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On a Voronoi aggregative process related to a bivariate Poisson process

Part of: Geometric probability and stochastic geometry Systems theory and control: General Discrete geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

S. G. Foss  and
S. A. Zuyev
S. G. Foss*
Affiliation:
Novosibirsk State University
S. A. Zuyev*
Affiliation:
INRIA, Sophia-Antipolis
*
∗ Postal address: Department of Mathematics, Novosibirsk State University, 630 090, Russia. e-mail: [email protected]
∗∗ Postal address: INRIA, 2004, R-te des Lucioles—B.P. 93-06902, Sophia-Antipolis Cedex, France. e-mail: [email protected]
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Abstract

We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.

Keywords

BIVARIATE POISSON PROCESSAGGREGATIVE PROCESSVORONOI TESSELLATIONPALM DISTRIBUTIONLARGE DEVIATIONSMODELING OF NETWORKS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 52C15: Packing and covering in $2$ dimensions
Secondary: 93A13: Hierarchical systems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 965 - 981
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

This work was completed while the authors were visiting INRIA, Sophia-Antipolis, France.

References

Baccelli, F., Klein, M., Lebourges, M. and Zuyev, S. (1996) Géométrie aléatoire et architecture de réseaux de télécommunications. Ann. Télécomm. 51, 158–179.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Dembo, A. and Zeitouni, O. (1993) Large Deviations Techniques and Applications. Jones and Barlett, Boston.Google Scholar
Foss, S. G. and Zuyev, S. A. (1993) On a certain segment process with Voronoi clustering. Rapport de Recherche No. 1993, INRIA. Google Scholar
Gilbert, E. N. (1962) Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
Kingman, J. F. C. (1993) Poisson Processes. Oxford University Press, Oxford.Google Scholar
Sallai, G. (1988) Optimal network structure with randomly distributed nodes. In 12th Int. Teletraffic Congress, Torino, June 1988. Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar
Zuyev, S. (1992) Estimates for distributions of the Voronoi polygon's geometric characteristics. Random Structures Algorithms 3, 149–162.Google Scholar