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Quermass-interaction processes: conditions for stability

Part of: Stochastic processes Inference from stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

W. S. Kendall ,
M. N. M. van Lieshout  and
A. J. Baddeley
W. S. Kendall*
Affiliation:
University of Warwick
M. N. M. van Lieshout*
Affiliation:
CWI
A. J. Baddeley*
Affiliation:
University of Western Australia
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Email address: [email protected]
∗∗ Postal address: Centre for Mathematics and Computer Science, PO Box 94079, 1090 GB, Amsterdam, The Netherlands.
∗∗∗ Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia.
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Abstract

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

Keywords

Area-interaction point processBoolean modelgerm-grain modelMarkov point processMinkowski functionalquermass integralsemi-Markov random closed setspatial point process

MSC classification

Primary: 62M30: Spatial processes 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 315 - 342
Copyright
Copyright © Applied Probability Trust 1999 

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