Annals of Applied Probability

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

Raphael Lachieze-Rey and Ilya Molchanov

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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.

Article information

Ann. Appl. Probab., Volume 25, Number 1 (2015), 116-149.

First available in Project Euclid: 16 December 2014

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42] 47B65: Positive operators and order-bounded operators 60G55: Point processes 74A40: Random materials and composite materials 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Point process correlation measure random closed set two-point covering probability contact distribution function realisability


Lachieze-Rey, Raphael; Molchanov, Ilya. Regularity conditions in the realisability problem with applications to point processes and random closed sets. Ann. Appl. Probab. 25 (2015), no. 1, 116--149. doi:10.1214/13-AAP990.

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