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Vershik’s Intermediate Level Standardness Criterion and the Scale of an Automorphism

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2078)

Abstract

In the case of r n -adic filtrations, Vershik’s standardness criterion takes a particular form, hereafter called Vershik’s intermediate level criterion. This criterion, whose nature is combinatorial, has been intensively used in the ergodic-theoretic literature, but it is not easily applicable by probabilists because it is stated in a language specific to the theory of measurable partitions and has not been translated into probabilistic terms. We aim to provide an easily applicable probabilistic statement of this criterion. Finally, Vershik’s intermediate level criterion is illustrated by revisiting Vershik’s definition of the scale of an invertible measure-preserving transformation.

Keywords

  • Level Criterion
  • Independent Random Variable
  • Polish Space
  • Bernoulli Shift
  • Splitting Sequence

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

I thank Michel Émery and Anatoly Vershik for the interest they have expressed in this work, and I also thank Michel Émery for helpful comments on a previous version of the paper.

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Correspondence to Stéphane Laurent .

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Laurent, S. (2013). Vershik’s Intermediate Level Standardness Criterion and the Scale of an Automorphism. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_3

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