Uniform entropy scalings of filtrations
Abstract
We study Vershik and Gorbulsky's notion of entropy scalings for filtrations in the particular case when the scaling is not epsilon-dependent, and is then termed as \emph{uniform scaling}. Our main result states 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 an example we derive a proper uniform entropy scaling for a next-jump time filtration. As a side note, we use the example of the next-jump time filtrations to write down a case study of Vershik's theory of intrinsic topology on Bratteli graphs.
Domains
Probability [math.PR]
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