This article is cited in 16 papers

Scaled entropy of filtrations of $\sigma$-fields

A. M. Vershik, A. D. Gorbul'skii

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We study the notion of the scaled entropy of a filtration of $\sigma$-fields (i.e., decreasing sequence of $\sigma$-fields) introduced in [A. M. Vershik, Russian Math. Surveys, 55 (2000), pp. 677–733]. We suggest a method for computing this entropy for the sequence of $\sigma$-fields of pasts of a Markov process determined by a random walk over the trajectories of a Bernoulli action of a commutative or nilpotent countable group. Since the scaled entropy is a metric invariant of the filtration, it follows that the sequences of $\sigma$-fields of pasts of random walks over the trajectories of Bernoulli actions of lattices (groups $\mathbf{Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the same $d$ but different values of the entropy of the Bernoulli scheme. We give a brief survey of the metric theory of filtrations; in particular, we formulate the standardness criterion and describe its connections with the scaled entropy and the notion of a tower of measures.

Keywords: filtration, $\sigma$-field of pasts, scaled entropy, random walks.

Received: 24.04.2007

DOI: 10.4213/tvp73

 English version: Theory of Probability and its Applications, 2008, 52:3, 493–508

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