The dimension of ergodic random sequences

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-Lof 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-Lof 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.

[1]  Aaron D. Wyner,et al.  Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression , 1989, IEEE Trans. Inf. Theory.

[2]  M. Hochman.  Upcrossing inequalities for stationary sequences and applications. , 2006, math/0608311.

[3]  Meir Smorodinsky,et al.  Ergodic Theory Entropy , 1971 .

[4]  Masahiro Nakamura.  Ergodic theorems for algorithmically random sequences , 2005, IEEE Information Theory Workshop, 2005..

[5]  Péter Gács,et al.  Exact Expressions for Some Randomness Tests , 1979, Math. Log. Q..

[6]  Benjamin Weiss,et al.  Entropy and data compression schemes , 1993, IEEE Trans. Inf. Theory.

[7]  Jack H. Lutz.  Effective fractal dimensions , 2005, Math. Log. Q..

[8]  Alexander Shen,et al.  Ergodic-Type Characterizations of Algorithmic Randomness , 2010, CiE.

[9]  Elvira Mayordomo,et al.  A Kolmogorov complexity characterization of constructive Hausdorff dimension , 2002, Inf. Process. Lett..

[10]  Jack H. Lutz,et al.  Effective Strong Dimension, Algorithmic Information, and Computational Complexity , 2002, ArXiv.

[11]  V. V'yugin.  Ergodic Theorems for Individual Random Sequences , 1998, Theor. Comput. Sci..

[12]  Jack H. Lutz,et al.  Dimension in complexity classes , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[13]  Mathieu Hoyrup,et al.  A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points , 2010, Inf. Comput..

[14]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[15]  Jack H. Lutz,et al.  Gales and the Constructive Dimension of Individual Sequences , 2000, ICALP.

[16]  Johanna N. Y. Franklin,et al.  Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets , 2012 .

[17]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.

[18]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[19]  C. Caramanis.  What is ergodic theory , 1963 .

[20]  Claus-Peter Schnorr,et al.  The process complexity and effective random tests. , 1972, STOC.

[21]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.