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On variances of partial volumes of the typical cell of a Poisson-Voronoi tessellation and large-dimensional volume degeneracy

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Yi-Ching Yao
Yi-Ching Yao*
Affiliation:
Academia Sinica and National Chengchi University
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, R.O.C. Email address: [email protected]
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Abstract

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For a typical cell of a homogeneous Poisson-Voronoi tessellation in ℝd, it is shown that the variance of the volume of the intersection of the typical cell with any measurable subset of ℝd is bounded by the variance of the volume of the typical cell. It is also shown that the variance of the volume of the intersection of the typical cell with a translation of itself is bounded by four times the variance of the volume of the typical cell. These bounds are applied to show large-dimensional volume degeneracy as d tends to ∞. An extension to the kth nearest-point Poisson-Voronoi tessellation for k ≥ 2 is also considered.

Keywords

Poisson processVoronoi tessellationtypical cellgeometric covariogramnearest neighbor

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 42 , Issue 2 , June 2010 , pp. 359 - 370
Copyright
Copyright © Applied Probability Trust 2010 

References

Alishahi, K. and Sharifitabar, M. (2008). Volume degeneracy of the typical cell and the chord length distribution for Poisson–Voronoi tessellations in high dimensions. Adv. App. Prob. 40, 919–938.CrossRefGoogle Scholar
Averkov, G. and Bianchi, G. (2007). Retrieving convex bodies from restricted covariogram functions. Adv. App. Prob. 39, 613–629.Google Scholar
Bianchi, G. (2005). Matheron's conjecture for the covariogram problem. J. Lond. Math. Soc. 71, 203–220.Google Scholar
Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.Google Scholar
Hinde, A. L. and Miles, R. E. (1980). Monte Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Statist. Comput. Simul. 10, 205–223.Google Scholar
Jensen, E. B. V. (1998). Local Stereology (Adv. Ser. Statist. Sci. Appl. Prob. 5). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
Newman, C. M. and Rinott, Y. (1985). Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions. Adv. App. Prob. 17, 794–809.CrossRefGoogle Scholar
Newman, C. M., Rinott, Y. and Tversky, A. (1983). Nearest neighbors and Voronoi regions in certain point processes. Adv. App. Prob. 15, 726–751.Google 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
Yao, Y.-C. and Simons, G. (1996). A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes. Ann. Appl. Prob. 6, 561–571.Google Scholar