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

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

2011-02-09

arxiv.org/abs/1102.1950v2

Mathematics

Probability

Submitted version

Scientific paper

The paper addresses the existence issue for a rather general random element whose distribution is only partially specified. 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 obtain existence results for 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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