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Stopping sets: Gamma-type results and hitting properties

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

Published online by Cambridge University Press:  01 July 2016

Sergei Zuyev
Sergei Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: Statistics and Modelling Science Dept, University of Strathclyde, 26 Richmond St, Livingston Tower, Glasgow G1 1XH, UK. Email address: sergei@stams.strath.ac.uk
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Abstract

Recently in the paper by Møller and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is Γ(n,λ) distributed, where λ is the intensity of the process. In this paper we give an alternative description of the class of random sets for which the Gamma-type results hold. We show that it corresponds to the class of stopping sets with respect to the natural filtration of the point process with certain scaling properties. The proof uses the martingale technique for directed processes, in particular, an analogue of Doob's optional sampling theorem proved in Kurtz (1980). As well as being compact, this approach provides a new insight into the nature of geometrical objects constructed with respect to a Poisson point process. We show, in particular, that in this framework the probability that a point is covered by a stopping set does not depend on whether it is a point of the process or not.

Keywords

Stopping setmartingalesdirected processesPalm distributionspoint processPoisson 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 31 , Issue 2 , June 1999 , pp. 355 - 366
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

The work was supported by CNET through the research grant CTI 1B 104.

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