Regularity conditions in the realisability problem with applications to point processes and random closed sets

Raphael Lachieze-Rey; Ilya Molchanov

doi:10.1214/13-AAP990

February 2015 Regularity conditions in the realisability problem with applications to point processes and random closed sets

Raphael Lachieze-Rey, Ilya Molchanov

Ann. Appl. Probab. 25(1): 116-149 (February 2015). DOI: 10.1214/13-AAP990

Abstract

We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive extension for linear functionals accompanied by additional conditions that ensure the regularity of the extension needed for interpreting it as a probability measure. It is shown in which case the extension can be chosen to possess some invariance properties.

The results are applied to the existence of point processes with given correlation measure and random closed sets with given two-point covering function or contact distribution function. It is shown that the regularity condition can be efficiently checked in many cases in order to ensure that the obtained point processes are indeed locally finite and random sets have closed realisations.

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Raphael Lachieze-Rey and Ilya Molchanov "Regularity conditions in the realisability problem with applications to point processes and random closed sets," The Annals of Applied Probability 25(1), 116-149, (February 2015). https://doi.org/10.1214/13-AAP990

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