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The iteration of random tessellations and a construction of a homogeneous process of cell divisions

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Joseph Mecke ,
Werner Nagel  and
Viola Weiss
Joseph Mecke*
Affiliation:
Friedrich-Schiller-Universität Jena
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
Viola Weiss*
Affiliation:
Fachhochschule Jena
*
∗ Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany.
∗ Postal address: Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany.
∗∗∗ Postal address: Fachhochschule Jena, D-07703 Jena, Germany.
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Abstract

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A random tessellation of ℝd is said to be homogeneous if its distribution is invariant under all shifts of ℝd. The iteration of homogeneous random tessellations is described in a new manner that makes it evident that the resulting random tessellation is homogeneous again. Furthermore, a tessellation-valued process is constructed, the random states of which are homogeneous random tessellations stable under iteration (STIT). It can be interpreted as a process of subsequent division of cells.

Keywords

Stochastic geometryrandom tessellationtessellation-valued stochastic processPoisson point processiteration of tessellationsnesting of tessellationsstability 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 40 , Issue 1 , March 2008 , pp. 49 - 59
Copyright
Copyright © Applied Probability Trust 2008 

References

Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge University Press.CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York, London.Google Scholar
Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.Google Scholar
Mecke, J., Nagel, W. and Weiss, V. (2008). A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics Internat. J. Prob. Stoch. Process. 80, 51–67.Google Scholar
Mecke, J., Nagel, W. and Weiss, V. (2007). Length distributions of edges of planar stationary and isotropic STIT tessellations. J. Contemp. Math. Anal. 42, 28–43.Google Scholar
Molchanov, I. S. (2005). Theory of Random Sets. Springer, London.Google Scholar
Nagel, W. and Weiss, V. (2003). Limits of sequences of stationary planar tessellations. Adv. Appl. Prob. 35, 123–138.Google Scholar
Nagel, W. and Weiss, V. (2005). Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Prob. 37, 859–883.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar