Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-31T04:33:11.459Z Has data issue: false hasContentIssue false
  • English
  • Français

On connected component Markov point processes

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Y. C. Chin  and
A. J. Baddeley
Y. C. Chin*
Affiliation:
The University of Western Australia
A. J. Baddeley*
Affiliation:
The University of Western Australia
*
∗ Postal address: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia.
∗ Postal address: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We note some interesting properties of the class of point processes which are Markov with respect to the ‘connected component’ relation. Results in the literature imply that this class is closed under random translation and independent cluster generation with almost surely non-empty clusters. We further prove that it is closed under superposition. A wide range of examples is also given.

Keywords

Cluster processescontinuum random cluster modelsuperposition of point processesnearest neighbour Markov point processes

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 279 - 282
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Baddeley, A. and Møller, J. (1989). Nearest neighbour Markov point processes and random sets. Internat. Statist. Rev. 57, 89–121.Google Scholar
Baddeley, A., Van Lieshout, M. N. M. and Møller, J. (1996). Markov properties of cluster processes. Adv. Appl. Prob. 28, 346–355.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Geyer, C. J. and Møller, J. (1994). Simulation and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
Häggström, O. and Van Lieshout, M. N. M. and Møller, J. (1999). Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point process. To appear in Bernoulli.Google Scholar
Møller, J. (1999). Markov chain {Monte Carlo and spatial point processes. In Stochastic Geometry: Likelihood and Computation, eds. Kendall, W. S., Van Lieshout, M. N. M. and Barndorff-Nielsen, O. E., Chapter 4, pp. 141–172. Chapman and Hall, London.Google Scholar
Ripley, B. D. and Kelly, F. P. (1977). Markov {p}oint {p}rocesses. J. Lond. Math. Soc. 15, 188–192.Google Scholar