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Problemy Peredachi Informatsii, 2003, Volume 39, Issue 1, Pages 134–165 (Mi ppi210)  

This article is cited in 3 scientific papers (total in 4 papers)

On the Role of the Law of Large Numbers in the Theory of Randomness

An. A. Muchnik, A. L. Semenov
References:
Abstract: In the first part of this article, we answer Kolmogorov's question (stated in 1963 in [1]) about exact conditions for the existence of random generators. Kolmogorov theory of complexity permits of a precise definition of the notion of randomness for an individual sequence. For infinite sequences, the property of randomness is a binary property, a sequence can be random or not. For finite sequences, we can solely speak about a continuous property, a measure of randomness. Is it possible to measure randomness of a sequence $t$ by the extent to which the law of large numbers is satisfied in all subsequences of $t$ obtained in an “admissible way”? The case of infinite sequences was studied in [2]. As a measure of randomness (or, more exactly, of nonrandomness) of a finite sequence, we consider the specific deficiency of randomness $\delta$ (Definition 5). In the second part of this paper, we prove that the function $\delta/\ln(1/\delta)$ characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied.
English version:
Problems of Information Transmission, 2003, Volume 39, Issue 1, Pages 119–147
DOI: https://doi.org/10.1023/A:1023638717091
Bibliographic databases:
UDC: 621.391.1:519.2
Language: Russian
Citation: An. A. Muchnik, A. L. Semenov, “On the Role of the Law of Large Numbers in the Theory of Randomness”, Probl. Peredachi Inf., 39:1 (2003), 134–165; Problems Inform. Transmission, 39:1 (2003), 119–147
Citation in format AMSBIB
\Bibitem{MucSem03}
\by An.~A.~Muchnik, A.~L.~Semenov
\paper On the Role of the Law of Large Numbers in the
Theory of Randomness
\jour Probl. Peredachi Inf.
\yr 2003
\vol 39
\issue 1
\pages 134--165
\mathnet{http://mi.mathnet.ru/ppi210}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2101670}
\zmath{https://zbmath.org/?q=an:1078.60005}
\transl
\jour Problems Inform. Transmission
\yr 2003
\vol 39
\issue 1
\pages 119--147
\crossref{https://doi.org/10.1023/A:1023638717091}
Linking options:
  • https://www.mathnet.ru/eng/ppi210
  • https://www.mathnet.ru/eng/ppi/v39/i1/p134
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Проблемы передачи информации Problems of Information Transmission
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