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On spatial thinning-replacement processes based on Voronoi cells

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

K. A. Borovkov  and
D. A. Odell
K. A. Borovkov*
Affiliation:
The University of Melbourne
D. A. Odell*
Affiliation:
MASCOS
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address: [email protected]
∗∗ Postal address: MASCOS, 139 Barry Street, Carlton, VIC 3010, Australia.
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Abstract

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We introduce a new class of spatial-temporal point processes based on Voronoi tessellations. At each step of such a process, a point is chosen at random according to a distribution determined by the associated Voronoi cells. The point is then removed, and a new random point is added to the configuration. The dynamics are simple and intuitive and could be applied to modelling natural phenomena. We prove ergodicity of these processes under wide conditions.

Keywords

Point processVoronoi tessellationMarkov chainergodicity

MSC classification

Primary: 60G55: Point processes
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60F99: None of the above, but in this section
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 293 - 306
Copyright
Copyright © Applied Probability Trust 2007 

References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Baddeley, A. J. and van Lieshout, M. N. M. (1996). A nonparametric measure of spatial interaction in point patterns. Statist. Neerlandica 50, 344–361.Google Scholar
Borovkov, K. A. and Odell, D. A. (2007). Simulation studies of some Voronoi point processes. Acta Appl. Math. 96, 87–97.CrossRefGoogle Scholar
Du, Q., Faber, V. and Gunzburger, M. (1999). Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41, 637–676 CrossRefGoogle Scholar
Hasegawa, M. and Tanemura, M. (1976). On the patterns of space division by territories. Ann. Inst. Statist. Math. 28, 509–519.CrossRefGoogle Scholar
Hotelling, H. (1929). Stability in competition. Econom. J. 39, 41–57.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.CrossRefGoogle Scholar
Nagel, K., Shubik, M., Paczuski, M. and Bak, P. (2000). Spatial competition and price formation. Physica A 287, 546–562.CrossRefGoogle Scholar
Okabe, A. and Suzuki, A. (1987). Stability of spatial competition for a large number of firms on a bounded two-dimensional space. Environ. Planning A 19, 1067–1082.CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley, Chichester.CrossRefGoogle Scholar
Ripley, B. D. (1977). Modelling spatial patterns. J. R. Statist. Soc. B 39, 172–212.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
Thönnes, E. and van Lieshout, M. N. M. (1995). A comparative study of the power of van Lieshout and Baddeley's J-function. Biom. J. 47, 721–734.Google Scholar
Zuyev, S. and Tchoumatchenko, K. (2001). Aggregate and fractal tessellations. Prob. Theory Relat. Fields 121, 198–218.Google Scholar