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Markov properties of cluster processes

Part of: Geometric probability and stochastic geometry Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

A. J. Baddeley ,
M. N. M. Van Lieshout  and
J. Møller
A. J. Baddeley*
Affiliation:
University of Western Australia and University of Leiden
M. N. M. Van Lieshout*
Affiliation:
University of Warwick
J. Møller*
Affiliation:
University of Aarhus
*
∗ Postal address: Department of Mathematics, University of Western Australia, Nedlands WA 6009, Australia.
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, U.K.
∗∗∗ Postal address: Department of Mathematics and Computer Science, University of Aalborg, F. Bajers Vej 7E, DK-9220 Aalborg Ø, Denmark.
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Abstract

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

Keywords

MARKOV POINT PROCESSNEAREST-NEIGHBOUR MARKOV PROCESSCLUSTER PROCESSCONNECTED COMPONENT RELATION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 62M30: Spatial processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 346 - 355
Copyright
Copyright © Applied Probability Trust 1996 

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References

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