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Limits of sequences of stationary planar tessellations

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-07740 Jena, Germany. Email address: [email protected]
∗∗ Postal address: Fachhochschule Jena, D-07703 Jena, Germany.
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Abstract

In order to increase the variety of feasible models for random stationary tessellations (mosaics), two operations acting on tessellations are studied: superposition and iteration (the latter is also referred to as nesting). The superposition of two planar tessellations is the superposition of the edges of the cells of both tessellations. The iteration of tessellations means that one tessellation is chosen as a ‘frame’ tessellation. The single cells of this ‘frame’ are simultaneously and independently subdivided by cut-outs of tessellations of an independent and identically distributed sequence of tessellations. In the present paper, we investigate the limits for sequences of tessellations that are generated by consecutive application of superposition or iteration respectively. Sequences of (renormalised) superpositions of stationary planar tessellations converge weakly to Poisson line tessellations. For consecutive iteration the notion of stability of distributions is adapted and necessary conditions are formulated for those tessellations which may occur as limits.

Keywords

Stochastic geometryrandom tessellationsuperposition of tessellationsiteration/nesting of tessellationsweak convergencestability of distributionsPoisson line processes

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 35 , Issue 1 , March 2003 , pp. 123 - 138
Copyright
Copyright © Applied Probability Trust 2003 

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