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Scaled entropy of filtrations of \(\sigma\)-fields. (English. Russian original) Zbl 1161.28005

Theory Probab. Appl. 52, No. 3, 493-508 (2008); translation from Teor. Veroyatn. Primen. 52, No. 3, 446-467 (2007).
Authors’ 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, Russ. Math. Surv. 55, No. 4, 667–733 (2000; Zbl 0991.37005)]. 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 \({\mathbb{Z}}^d\)) 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.

MSC:

28D20 Entropy and other invariants
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
60G10 Stationary stochastic processes

Citations:

Zbl 0991.37005
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