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Areas of components of a Voronoi polygon in a homogeneous Poisson process in the plane

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

A. Hayen  and
M. P. Quine
A. Hayen*
Affiliation:
University of Sydney
M. P. Quine*
Affiliation:
University of Sydney
*
∗ 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.
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Abstract

We study the contribution made by three or four points to certain areas associated with a typical polygon in a Voronoi tessellation of a planar Poisson process. We obtain some new results about moments and distributions and give simple proofs of some known results. We also use Robbins' formula to obtain the first three moments of the area of a typical polygon and hence the variance of the area of the polygon covering the origin.

Keywords

PoissonVoronoitessellationdistribution functionmoments

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 2 , June 2002 , pp. 281 - 291
Copyright
Copyright © Applied Probability Trust 2002 

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References

Berntsen, J., Espelid, T. O. and Genz, A. (1991a). An adaptive algorithm for the approximate calculation of multiple integrals. Assoc. Comput. Mach. Trans. Math. Software 17, 437–451.Google Scholar
Berntsen, J., Espelid, T. O. and Genz, A. (1991b). Algorithm 698: DCUHRE: an adaptive multidimensional integration routine for a vector of integrals. Assoc. Comput. Mach. Trans. Math. Software 17, 452–456.Google Scholar
Feller, W. (1971). Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.Google Scholar
Hayen, A. and Quine, M. P. (2000a). The proportion of triangles in a Poisson–Voronoi tessellation of the plane. Adv. Appl. Prob. 32, 67–74.Google Scholar
Hayen, A. and Quine, M. P. (2000b). Calculating the proportion of triangles in a Poisson–Voronoi tessellation of the plane. J. Statist. Comput. Simul. 67, 351–358.CrossRefGoogle Scholar
Kendall, M. G. and Moran, P. A. P. (1963). Geometrical Probability. Griffin, London.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson process. Math. Biosci. 6, 85–127.Google Scholar
Miles, R. E. (1974). A synopsis of Poisson flats in Euclidean spaces. In Stochastic Geometry, eds Harding, E. F. and Kendall, D. G., John Wiley, London, pp. 202–227.Google Scholar
Miles, R. E. and Maillardet, R. J. (1982). The basic structures of Voronoi and generalized Voronoi polygons. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 97–111.Google Scholar
Quine, M. P. and Watson, D. F. (1984). Radial generation of n-dimensional Poisson processes. J. Appl. Prob. 21, 548–557.CrossRefGoogle Scholar
Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes and Point Fields. John Wiley, Chichester.Google Scholar