×

Stationary iterated tessellations. (English) Zbl 1041.60012

Summary: The iteration of random tessellations in \(\mathbb R^{d}\) is considered, where each cell of an initial tessellation is further subdivided into smaller cells by so-called component tessellations. Sufficient conditions for stationarity and isotropy of iterated tessellations are given. Formulae are derived for the intensities of their facet processes, and for the expected intrinsic volumes of their typical facets. Particular emphasis is put on two special cases: superposition and nesting of tessellations. Bernoulli thinning of iterated tessellations is also considered.

MSC:

60D05 Geometric probability and stochastic geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambartzumian, R. V. (1990). Factorization Calculus and Geometric Probability . Cambridge University Press. · Zbl 0715.53049
[2] Baccelli, F., Gloaguen, C. and Zuyev, S. (2000). Superposition of Planar Voronoi Tessellations. Commun. Statist. Stoch. Models 16 , 69–98. · Zbl 0978.90013 · doi:10.1080/15326340008807577
[3] Kendall, W. S. and Mecke, J. (1987). The range of the mean-value quantities of planar tessellations. J. Appl. Prob. 24 , 411–421. · Zbl 0624.60018 · doi:10.2307/3214265
[4] Maier, R. and Schmidt, V. (2002). Stationary iterated tessellations. Tech. Rep., University of Ulm. Available at http://www.mathematik.uni-ulm.de/stochastik/. · Zbl 1041.60012
[5] Matheron, G. (1975). Random Sets and Integral Geometry . John Wiley, New York. · Zbl 0321.60009
[6] Mecke, J. (1981). Formulas for stationary planar fibre processes III – Intersections with fibre systems. Math. Operationsforsch. Statist. Ser. Statist. 12 , 201–210. · Zbl 0472.60016 · doi:10.1080/02331888108801582
[7] Mecke, J. (1984). Parametric representation of mean values for stationary random mosaics. Math. Operationsforsch. Statist. Ser. Statist. 15 , 437–442. · Zbl 0547.60019 · doi:10.1080/02331888408801794
[8] Miles, R. E. (1998). A large class of random tessellations with the class Poisson polygon distributions (abstract). Adv. Appl. Prob. 30 , 285.
[9] Miles, R. E. and Mackisack, M. S. (1996). Further random tessellations with the class Poisson polygon distributions (abstract). Adv. Appl. Prob. 28 , 338–339.
[10] Miles, R. E. and Mackisack, M. S. (2002). A large class of random tessellations with the class Poisson polygon distributions. Forma 17 , 1–17.
[11] Møller, J. (1989). Random tessellations in \(\R^d\). Adv. Appl. Prob. 21 , 37–73. · Zbl 0684.60007 · doi:10.2307/1427197
[12] Nagel, W. and Weiss, V. (2003). Limits of sequences of stationary planar tessellations. Adv. Appl. Prob. 35 , 123–138. · Zbl 1023.60015 · doi:10.1239/aap/1046366102
[13] Santaló, L. A. (1984). Mixed random mosaics. Math. Nachr. 117 , 129–133. · Zbl 0554.60021 · doi:10.1002/mana.3211170108
[14] Schneider, R. and Weil, W. (2000). Stochastische Geometrie . Teubner, Stuttgart. · Zbl 0964.52009
[15] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications . John Wiley, Chichester. · Zbl 0838.60002
[16] Weiss, V. and Nagel, W. (1999). Interdependences of directional quantities of planar tessellations. Adv. Appl. Prob. 32 , 664–678. · Zbl 0949.60022 · doi:10.1239/aap/1029955198
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.