ABSTRACT
This work is a synthesis of recent advances in computable analysis with the theory of algorithmic randomness. In this theory, we try to strengthen probabilistic laws, i.e., laws which hold with probability 1, to laws which hold in their pointwise effective form - i.e., laws which hold for every individual constructively random point. In a tour-de-force, V'yugin proved an effective version of the Ergodic Theorem which holds when the probability space, the transformation and the random variable are computable. However, V'yugin's Theorem cannot be directly applied to many examples, because all computable functions are continuous, and many applications use discontinuous functions.
We prove a stronger effective ergodic theorem to include a restriction of Braverman's "graph-computable functions". We then use this to give effective ergodic proofs of the effective versions of Levy-Kuzmin and Khinchin Theorems relating to continued fractions.
- P. Billingsley. Ergodic theory and information. John Wiley and Sons, New York, 1965.Google Scholar
- G. D. Birkhoff. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences, USA, 17:656--660, 1931.Google Scholar
Cross Ref
- E. Bishop. Foundations of constructive analysis. McGraw Hill, New York, 1967.Google Scholar
- M. Braverman. On the complexity of real functions. In Proceedings of the forty sixth IEEE Annual Symposium on the Foundations of Computer Science, pages 155--164, 2005. Google Scholar
Digital Library
- K. Dajani and C. Kraaikamp. Ergodic Theory of Numbers. Mathematical Association of America, 2004.Google Scholar
- P. Gacs. Unifrom test of algorithmic randomness over a general space. Theoretical Computer Science, 341:91--137, 2005. Google Scholar
Digital Library
- P. Martin--Lof. The de^Lnition of random sequences. Information and CGoogle Scholar
Index Terms
An effective ergodic theorem and some applications
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