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Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 3, Pages 446–467 (Mi tvp73)  

This article is cited in 13 scientific papers (total in 14 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.

DOI: https://doi.org/10.4213/tvp73

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

Bibliographic databases:

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

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Lodkin, I. E. Manaev, A. R. Minabutdinov, “Asymptotic behavior of the scaling entropy of the Pascal adic transformation”, J. Math. Sci. (N. Y.), 174:1 (2011), 28–35  mathnet  crossref
    2. Theory Probab. Appl., 55:1 (2011), 54–76  mathnet  crossref  crossref  mathscinet  isi
    3. A. M. Vershik, “Scailing entropy and automorphisms with pure pointspectrum”, St. Petersburg Math. J., 23:1 (2012), 75–91  mathnet  crossref  mathscinet  zmath  isi  elib
    4. Vershik A.M., Zatitskiy P.B., Petrov F.V., “Geometry and Dynamics of Admissible Metrics in Measure Spaces”, Cent. Eur. J. Math., 11:3 (2013), 379–400  crossref  mathscinet  zmath  isi  elib  scopus
    5. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. P. B. Zatitskii, “On a Scaling Entropy Sequence of a Dynamical System”, Funct. Anal. Appl., 48:4 (2014), 291–294  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. P. B. Zatitskiy, “Scaling entropy sequence: invariance and examples”, J. Math. Sci. (N. Y.), 209:6 (2015), 890–909  mathnet  crossref
    8. A. M. Vershik, “Standardness as an Invariant Formulation of Independence”, Funct. Anal. Appl., 49:4 (2015), 253–263  mathnet  crossref  crossref  isi  elib
    9. P. B. Zatitskiy, “On the possible growth rate of a scaling entropy sequence”, J. Math. Sci. (N. Y.), 215:6 (2016), 715–733  mathnet  crossref  mathscinet
    10. P. B. Zatitskiy, F. V. Petrov, “On the subadditivity of a scaling entropy sequence”, J. Math. Sci. (N. Y.), 215:6 (2016), 734–737  mathnet  crossref  mathscinet
    11. Vershik A.M., “Asymptotic theory of path spaces of graded graphs and its applications”, Jap. J. Math., 11:2 (2016), 151–218  crossref  mathscinet  zmath  isi  scopus
    12. A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. A. M. Vershik, P. B. Zatitskii, “Universal adic approximation, invariant measures and scaled entropy”, Izv. Math., 81:4 (2017), 734–770  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. A. M. Vershik, P. B. Zatitskii, “Combinatorial Invariants of Metric Filtrations and Automorphisms; the Universal Adic Graph”, Funct. Anal. Appl., 52:4 (2018), 258–269  mathnet  crossref  crossref  isi  elib
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