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

TL;DR

This paper investigates conditions for the existence of random elements with specified distributions, focusing on regularity conditions that ensure the resulting point processes and random closed sets are well-defined and possess desired properties.

Contribution

It introduces a framework for verifying regularity conditions in the realizability problem, enabling the construction of point processes and random closed sets with prescribed characteristics.

Findings

Regularity conditions can be efficiently checked for realizability.

The approach ensures the constructed point processes are locally finite.

Random closed sets can be realized with closed realizations under these conditions.

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.