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Funktsional. Anal. i Prilozhen., 2015, Volume 49, Issue 4, Pages 18–32 (Mi faa3213)  

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

Funding Agency Grant Number
Russian Science Foundation 14-50-00150


DOI: https://doi.org/10.4213/faa3213

Full text: PDF file (213 kB)
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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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. 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
    3. 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
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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