Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T19:23:56.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Limits of sequences of stationary planar tessellations

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-07740 Jena, Germany. Email address: [email protected]
∗∗ Postal address: Fachhochschule Jena, D-07703 Jena, Germany.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambartzumian, R. V. (1990). Factorization Calculus and Geometric Probability. Cambridge University Press.Google Scholar
Arak, T., Clifford, P. and Surgailis, D. (1993). Point-based polygonal models for random graphs. Adv. Appl. Prob. 25, 348–372.Google 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
Kendall, W. S. and Mecke, J. (1987). The range of the mean-value quantities of planar tessellations. J. Appl. Prob. 24, 411–421.Google Scholar
Maier, R. and Schmidt, V. (2003). Stationary iterated tessellations. Preprint, University of Ulm. To appear in Adv. Appl. Prob.Google Scholar
Maier, R., Mayer, J. and Schmidt, V. (2002). Distributional properties of the typical cell of stationary iterated tessellations. Preprint, University of Ulm.Google Scholar
Mathéron, G., (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Mayer, J., Schmidt, V. and Schweiggert, F. (2002). A generic framework design for the simulation of spatial stochastic models. Submitted. Preprint, University of Ulm.Google Scholar
Mecke, J. (1984). Parametric representation of mean values for stationary random mosaics. Math. Operationsforsch. Statist. Ser. Statist. 15, 437–442.Google Scholar
Mecke, J. and Stoyan, D. (1980). A general approach to Buffon's needle and Wicksell's corpuscle problem. In Combinatorial Principles in Stochastic Geometry, ed. Ambartzumian, R. V. (in Russian), Izdatel´stvo AN Armianskoi SSR, pp. 164–171.Google Scholar
Mecke, J., Schneider, R., Stoyan, D. and Weil, W. (1990). Stochastische Geometrie. Birkhäuser, Basel.CrossRefGoogle 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
Molchanov, I. S. (1993). Limit Theorems for Unions of Random Closed Sets (Lecture Notes Math. 1561). Springer, Berlin.Google Scholar
Möller, J., (1989). Random tessellations in Rd . Adv. Appl. Prob. 24, 37–73.CrossRefGoogle Scholar
Norberg, T. (1984). Convergence and existence of random set distributions. Ann. Prob. 12, 726–732.Google Scholar
Santaló, L., (1984). Mixed random mosaics. Math. Nachr. 117, 129–133.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Tchoumatchenko, K. and Zuev, S. (2001). Aggregate and fractal tessellations. Prob. Theory Relat. Fields 121, 198–218.Google Scholar
Weiss, V. and Nagel, W. (1999). Interdependences of directional quantities of planar tessellations. Adv. Appl. Prob. 31, 664–678.Google Scholar