## The Annals of Applied Probability

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 16 December 2014

https://projecteuclid.org/euclid.aoap/1418740181

Digital Object Identifier
doi:10.1214/13-AAP990

Mathematical Reviews number (MathSciNet)
MR3297768

Zentralblatt MATH identifier
1358.60022

#### Citation

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. https://projecteuclid.org/euclid.aoap/1418740181

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