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Decomposition of gamma-distributed domains constructed from Poisson point processes

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

Published online by Cambridge University Press:  01 July 2016

Richard Cowan ,
Malcolm Quine  and
Sergei Zuyev
Richard Cowan*
Affiliation:
University of Sydney
Malcolm Quine*
Affiliation:
University of Sydney
Sergei Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.
∗∗ Postal address: Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, UK. Email address: [email protected]
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Abstract

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

Keywords

Stopping setgamma-type resultsSlivnyak theoremset-indexed martingaleVoronoi tessellationDelaunay triangulationPoisson process

MSC classification

Primary: 60G48: Generalizations of martingales 60D05: Geometric probability and stochastic geometry
Secondary: 62M30: Spatial processes 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 56 - 69
Copyright
Copyright © Applied Probability Trust 2003 

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