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A more comprehensive complementary theorem for the analysis of Poisson point processes

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Richard Cowan
Richard Cowan*
Affiliation:
University of Sydney
*
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: rcowan@mail.usyd.edu.au
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Abstract

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In this paper we discuss the complementary theorem applied to the typical n-tuple of a Poisson point process. The theorem was first presented by Miles in 1970 and discussed by Santaló in 1976 and, within a Palm measure framework, by Møller and Zuyev in 1996. The theorems put forward by these authors are not correct for all the examples that they present, suggesting that further consideration of their work is needed if one wishes to bring all those examples within the ambit of the complementary theorem. We give alternative analyses of the errant examples and, with a modification of the technicalities in the work of the above authors, move toward a more comprehensive complementary theorem. Some open issues still remain.

Keywords

Random geometryPoisson point processPoisson flat processcomplementary theoremgamma distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 581 - 601
Copyright
Copyright © Applied Probability Trust 2006 

References

Cowan, R., Quine, M. and Zuyev, S. (2003). Decomposition of Gamma-distributed domains constructed from Poisson point processes. Adv. Appl. Prob. 35, 56–69.Google Scholar
Hall, G. R. (1982). Acute triangles in the n-ball. J. Appl. Prob. 19, 712–715.Google Scholar
Kallenberg, O. (1976). Random Measures. Springer, Berlin.Google Scholar
Mecke, J. (1967). Stationäre zufällige Mass e auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson process. Math. Biosci. 6, 85–127.CrossRefGoogle Scholar
Miles, R. E. (1971). Poisson flats in Euclidean spaces. II. Homogeneous Poisson flats and the complementary theorem. Adv. Appl. Prob. 3, 1–43.CrossRefGoogle Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Møller, J. and Zuyev, S. (1996). Gamma-type results and other related properties of Poisson processes. Adv. Appl. Prob. 28, 662–673.Google Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability (Encyclopaedia Math. Appl. 1). Addison-Wesley, Reading, MA.Google Scholar
Slivnyak, I. M. (1962). Some properties of stationary flows of homogeneous random events. Theory Prob. Appl. 7, 336–341.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Zuyev, S. (1999). Stopping sets: gamma-type results and hitting properties. Adv. Appl. Prob. 31, 355–366.Google Scholar