When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2009.1828
URN: urn:nbn:de:0030-drops-18280
URL: http://drops.dagstuhl.de/opus/volltexte/2009/1828/
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### Randomness on Computable Probability Spaces - A Dynamical Point of View

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### Abstract

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a \emph{dynamical} notion of randomness: typicality. Roughly, a point is \emph{typical} for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every \emph{mixing} computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

### BibTeX - Entry

@InProceedings{gacs_et_al:LIPIcs:2009:1828,
author =	{Peter Gacs and Mathieu Hoyrup and Cristobal Rojas},
title =	{{Randomness on Computable Probability Spaces - A Dynamical Point of View}},
booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
pages =	{469--480},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-09-5},
ISSN =	{1868-8969},
year =	{2009},
volume =	{3},
editor =	{Susanne Albers and Jean-Yves Marion},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/1828},
URN =		{urn:nbn:de:0030-drops-18280},
doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1828},
annote =	{Keywords: Schnorr randomness, Birkhoff's ergodic theorem, Computable measures}
}


 Keywords: Schnorr randomness, Birkhoff's ergodic theorem, Computable measures Seminar: 26th International Symposium on Theoretical Aspects of Computer Science Issue Date: 2009 Date of publication: 19.02.2009

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