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 Probl. Peredachi Inf., 2003, Volume 39, Issue 1, Pages 134–165 (Mi ppi210)

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

An. A. Muchnik, A. L. Semenov

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.

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English version:
Problems of Information Transmission, 2003, 39:1, 119–147

Bibliographic databases:

UDC: 621.391.1:519.2

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://www.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} 

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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. K. Yu. Gorbunov, “Estimation of the Number of Elements in a Covering of an Arbitrary Randomness Test by Frequency Tests”, Problems Inform. Transmission, 43:1 (2007), 48–56
2. S. I. Adian, A. L. Semenov, V. A. Uspenskii, “Andrei Al'bertovich Muchnik (obituary)”, Russian Math. Surveys, 62:4 (2007), 775–779
3. Uspensky V.A., V'yugin V.V., “Development of the algorithmic information theory in Russia”, Journal of Communications Technology and Electronics, 56:6 (2011), 739–747
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