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Spatial Stit Tessellations: Distributional Results for I-Segments

Part of: Stochastic processes Distribution theory - Probability Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

Christoph Thäle ,
Viola Weiss  and
Werner Nagel
Christoph Thäle*
Affiliation:
Universität Osnabrück
Viola Weiss*
Affiliation:
Fachhochschule Jena
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
*
∗ Postal address: Institut für Mathematik, Universität Osnabrück, Albrechtstr. 28a, D-49076 Osnabrück, Germany. Email address: christoph.thaele@uni-osnabrueck.de
∗∗ Postal address: Fachhochschule Jena, Fachbereich Grundlagenwissenschaften, Carl-Zeiss-Promenade 2, D-07745 Jena, Germany. Email address: viola.weiss@fh-jena.de
∗∗∗ Postal address: Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany. Email address: werner.nagel@uni-jena.de
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Abstract

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In this paper we consider three-dimensional random tessellations that are stable under iteration (STIT tessellations). STIT tessellations arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the cell-dividing polygons are the so-called I-segments of the tessellation. The main result is an explicit formula for the distribution of the number of vertices in the relative interior of the typical I-segment. In preparation for its proof, we obtain other distributional identities for the typical I-segment and the length-weighted typical I-segment, which provide new insight into the spatiotemporal construction process.

Keywords

Cell division processiteration/nestingmarked point processrandom tessellationstability under iterationstochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 60E05: Distributions
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 635 - 654
Copyright
© Applied Probability Trust 

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