Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T06:51:53.561Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing properties for STIT tessellations

Part of: Geometric probability and stochastic geometry Designs and configurations Ergodic theory

Published online by Cambridge University Press:  01 July 2016

R. Lachièze-Rey
R. Lachièze-Rey*
Affiliation:
Université des Sciences et Technologies de Lille
*
∗ Postal address: Laboratoire de Statistique et Probabilités, UFR de Mathematiques Bat. M2, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq, France. Email address: lr.raphael@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The so-called STIT tessellations form a class of homogeneous (spatially stationary) tessellations in Rd which are stable under the nesting/iteration operation. In this paper we establish the mixing property for these tessellations and give the decay rate of P(A ∩ M = ∅, ThB ∩ M = ∅) / P(A ∩ Y = ∅)P(B ∩ Y = ∅) − 1, where A and B are both compact connected sets, h is a vector of Rd, Th is the corresponding translation operator, and M is a STIT tessellation.

Keywords

Stochastic geometryrandom tessellationSTIT tessellationspace ergodicitymixing property

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05B45: Tessellation and tiling problems 37A25: Ergodicity, mixing, rates of mixing
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 40 - 48
Copyright
Copyright © Applied Probability Trust 2011 

References

Cowan, R. (1978). The use of the ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 47–57.CrossRefGoogle Scholar
Cowan, R. (1980). Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.CrossRefGoogle Scholar
Cowan, R. (1984). A collection of problems in random geometry. In Stochastic Geometry, Geometric Statistics, Stereology, eds Ambartzuminan, R. V. and Weil, W., Teubner, Leipzig, pp. 64–68.Google Scholar
Heinrich, L. (1992). On existence and mixing properties of germ-grain models. Statistics 23, 271–286.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Mecke, J., Nagel, W. and Weiss, V. (2008). A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics 80, 51–67.CrossRefGoogle Scholar
Mecke, J., Nagel, W. and Weiss, V. (2008). The iteration of random tessellations and a construction of a homogeneous process of cell divisions. Adv. Appl. Prob. 40, 49–59.CrossRefGoogle 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.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar