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A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points

Identifieur interne : 001D03 ( Main/Curation ); précédent : 001D02; suivant : 001D04

A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points

Auteurs : Laurent Bienvenu [France] ; Adam Day [États-Unis] ; Mathieu Hoyrup [France] ; Ilya Mezhirov [Allemagne] ; Alexander Shen [France]

Source :

Information and Computation [ 0890-5401 ] ; 2012.

RBID : Hal:hal-00643629

English descriptors

mix :
- Algorithmic randomness, Birkhoff's ergodic theorem, Martin-Löf randomness, Poincaré recurrence theorem.

Abstract

A theorem of Kucera states that given a Martin-Löf random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.

Url:

https://hal.inria.fr/hal-00643629

DOI: 10.1016/j.ic.2011.10.006

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<orgName type="acronym">ESCAPE</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.lirmm.fr/recherche/equipes/escape</ref>
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<tutelles><tutelle active="#struct-181" type="direct"><org type="laboratory" xml:id="struct-181" status="VALID"><idno type="RNSR">199111950H</idno>
<orgName>Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier</orgName>
<orgName type="acronym">LIRMM</orgName>
<date type="start">1995</date>
<desc><address><addrLine>CC 477, 161 rue Ada, 34095 Montpellier Cedex 5</addrLine>
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</address>
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<tutelle name="UMR5506" active="#struct-410122" type="indirect"><org type="institution" xml:id="struct-410122" status="VALID"><orgName>Université de Montpellier</orgName>
<orgName type="acronym">UM</orgName>
<desc><address><addrLine>163 rue Auguste Broussonnet - 34090 Montpellier</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.umontpellier.fr/</ref>
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<tutelle name="UMR5506" active="#struct-441569" type="indirect"><org type="institution" xml:id="struct-441569" status="VALID"><idno type="IdRef">02636817X</idno>
<idno type="ISNI">0000000122597504</idno>
<orgName>Centre National de la Recherche Scientifique</orgName>
<orgName type="acronym">CNRS</orgName>
<date type="start">1939-10-19</date>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.cnrs.fr/</ref>
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<country>France</country>
</affiliation>
</author>
</analytic>
<idno type="DOI">10.1016/j.ic.2011.10.006</idno>
<series><title level="j">Information and Computation</title>
<idno type="ISSN">0890-5401</idno>
<imprint><date type="datePub">2012</date>
</imprint>
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<profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>Algorithmic randomness</term>
<term>Birkhoff's ergodic theorem</term>
<term>Martin-Löf randomness</term>
<term>Poincaré recurrence theorem</term>
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<front><div type="abstract" xml:lang="en">A theorem of Kucera states that given a Martin-Löf random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.</div>
</front>
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A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points

A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points

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