A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points
Identifieur interne : 000632 ( Hal/Corpus ); précédent : 000631; suivant : 000633A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points
Auteurs : Laurent Bienvenu ; Adam Day ; Mathieu Hoyrup ; Ilya Mezhirov ; Alexander ShenSource :
- Information and Computation [ 0890-5401 ] ; 2012.
English descriptors
- mix :
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:
DOI: 10.1016/j.ic.2011.10.006
Links to Exploration step
Hal:hal-00643629Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points</title>
<author><name sortKey="Bienvenu, Laurent" sort="Bienvenu, Laurent" uniqKey="Bienvenu L" first="Laurent" last="Bienvenu">Laurent Bienvenu</name>
<affiliation><hal:affiliation type="laboratory" xml:id="struct-1033" status="OLD"><orgName>Laboratoire d'informatique Algorithmique : Fondements et Applications</orgName>
<orgName type="acronym">LIAFA</orgName>
<date type="end">2015-12-31</date>
<desc><address><addrLine>Université Paris Diderot, Bât. Sophie Germain, case postale 7014, 75205 Paris cedex 13</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.liafa.jussieu.fr/</ref>
</desc>
<listRelation><relation active="#struct-300301" type="direct"></relation>
<relation name="UMR7089" active="#struct-441569" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-300301" type="direct"><org type="institution" xml:id="struct-300301" status="VALID"><orgName>Université Paris Diderot - Paris 7</orgName>
<orgName type="acronym">UP7</orgName>
<desc><address><addrLine>5 rue Thomas-Mann - 75205 Paris cedex 13</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.univ-paris-diderot.fr</ref>
</desc>
</org>
</tutelle>
<tutelle name="UMR7089" active="#struct-441569" type="direct"><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>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Day, Adam" sort="Day, Adam" uniqKey="Day A" first="Adam" last="Day">Adam Day</name>
<affiliation><hal:affiliation type="laboratory" xml:id="struct-83772" status="VALID"><orgName>Miller Institute for Basic Research in Science</orgName>
<desc><address><country key="US"></country>
</address>
<ref type="url">http://millerinstitute.berkeley.edu/</ref>
</desc>
<listRelation><relation active="#struct-300674" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-300674" type="direct"><org type="institution" xml:id="struct-300674" status="VALID"><orgName>University of California [Berkeley]</orgName>
<desc><address><addrLine>Berkeley, CA</addrLine>
<country key="US"></country>
</address>
<ref type="url">http://www.berkeley.edu/</ref>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Hoyrup, Mathieu" sort="Hoyrup, Mathieu" uniqKey="Hoyrup M" first="Mathieu" last="Hoyrup">Mathieu Hoyrup</name>
<affiliation><hal:affiliation type="researchteam" xml:id="struct-29797" status="VALID"><idno type="RNSR">200918992J</idno>
<orgName>Theoretical adverse computations, and safety</orgName>
<orgName type="acronym">CARTE</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/carte</ref>
</desc>
<listRelation><relation active="#struct-129671" type="direct"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-423084" type="direct"></relation>
<relation active="#struct-206040" type="indirect"></relation>
<relation active="#struct-413289" type="indirect"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
</listRelation>
<tutelles><tutelle active="#struct-129671" type="direct"><org type="laboratory" xml:id="struct-129671" status="VALID"><idno type="RNSR">198618246Y</idno>
<orgName>INRIA Nancy - Grand Est</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/nancy</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-423084" type="direct"><org type="department" xml:id="struct-423084" status="VALID"><orgName>Department of Formal Methods </orgName>
<orgName type="acronym">LORIA - FM</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr/la-recherche-en/departements/formal-methods</ref>
</desc>
<listRelation><relation active="#struct-206040" type="direct"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-413289" type="indirect"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
</listRelation>
</org>
</tutelle>
<tutelle active="#struct-206040" type="indirect"><org type="laboratory" xml:id="struct-206040" status="VALID"><idno type="IdRef">067077927</idno>
<idno type="RNSR">198912571S</idno>
<idno type="IdUnivLorraine">[UL]RSI--</idno>
<orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<date type="start">2012-01-01</date>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-413289" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle active="#struct-413289" type="indirect"><org type="institution" xml:id="struct-413289" status="VALID"><idno type="IdRef">157040569</idno>
<idno type="IdUnivLorraine">[UL]100--</idno>
<orgName>Université de Lorraine</orgName>
<orgName type="acronym">UL</orgName>
<date type="start">2012-01-01</date>
<desc><address><addrLine>34 cours Léopold - CS 25233 - 54052 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.univ-lorraine.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle name="UMR7503" 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>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Mezhirov, Ilya" sort="Mezhirov, Ilya" uniqKey="Mezhirov I" first="Ilya" last="Mezhirov">Ilya Mezhirov</name>
<affiliation><hal:affiliation type="laboratory" xml:id="struct-194149" status="VALID"><orgName>Technical University of Kaiserslautern</orgName>
<orgName type="acronym">TU Kaiserslautern</orgName>
<desc><address><addrLine>PO Box 3 049 67.653 Kaiserslautern</addrLine>
<country key="DE"></country>
</address>
</desc>
<listRelation><relation active="#struct-368320" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-368320" type="direct"><org type="institution" xml:id="struct-368320" status="INCOMING"><orgName>Technical University of Kaiserslautern</orgName>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Shen, Alexander" sort="Shen, Alexander" uniqKey="Shen A" first="Alexander" last="Shen">Alexander Shen</name>
<affiliation><hal:affiliation type="researchteam" xml:id="struct-394760" status="VALID"><orgName>Systèmes complexes, automates et pavages</orgName>
<orgName type="acronym">ESCAPE</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.lirmm.fr/recherche/equipes/escape</ref>
</desc>
<listRelation><relation active="#struct-181" type="direct"></relation>
<relation name="UMR5506" active="#struct-410122" type="indirect"></relation>
<relation name="UMR5506" active="#struct-441569" type="indirect"></relation>
</listRelation>
<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>
<country key="FR"></country>
</address>
<ref type="url">http://www.lirmm.fr</ref>
</desc>
<listRelation><relation name="UMR5506" active="#struct-410122" type="direct"></relation>
<relation name="UMR5506" active="#struct-441569" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<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>
</desc>
</org>
</tutelle>
<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>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">HAL</idno>
<idno type="RBID">Hal:hal-00643629</idno>
<idno type="halId">hal-00643629</idno>
<idno type="halUri">https://hal.inria.fr/hal-00643629</idno>
<idno type="url">https://hal.inria.fr/hal-00643629</idno>
<idno type="doi">10.1016/j.ic.2011.10.006</idno>
<date when="2012">2012</date>
<idno type="wicri:Area/Hal/Corpus">000632</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en">A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points</title>
<author><name sortKey="Bienvenu, Laurent" sort="Bienvenu, Laurent" uniqKey="Bienvenu L" first="Laurent" last="Bienvenu">Laurent Bienvenu</name>
<affiliation><hal:affiliation type="laboratory" xml:id="struct-1033" status="OLD"><orgName>Laboratoire d'informatique Algorithmique : Fondements et Applications</orgName>
<orgName type="acronym">LIAFA</orgName>
<date type="end">2015-12-31</date>
<desc><address><addrLine>Université Paris Diderot, Bât. Sophie Germain, case postale 7014, 75205 Paris cedex 13</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.liafa.jussieu.fr/</ref>
</desc>
<listRelation><relation active="#struct-300301" type="direct"></relation>
<relation name="UMR7089" active="#struct-441569" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-300301" type="direct"><org type="institution" xml:id="struct-300301" status="VALID"><orgName>Université Paris Diderot - Paris 7</orgName>
<orgName type="acronym">UP7</orgName>
<desc><address><addrLine>5 rue Thomas-Mann - 75205 Paris cedex 13</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.univ-paris-diderot.fr</ref>
</desc>
</org>
</tutelle>
<tutelle name="UMR7089" active="#struct-441569" type="direct"><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>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Day, Adam" sort="Day, Adam" uniqKey="Day A" first="Adam" last="Day">Adam Day</name>
<affiliation><hal:affiliation type="laboratory" xml:id="struct-83772" status="VALID"><orgName>Miller Institute for Basic Research in Science</orgName>
<desc><address><country key="US"></country>
</address>
<ref type="url">http://millerinstitute.berkeley.edu/</ref>
</desc>
<listRelation><relation active="#struct-300674" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-300674" type="direct"><org type="institution" xml:id="struct-300674" status="VALID"><orgName>University of California [Berkeley]</orgName>
<desc><address><addrLine>Berkeley, CA</addrLine>
<country key="US"></country>
</address>
<ref type="url">http://www.berkeley.edu/</ref>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Hoyrup, Mathieu" sort="Hoyrup, Mathieu" uniqKey="Hoyrup M" first="Mathieu" last="Hoyrup">Mathieu Hoyrup</name>
<affiliation><hal:affiliation type="researchteam" xml:id="struct-29797" status="VALID"><idno type="RNSR">200918992J</idno>
<orgName>Theoretical adverse computations, and safety</orgName>
<orgName type="acronym">CARTE</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/equipes/carte</ref>
</desc>
<listRelation><relation active="#struct-129671" type="direct"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-423084" type="direct"></relation>
<relation active="#struct-206040" type="indirect"></relation>
<relation active="#struct-413289" type="indirect"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
</listRelation>
<tutelles><tutelle active="#struct-129671" type="direct"><org type="laboratory" xml:id="struct-129671" status="VALID"><idno type="RNSR">198618246Y</idno>
<orgName>INRIA Nancy - Grand Est</orgName>
<desc><address><addrLine>615 rue du Jardin Botanique 54600 Villers-lès-Nancy</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/nancy</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle active="#struct-300009" type="indirect"><org type="institution" xml:id="struct-300009" status="VALID"><orgName>Institut National de Recherche en Informatique et en Automatique</orgName>
<orgName type="acronym">Inria</orgName>
<desc><address><addrLine>Domaine de VoluceauRocquencourt - BP 10578153 Le Chesnay Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.inria.fr/en/</ref>
</desc>
</org>
</tutelle>
<tutelle active="#struct-423084" type="direct"><org type="department" xml:id="struct-423084" status="VALID"><orgName>Department of Formal Methods </orgName>
<orgName type="acronym">LORIA - FM</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr/la-recherche-en/departements/formal-methods</ref>
</desc>
<listRelation><relation active="#struct-206040" type="direct"></relation>
<relation active="#struct-300009" type="indirect"></relation>
<relation active="#struct-413289" type="indirect"></relation>
<relation name="UMR7503" active="#struct-441569" type="indirect"></relation>
</listRelation>
</org>
</tutelle>
<tutelle active="#struct-206040" type="indirect"><org type="laboratory" xml:id="struct-206040" status="VALID"><idno type="IdRef">067077927</idno>
<idno type="RNSR">198912571S</idno>
<idno type="IdUnivLorraine">[UL]RSI--</idno>
<orgName>Laboratoire Lorrain de Recherche en Informatique et ses Applications</orgName>
<orgName type="acronym">LORIA</orgName>
<date type="start">2012-01-01</date>
<desc><address><addrLine>Campus Scientifique BP 239 54506 Vandoeuvre-lès-Nancy Cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.loria.fr</ref>
</desc>
<listRelation><relation active="#struct-300009" type="direct"></relation>
<relation active="#struct-413289" type="direct"></relation>
<relation name="UMR7503" active="#struct-441569" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<tutelle active="#struct-413289" type="indirect"><org type="institution" xml:id="struct-413289" status="VALID"><idno type="IdRef">157040569</idno>
<idno type="IdUnivLorraine">[UL]100--</idno>
<orgName>Université de Lorraine</orgName>
<orgName type="acronym">UL</orgName>
<date type="start">2012-01-01</date>
<desc><address><addrLine>34 cours Léopold - CS 25233 - 54052 Nancy cedex</addrLine>
<country key="FR"></country>
</address>
<ref type="url">http://www.univ-lorraine.fr/</ref>
</desc>
</org>
</tutelle>
<tutelle name="UMR7503" 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>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Mezhirov, Ilya" sort="Mezhirov, Ilya" uniqKey="Mezhirov I" first="Ilya" last="Mezhirov">Ilya Mezhirov</name>
<affiliation><hal:affiliation type="laboratory" xml:id="struct-194149" status="VALID"><orgName>Technical University of Kaiserslautern</orgName>
<orgName type="acronym">TU Kaiserslautern</orgName>
<desc><address><addrLine>PO Box 3 049 67.653 Kaiserslautern</addrLine>
<country key="DE"></country>
</address>
</desc>
<listRelation><relation active="#struct-368320" type="direct"></relation>
</listRelation>
<tutelles><tutelle active="#struct-368320" type="direct"><org type="institution" xml:id="struct-368320" status="INCOMING"><orgName>Technical University of Kaiserslautern</orgName>
<desc><address><country key="FR"></country>
</address>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</affiliation>
</author>
<author><name sortKey="Shen, Alexander" sort="Shen, Alexander" uniqKey="Shen A" first="Alexander" last="Shen">Alexander Shen</name>
<affiliation><hal:affiliation type="researchteam" xml:id="struct-394760" status="VALID"><orgName>Systèmes complexes, automates et pavages</orgName>
<orgName type="acronym">ESCAPE</orgName>
<desc><address><country key="FR"></country>
</address>
<ref type="url">http://www.lirmm.fr/recherche/equipes/escape</ref>
</desc>
<listRelation><relation active="#struct-181" type="direct"></relation>
<relation name="UMR5506" active="#struct-410122" type="indirect"></relation>
<relation name="UMR5506" active="#struct-441569" type="indirect"></relation>
</listRelation>
<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>
<country key="FR"></country>
</address>
<ref type="url">http://www.lirmm.fr</ref>
</desc>
<listRelation><relation name="UMR5506" active="#struct-410122" type="direct"></relation>
<relation name="UMR5506" active="#struct-441569" type="direct"></relation>
</listRelation>
</org>
</tutelle>
<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>
</desc>
</org>
</tutelle>
<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>
</desc>
</org>
</tutelle>
</tutelles>
</hal:affiliation>
</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>
</series>
</biblStruct>
</sourceDesc>
</fileDesc>
<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>
</keywords>
</textClass>
</profileDesc>
</teiHeader>
<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>
</TEI>
<hal api="V3"><titleStmt><title xml:lang="en">A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points</title>
<author role="aut"><persName><forename type="first">Laurent</forename>
<surname>Bienvenu</surname>
</persName>
<email>laurent.bienvenu@liafa.jussieu.fr</email>
<idno type="halauthor">460891</idno>
<orgName ref="#struct-441569"></orgName>
<affiliation ref="#struct-1033"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Adam</forename>
<surname>Day</surname>
</persName>
<email></email>
<idno type="halauthor">661299</idno>
<orgName ref="#struct-300674"></orgName>
<affiliation ref="#struct-83772"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Mathieu</forename>
<surname>Hoyrup</surname>
</persName>
<email>Mathieu.Hoyrup@loria.fr</email>
<idno type="idhal">mathieu-hoyrup</idno>
<idno type="halauthor">601200</idno>
<orgName ref="#struct-300009"></orgName>
<affiliation ref="#struct-29797"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Ilya</forename>
<surname>Mezhirov</surname>
</persName>
<email></email>
<idno type="halauthor">661300</idno>
<orgName ref="#struct-368320"></orgName>
<affiliation ref="#struct-194149"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Alexander</forename>
<surname>Shen</surname>
</persName>
<email>alexander.shen@lirmm.fr</email>
<ptr type="url" target="www.lirmm.fr/~ashen"></ptr>
<idno type="halauthor">1251218</idno>
<orgName ref="#struct-181"></orgName>
<affiliation ref="#struct-394760"></affiliation>
</author>
<editor role="depositor"><persName><forename>Mathieu</forename>
<surname>Hoyrup</surname>
</persName>
<email>Mathieu.Hoyrup@loria.fr</email>
</editor>
</titleStmt>
<editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2011-11-22 14:20:24</date>
<date type="whenWritten">2012-01-31</date>
<date type="whenModified">2016-03-21 17:48:31</date>
<date type="whenReleased">2011-11-22 16:36:48</date>
<date type="whenProduced">2012</date>
<date type="whenEndEmbargoed">2011-11-22</date>
<ref type="file" target="https://hal.inria.fr/hal-00643629/document"><date notBefore="2011-11-22"></date>
</ref>
<ref type="file" subtype="author" n="1" target="https://hal.inria.fr/hal-00643629/file/ergodic-edited.pdf"><date notBefore="2011-11-22"></date>
</ref>
</edition>
<respStmt><resp>contributor</resp>
<name key="131574"><persName><forename>Mathieu</forename>
<surname>Hoyrup</surname>
</persName>
<email>Mathieu.Hoyrup@loria.fr</email>
</name>
</respStmt>
</editionStmt>
<publicationStmt><distributor>CCSD</distributor>
<idno type="halId">hal-00643629</idno>
<idno type="halUri">https://hal.inria.fr/hal-00643629</idno>
<idno type="halBibtex">bienvenu:hal-00643629</idno>
<idno type="halRefHtml">Information and Computation, Elsevier, 2012, 210, pp.021-030. <http://www.sciencedirect.com/science/article/pii/S0890540111001465>. <10.1016/j.ic.2011.10.006></idno>
<idno type="halRef">Information and Computation, Elsevier, 2012, 210, pp.021-030. <http://www.sciencedirect.com/science/article/pii/S0890540111001465>. <10.1016/j.ic.2011.10.006></idno>
</publicationStmt>
<seriesStmt><idno type="stamp" n="UNIV-PARIS7">Université Denis Diderot - Paris VII</idno>
<idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
<idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
<idno type="stamp" n="INSMI">CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions</idno>
<idno type="stamp" n="LORIA2">Publications du LORIA</idno>
<idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno>
<idno type="stamp" n="LORIA-FM" p="LORIA">Méthodes formelles</idno>
<idno type="stamp" n="LIAFA">Laboratoire d'Informatique Algorithmique, Fondements et Applications</idno>
<idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno>
<idno type="stamp" n="ESCAPE" p="LIRMM">Systèmes complexes, automates et pavages</idno>
<idno type="stamp" n="INRIA_TEST">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
<idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno>
<idno type="stamp" n="LIRMM">Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier</idno>
<idno type="stamp" n="INRIA2">INRIA 2</idno>
<idno type="stamp" n="TDS-MACS">Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes</idno>
</seriesStmt>
<notesStmt><note type="commentary">Improved version of the CiE'10 paper, with the strong form of Birkhoff's ergodic theorem for random points</note>
<note type="audience" n="2">International</note>
<note type="popular" n="0">No</note>
<note type="peer" n="1">Yes</note>
</notesStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en">A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points</title>
<author role="aut"><persName><forename type="first">Laurent</forename>
<surname>Bienvenu</surname>
</persName>
<email>laurent.bienvenu@liafa.jussieu.fr</email>
<idno type="halAuthorId">460891</idno>
<orgName ref="#struct-441569"></orgName>
<affiliation ref="#struct-1033"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Adam</forename>
<surname>Day</surname>
</persName>
<idno type="halAuthorId">661299</idno>
<orgName ref="#struct-300674"></orgName>
<affiliation ref="#struct-83772"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Mathieu</forename>
<surname>Hoyrup</surname>
</persName>
<email>Mathieu.Hoyrup@loria.fr</email>
<idno type="idHal">mathieu-hoyrup</idno>
<idno type="halAuthorId">601200</idno>
<orgName ref="#struct-300009"></orgName>
<affiliation ref="#struct-29797"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Ilya</forename>
<surname>Mezhirov</surname>
</persName>
<idno type="halAuthorId">661300</idno>
<orgName ref="#struct-368320"></orgName>
<affiliation ref="#struct-194149"></affiliation>
</author>
<author role="aut"><persName><forename type="first">Alexander</forename>
<surname>Shen</surname>
</persName>
<email>alexander.shen@lirmm.fr</email>
<ptr type="url" target="www.lirmm.fr/~ashen"></ptr>
<idno type="halAuthorId">1251218</idno>
<orgName ref="#struct-181"></orgName>
<affiliation ref="#struct-394760"></affiliation>
</author>
</analytic>
<monogr><idno type="halJournalId" status="VALID">14180</idno>
<idno type="issn">0890-5401</idno>
<idno type="eissn">1090-2651</idno>
<title level="j">Information and Computation</title>
<imprint><publisher>Elsevier</publisher>
<biblScope unit="volume">210</biblScope>
<biblScope unit="pp">021-030</biblScope>
<date type="datePub">2012</date>
</imprint>
</monogr>
<idno type="doi">10.1016/j.ic.2011.10.006</idno>
<ref type="publisher">http://www.sciencedirect.com/science/article/pii/S0890540111001465</ref>
</biblStruct>
</sourceDesc>
<profileDesc><langUsage><language ident="en">English</language>
</langUsage>
<textClass><keywords scheme="author"><term xml:lang="en">Algorithmic randomness</term>
<term xml:lang="en">Birkhoff's ergodic theorem</term>
<term xml:lang="en">Poincaré recurrence theorem</term>
<term xml:lang="en">Martin-Löf randomness</term>
</keywords>
<classCode scheme="halDomain" n="info.info-lo">Computer Science [cs]/Logic in Computer Science [cs.LO]</classCode>
<classCode scheme="halDomain" n="math.math-ds">Mathematics [math]/Dynamical Systems [math.DS]</classCode>
<classCode scheme="halDomain" n="math.math-pr">Mathematics [math]/Probability [math.PR]</classCode>
<classCode scheme="halTypology" n="ART">Journal articles</classCode>
</textClass>
<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.</abstract>
</profileDesc>
</hal>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Hal/Corpus
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 000632 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Hal/Corpus/biblio.hfd -nk 000632 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Lorraine |area= InforLorV4 |flux= Hal |étape= Corpus |type= RBID |clé= Hal:hal-00643629 |texte= A constructive version of Birkhoff's ergodic theorem for Martin-Lof random points }}
![]() | This area was generated with Dilib version V0.6.33. | ![]() |