Hoyrup, Mathieu
The dimension of ergodic random sequences
Abstract
Let m be a computable ergodic shiftinvariant measure over the set of infinite binary sequences. Providing a constructive proof of ShannonMcMillanBreiman theorem, V'yugin proved that if x is a MartinLöf random binary sequence w.r.t. m then its strong effective dimension Dim(x) equals the entropy of m. Whether its effective dimension dim(x) also equals the entropy was left as an open problem. In this paper we settle this problem, providing a positive answer. A key step in the proof consists in extending recent results on Birkhoff's ergodic theorem for MartinLöf random sequences. At the same time, we present extensions of some previous results.
As pointed out by a referee the main result can also be derived from results by Hochman [Upcrossing inequalities for stationary sequences and applications. The Annals of Probability, 37(6):21352149, 2009], using rather different considerations.
BibTeX  Entry
@InProceedings{hoyrup:LIPIcs:2012:3391,
author = {Mathieu Hoyrup},
title = {{The dimension of ergodic random sequences}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {567576},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897354},
ISSN = {18688969},
year = {2012},
volume = {14},
editor = {Christoph D{\"u}rr and Thomas Wilke},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2012/3391},
URN = {urn:nbn:de:0030drops33917},
doi = {http://dx.doi.org/10.4230/LIPIcs.STACS.2012.567},
annote = {Keywords: ShannonMcMillanBreiman theorem, MartinL{\"o}f random sequence, effective Hausdorff dimension, compression rate, entropy }
}
2012
Keywords: 

ShannonMcMillanBreiman theorem, MartinLöf random sequence, effective Hausdorff dimension, compression rate, entropy 
Seminar: 

29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Related Scholarly Article: 


Issue date: 

2012 
Date of publication: 

2012 