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Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

Part of: Geometric probability and stochastic geometry Limit theorems

Published online by Cambridge University Press:  01 July 2016

Kasra Alishahi  and
Mohsen Sharifitabar
Kasra Alishahi*
Affiliation:
Sharif University of Technology
Mohsen Sharifitabar*
Affiliation:
Sharif University of Technology
*
∗ Postal address: Department of Mathematics, Sharif University of Technology, Tehran, Iran.
∗ Postal address: Department of Mathematics, Sharif University of Technology, Tehran, Iran.
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Abstract

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This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, …. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ−1(1 − e−λu). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

Keywords

Poisson-Voronoi tessellationtypical cellBlaschke-Petkantschin formulahigh dimensionchord length distributionlinear contact distributiongeometric covariogram

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60F25: $L^p$-limit theorems 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 919 - 938
Copyright
Copyright © Applied Probability Trust 2008 

References

Blaschke, W. (1935). Integralgeometrie 1. Ermittlung der Dichten für Linear Unterräume im En. Hermann, Paris.Google Scholar
Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
Hug, D. (2007). Random mosaics. In Stochastic Geometry, ed. Weil, W., Springer, Berlin, pp. 247–266.Google Scholar
Jensen, E. B. V. (1998). Local Stereology (Adv. Ser. Statist. Sci. Appl. Prob. 5). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Lantuejoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin.CrossRefGoogle Scholar
Mathai, A. M. (1999). An Introduction to Geometrical Probability. Gordon and Breach, Amsterdam.Google Scholar
Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.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, New York, pp. 202–227.Google Scholar
Miles, R. E. (1984). Sectional Voronoi tessellations. Revista de la Unión Matemática Argentina 29, 310–327.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
Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.CrossRefGoogle Scholar
Møller, J. (1987). A simple derivation of a formula of Blaschke and Petkantschin. Res. Rep. 138, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus.Google Scholar
Møller, J. (1989). Random tessellations in R n . Adv. Appl. Prob. 21, 37–73.CrossRefGoogle Scholar
Muche, L. and Stoyan, D. (1992). Contact and chord length distributions of the Poisson–Voronoi tessellation. J. Appl. Prob. 29, 467–471.CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
Petkantschin, B. (1936). Integralgeometrie 6. Zusammenhänge zwischen den Dichten der linearen Unterräume im n-dimensionalen Raum. Abh. Math. Seminar Hamburg 11, 249–310.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar