionals of spatial point processes having a density with respect to the Poisson process. Kybernetika, Tome 50 (2014) no. 6, pp. 896-913. doi: 10.14736/kyb-2014-6-0896

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