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On a Voronoi aggregative process related to a bivariate Poisson process

Part of: Systems theory and control: General Discrete geometry Geometric probability and stochastic 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: foss@math.nsk.su
∗∗ Postal address: INRIA, 2004, R-te des Lucioles—B.P. 93-06902, Sophia-Antipolis Cedex, France. e-mail: sergei@sophia.inria.fr
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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.

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