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 Probl. Peredachi Inf., 2001, Volume 37, Issue 2, Pages 27–39 (Mi ppi515)

Large Systems

Nonrobustness Property of the Individual Ergodic Theorem

V. V. V'yugin

Abstract: Main laws of probability theory, when applied to individual sequences, have a “robustness” property under small violations of randomness. For example, the law of large numbers for the symmetric Bernoulli scheme holds for a sequence where the randomness deficiency of its initial fragment of length $n$ grows as $o(n)$. The law of iterated logarithm holds if the randomness deficiency grows as $o(\log\log n)$. We prove that Birkhoff's individual ergodic theorem is nonrobust in this sense. If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, this theorem can be violated. An analogous nonrobustness property holds for the Shannon–McMillan–Breiman theorem.

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English version:
Problems of Information Transmission, 2001, 37:2, 108–119

Bibliographic databases:

UDC: 621.391.1:519.2

Citation: V. V. V'yugin, “Nonrobustness Property of the Individual Ergodic Theorem”, Probl. Peredachi Inf., 37:2 (2001), 27–39; Problems Inform. Transmission, 37:2 (2001), 108–119

Citation in format AMSBIB
\Bibitem{Vyu01} \by V.~V.~V'yugin \paper Nonrobustness Property of the Individual Ergodic Theorem \jour Probl. Peredachi Inf. \yr 2001 \vol 37 \issue 2 \pages 27--39 \mathnet{http://mi.mathnet.ru/ppi515} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2099896} \zmath{https://zbmath.org/?q=an:1012.37003} \transl \jour Problems Inform. Transmission \yr 2001 \vol 37 \issue 2 \pages 108--119 \crossref{https://doi.org/10.1023/A:1010418008049} 

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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. V. V. V'yugin, “Problems of Robustness for Universal Coding Schemes”, Problems Inform. Transmission, 39:1 (2003), 32–46
2. V'yugin V.V., “on the Stability Property of Asymptotic Laws of Ergodic Theory and Universal Compression Schemes”, Dokl. Math., 92:2 (2015), 556–558
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