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This article is cited in 16 scientific papers (total in 17 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
Citation:
A. M. Vershik, A. D. Gorbul'skii, “Scaled entropy of filtrations of $\sigma$-fields”, Teor. Veroyatnost. i Primenen., 52:3 (2007), 446–467; Theory Probab. Appl., 52:3 (2008), 493–508
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https://www.mathnet.ru/eng/tvp73https://doi.org/10.4213/tvp73 https://www.mathnet.ru/eng/tvp/v52/i3/p446
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