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Stit Tessellations have Trivial Tail σ-Algebra

Part of: Markov processes Ergodic theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Servet Martínez  and
Werner Nagel
Servet Martínez*
Affiliation:
Universidad de Chile
Werner Nagel*
Affiliation:
Friedrich-Schiller-Universität Jena
*
∗ Postal address: Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile. Email address: [email protected]
∗∗ Postal address: Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany. Email address: [email protected]
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Abstract

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We consider homogeneous STIT tessellations Y in the ℓ-dimensional Euclidean space ℝℓ and show the triviality of the tail σ-algebra. This is a sharpening of the mixing result by Lachièze-Rey (2001).

Keywords

Stochastic geometryrandom process of tessellationsergodic theorytail σ-algebra

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 37A25: Ergodicity, mixing, rates of mixing
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 643 - 660
Copyright
© Applied Probability Trust 

References

Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bradley, R. C. (2007). Introduction to Strong Mixing Conditions, Vol. 1. Kendrick Press, Heber City, UT.Google Scholar
Breiman, L. (1992). Probability, Corrected reprint of the 1968 original. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, General Theory and Structure, 2nd edn. Springer, New York.Google Scholar
Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter, Berlin.CrossRefGoogle Scholar
Heinrich, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Meth. Statist. 3, 1–24.Google Scholar
Heinrich, L. (2013). Asymptotic methods in statistics of random point processes. In Stochastic Geometry, Spatial Statistics and Random Fields (Lecture Notes Math. 2068), Springer, Heidelberg, pp. 115–150.CrossRefGoogle Scholar
Heinrich, L. and Molchanov, I. S. (1999). Central limit theorem for a class of random measures associated with germ–grain models. Adv. Appl. Prob. 31, 283–314.CrossRefGoogle Scholar
Heinrich, L., Schmidt, H. and Schmidt, V. (2007). Limit theorems for functionals on the facets of stationary random tessellations. Bernoulli 13, 868–891.CrossRefGoogle Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.Google Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Lachièze-Rey, R. (2011). Mixing properties for STIT tessellations. Adv. Appl. Prob. 43, 40–48.CrossRefGoogle Scholar
Martı´nez, S. and Nagel, W. (2012). Ergodic description of STIT tessellations. Stochastics 84, 113–134.CrossRefGoogle Scholar
Mecke, J., Nagel, W. and Weiss, V. (2007). Length distributions of edges in planar stationary and isotropic STIT tessellations. J. Contemp. Math. Analysis 42, 28–43.CrossRefGoogle 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
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
Parry, W. (1981). Topics in Ergodic Theory. Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Schreiber, T. and Thäle, C. (2010). Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv. Appl. Prob. 42, 913–935.CrossRefGoogle Scholar
Schreiber, T. and Thäle, C. (2011). Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations. Stoch. Process. Appl. 121, 989–1012.CrossRefGoogle Scholar
Schreiber, T. and Thäle, C. (2013). Limit theorems for iteration stable tessellations. Ann. Prob. 41, 2261–2278.CrossRefGoogle Scholar