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Functionals of spatial point process having density with respect to the Poisson process. (English) Zbl 1316.60069
Summary: \(U\)-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-Itō 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.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
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