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Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Werner Nagel  and
Viola Weiss
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
Viola Weiss*
Affiliation:
Fachhochschule Jena
*
∗ Postal address: Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, D-07737 Jena, Germany. Email address: nagel@minet.uni-jena.de
∗∗ Postal address: Fachhochschule Jena, D-07703 Jena, Germany.
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Abstract

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Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space with the following stability property: their distribution is invariant with respect to the operation of iteration (or nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary tessellations is fully characterized.

Keywords

Stochastic geometryrandom tessellationiteration of tessellationsnesting of tessellationsweak convergencestability of distributions

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 37 , Issue 4 , December 2005 , pp. 859 - 883
Copyright
Copyright © Applied Probability Trust 2005 

References

Ambartzumian, R. V. (1990). Factorization Calculus and Geometric Probability. Cambridge University Press.CrossRefGoogle Scholar
Arak, T., Clifford, P. and Surgailis, D. (1993). Point-based polygonal models for random graphs. Adv. Appl. Prob. 25, 348–372.CrossRefGoogle Scholar
Cowan, R. (1984). A collection of problems in random geometry. In Stochastic Geometry, Geometric Statistics, Stereology, eds Ambartzumjan, R. V. and Weil, W., Teubner, Leipzig, pp. 64–68.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Fisz, M. (1989). Wahrscheinlichkeitsrechnung und Mathematische Statistik, 11th edn. Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
Gould, R. (1988). Graph Theory. The Benjamin/Cummings Publishing Company, Menlo Park, CA.Google Scholar
Gross, J. L. and Yellen, J. (eds) (2004). Handbook of Graph Theory. CRC, Boca Raton, FL.Google Scholar
Mathéron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Miles, R. E. and Mackisack, M. S. (2002). A large class of random tessellations with the classic Poisson polygon distributions. Forma 17, 1–17.Google Scholar
Nagel, W. and Weiss, V. (2003). Limits of sequences of stationary planar tessellations. Adv. Appl. Prob. 35, 123–138.CrossRefGoogle Scholar
Nagel, W. and Weiss, V. (2004). A planar crack tessellation which is stable with respect to iteration. Jenaer Schriften zur Mathematik und Informatik 12/04 (Tech. Rep.), Jena. Available at: http://www.fh-jena.de/∼weiss/doc/prepstit.pdf.Google Scholar
Nagel, W. and Weiss, V. (2005). Some geometric features of crack STIT tessellations in the plane. Submitted.Google Scholar
Resnick, S. I. (1992). Adventures in Stochastic Processes. Birkhäuser, Boston, MA.Google Scholar
Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar