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

Part of: Stochastic processes Geometric probability and stochastic geometry

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 

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