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 ǫ-dependent, and is then termed as 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 a case-study we consider a family of next-jump time filtrations. We also rephrase some old theorems by Vershik and provide a generalization of a theorem by Gorbulsky about the case of poly-adic filtrations.
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