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An effective ergodic theorem and some applications

Published:17 May 2008Publication History

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.

References

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  4. 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 ScholarGoogle ScholarDigital LibraryDigital Library
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          cover image ACM Conferences
          STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
          May 2008
          712 pages
          ISBN:9781605580470
          DOI:10.1145/1374376

          Copyright © 2008 ACM

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          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 17 May 2008

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          Acceptance Rates

          STOC '08 Paper Acceptance Rate80of325submissions,25%Overall Acceptance Rate1,469of4,586submissions,32%

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