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Contact and chord length distribution of a stationary Voronoi tessellation

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Lothar Heinrich
Lothar Heinrich*
Affiliation:
University of Augsburg
*
∗ Postal address: Institute of Mathematics, University of Augsburg, Universtätsstr. 14, D-86135 Augsburg, Germany. Email address: [email protected]
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Abstract

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

Keywords

Voronoi tessellationchord lengthcontact distributionmotion-invariant point processtwo-point Palm distributionPoisson cluster processGibbsian process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60G10: Stationary processes 60G60: Random fields
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 603 - 618
Copyright
Copyright © Applied Probability Trust 1998 

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References

Ambartzumian, R. B. (1975). Homogeneous and isotropic random point fields in the plane (in Russian). Math. Nachr. 70, 365–385.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
Hahn, U. and Lorz, U. (1994). Stereological analysis of the spatial Poisson–Voronoi tessellation. J. Microscopy 175, 176–185.Google Scholar
Hanisch, K.-H. (1982). On inversion formulae for n-fold Palm distributions of point processes in LCS-spaces. Math. Nachr. 106, 171–179.CrossRefGoogle Scholar
Hansen, M. B., Gill, R. D. and Baddeley, A. (1994). Some regularity properties for first contact distributions. Unpublished.Google Scholar
Heinrich, L. (1988). Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary Poisson cluster processes. Math. Nachr. 136, 131–148.Google Scholar
Heinrich, L. (1993). Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 31, 349–360.Google Scholar
Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Meth. Statist. 3, 1–24.Google Scholar
Heinrich, L. and Muche, L. (1994). On the pair correlation function of the point process of nodes in a Voronoi tessellation. Preprint 94-07. TU Bergakademie Freiberg, Fak. für Mathematik und Informatik, pp. 1–30.Google Scholar
Heinrich, L. and Schüle, E. (1995). Generation of the typical cell of a non-Poissonian Johnson–Mehl tessellation. Commun. Statist.–Stoch. Models 11, 541–560.Google Scholar
Heinrich, L. and Stoyan, D. (1984). On generalized Palm–Khinchin formulae (in Russian). Izvest. Acad. Sci. Armenian SSR, Ser. Mat. 19, 280–287.Google Scholar
Kallenberg, O. (1988). Random Measures. Academic Press, London.Google Scholar
Last, G. and Schassberger, R. (1996). On the spherical contact distribution of stationary random set. Preprint 96-06, TU Braunschweig, Institut für Mathematik, pp. 1–18.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson process. Math. Biosci. 6, 85–127.CrossRefGoogle 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 (Special Volume J. Appl. Prob. 19A), ed. Gani, J. and Hannan, E. J.. Applied Probability Trust, Sheffield, pp. 97–111.Google Scholar
Møller, J., (1989). Random tessellations in Rd . Adv. Appl. Prob. 21, 37–73.Google Scholar
Møller, J., (1994). Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics 87, Springer, New York.Google 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. and Sugihara, K. (1992). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. Akademie-Verlag, Berlin.Google Scholar