Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-30T21:55:07.496Z Has data issue: false hasContentIssue false
  • English
  • Français

STIT tessellations are Bernoulli and standard

Published online by Cambridge University Press:  29 November 2012

SERVET MARTÍNEZ
SERVET MARTÍNEZ*
Affiliation:
Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile (email: smartine@dim.uchile.cl)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (Yt:t>0) be a STIT tessellation process and a>1. We prove that the random sequence (anYan:n∈ℤ) is a non-anticipating factor of a Bernoulli shift. We deduce that the continuous time process (atYat:t∈ℝ) is a Bernoulli flow. We use the techniques and results in Martínez and Nagel [Ergodic description of STIT tessellations. Stochastics 84(1) (2012), 113–134]. We also show that the filtration associated to the non-anticipating factor is standard in Vershik’s sense.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 876 - 892
Copyright
©2012 Cambridge University Press 

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.)

Article purchase

Temporarily unavailable

References

[1]Blumenthal, R. M. and Getoor, R. K.. Markov Processes and Potential Theory. Academic Press, New York and London, 1968.Google Scholar
[2]de la Rue, T.. Espaces de Lebesgue. Séminaire de Probabilités de Strasbourg XXVII (Lectures Notes in Mathematics, 1557). Springer, Berlin, 1993, pp. 15–21.CrossRefGoogle Scholar
[3]Émery, M. and Schachermayer, W.. On Vershik’s standardness criterion and Tsirelson’s notion of cosiness. Séminaire de Probabilités, XXXV (Lecture Notes in Mathematics, 1755). Springer, Berlin, 2001, pp. 265–305.CrossRefGoogle Scholar
[4]Ethier, S. N. and Kurtz, T. G.. Markov Processes. John Wiley & Sons, New York, 1985.Google Scholar
[5]Feldman, J. and Smorodinsky, M.. Decreasing sequences of measurable partitions: product type, standard and prestandard. Ergod. Th. & Dynam. Syst. 20(4) (2000), 1079–1090.Google Scholar
[6]Lachièze-Rey, R.. Strong mixing property for STIT tessellations. Adv. Appl. Prob. 43(1) (2011), 40–48.CrossRefGoogle Scholar
[7]Martínez, S. and Nagel, W.. Ergodic description of STIT tessellations. Stochastics 84(1) (2012), 113–134.Google Scholar
[8]Mecke, J., Nagel, W. and Weiß, V.. A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics 80 (2008), 51–67.Google Scholar
[9]Nagel, W. and Weiß, V.. Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Prob. 37 (2005), 859–883.Google Scholar
[10]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337–352.Google Scholar
[11]Ornstein, D.. Two Bernoulli shifts with infinite entropy are isomorphic. Adv. Math. 5 (1970), 339–348.CrossRefGoogle Scholar
[12]Ornstein, D.. Factors of Bernoulli shifts are Bernoulli shifts. Adv. Math. 5 (1970), 349–364.Google Scholar
[13]Ornstein, D.. Ergodic Theory, Randomness and Dynamical Systems. Yale University Press, New Haven, CT, 1974.Google Scholar
[14]Schneider, R. and Weil, W.. Stochastic and Integral Geometry. Springer, Berlin, 2008.Google Scholar
[15]Vershik, A. M.. The theory of decreasing sequences of measurable partitions. St. Petersburg Math. J. 16(2) (1995), 705–761. Expandend version of doctoral thesis, Leningrad University, 1973.Google Scholar