The dimension of ergodic random sequences - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2011

The dimension of ergodic random sequences

Abstract

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.
Fichier principal
Vignette du fichier
paperbiblio.pdf (123.71 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

inria-00606457 , version 1 (06-07-2011)
inria-00606457 , version 2 (06-07-2011)
inria-00606457 , version 3 (11-07-2011)
inria-00606457 , version 4 (21-07-2011)

Identifiers

  • HAL Id : inria-00606457 , version 4
  • ARXIV : 1107.1149

Cite

Mathieu Hoyrup. The dimension of ergodic random sequences. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.567-576. ⟨inria-00606457v4⟩
182 View
209 Download

Altmetric

Share

Gmail Facebook X LinkedIn More