eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2012-02-24
567
576
10.4230/LIPIcs.STACS.2012.567
article
The dimension of ergodic random sequences
Hoyrup, Mathieu
Let m be a computable ergodic shift-invariant measure over the set of infinite binary sequences. Providing a constructive proof of Shannon-McMillan-Breiman theorem, V'yugin proved that if x is a Martin-Löf 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 Martin-Löf 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):2135--2149, 2009], using rather different considerations.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol014-stacs2012/LIPIcs.STACS.2012.567/LIPIcs.STACS.2012.567.pdf
Shannon-McMillan-Breiman theorem
Martin-Löf random sequence
effective Hausdorff dimension
compression rate
entropy