Uniform Entropy Scalings of Filtrations

We study Vershik and Gorbulsky’s notion of entropy scalings for filtrations in the particular case when the scaling is not 𝜖-dependent, and is then termed as uniform scaling. Among our main results, we prove that the scaled entropy of the filtration generated by the Vershik progressive predictions o...

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Bibliographic Details
Published inSéminaire de Probabilités L pp. 83 - 126
Main Author Laurent, Stéphane
Format Book Chapter
LanguageEnglish
Published Cham Springer International Publishing
SeriesLecture Notes in Mathematics
Subjects
Entropy
Standardness
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Summary:We study Vershik and Gorbulsky’s notion of entropy scalings for filtrations in the particular case when the scaling is not 𝜖-dependent, and is then termed as uniform scaling. Among our main results, we prove that the scaled entropy of the filtration generated by the Vershik progressive predictions of a random variable is equal to the scaled entropy of this random variable. Standardness of a filtration is the case when the scaled entropy with a constant scaling is zero, thus our results generalize some known results about standardness. As a case-study we consider a family of next-jump time filtrations. We also provide some results about the entropy of poly-adic filtrations, rephrasing or generalizing some old results.
ISBN:3030285340
9783030285340
ISSN:0075-8434
1617-9692
DOI:10.1007/978-3-030-28535-7_7