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

TL;DR

This paper investigates U-statistics of spatial point processes with densities relative to Poisson processes, deriving moment relations and explicit results for models in stochastic geometry, including connections to Gibbs models and a CLT.

Contribution

It introduces new moment relations for U-statistics of spatial point processes and applies them to parametric models, linking to Gibbs models and establishing a CLT for Poisson cases.

Findings

Derived general relations for moments of functionals using Wiener-Ito chaos kernels.

Obtained explicit results for parametric models in stochastic geometry.

Established a version of the central limit theorem for Poisson functionals.

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-Ito 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 logaritmic 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.