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This article is cited in 3 scientific papers (total in 3 papers)
Standardness as an Invariant Formulation of Independence
A. M. Vershikabc
a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
b Chebyshev Laboratory, St. Petersburg State University, Department of Mathematics and Mechanics
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
Abstract:
The notion of a homogeneous standard filtration of $\sigma$-algebras was introduced by the author in 1970. The main theorem asserted that a homogeneous filtration is standard, i.e., generated by a sequence of independent random variables (is Bernoulli), if and only if a standardness criterion is satisfied. The author has recently generalized the notion of standardness to arbitrary filtrations. In this paper we give detailed definitions and characterizations of Markov standard filtrations. The notion of standardness is essential for applications of probabilistic, combinatorial, and algebraic nature. At the end of the paper we present new notions related to nonstandard filtrations.
Keywords:
filtration, standardness, intrinsic metric, virtual metric space with measure
DOI:
https://doi.org/10.4213/faa3213
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English version:
Functional Analysis and Its Applications, 2015, 49:4, 253–263
Bibliographic databases:
  
UDC:
519.2
Received: 13.09.2015
Citation:
A. M. Vershik, “Standardness as an Invariant Formulation of Independence”, Funktsional. Anal. i Prilozhen., 49:4 (2015), 18–32; Funct. Anal. Appl., 49:4 (2015), 253–263
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/faa3213https://doi.org/10.4213/faa3213 http://mi.mathnet.ru/eng/faa/v49/i4/p18
Citing articles on Google Scholar:
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English citations
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Russian articles,
English articles
This publication is cited in the following articles:
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Vershik A.M., “Asymptotic theory of path spaces of graded graphs and its applications”, Jap. J. Math., 11:2 (2016), 151–218
  -
A. M. Vershik, “The theory of filtrations of subalgebras, standardness, and independence”, Russian Math. Surveys, 72:2 (2017), 257–333
  -
A. M. Vershik, P. B. Zatitskii, “Universal adic approximation, invariant measures and scaled entropy”, Izv. Math., 81:4 (2017), 734–770
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