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An effective ergodic theorem and some applications

Published: 17 May 2008 Publication 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

[1]
P. Billingsley. Ergodic theory and information. John Wiley and Sons, New York, 1965.
[2]
G. D. Birkhoff. Proof of the ergodic theorem. Proceedings of the National Academy of Sciences, USA, 17:656--660, 1931.
[3]
E. Bishop. Foundations of constructive analysis. McGraw Hill, New York, 1967.
[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.
[5]
K. Dajani and C. Kraaikamp. Ergodic Theory of Numbers. Mathematical Association of America, 2004.
[6]
P. Gacs. Unifrom test of algorithmic randomness over a general space. Theoretical Computer Science, 341:91--137, 2005.
[7]
P. Martin--Lof. The de^Lnition of random sequences. Information and C

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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
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 May 2008

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Author Tags

  1. algorithmic randomness
  2. ergodic theorem

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  • Research-article

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STOC '08
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STOC '08: Symposium on Theory of Computing
May 17 - 20, 2008
British Columbia, Victoria, Canada

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STOC '08 Paper Acceptance Rate 80 of 325 submissions, 25%;
Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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STOC '25
57th Annual ACM Symposium on Theory of Computing (STOC 2025)
June 23 - 27, 2025
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  • (2022)On continued fraction randomness and normalityInformation and Computation10.1016/j.ic.2022.104876285(104876)Online publication date: May-2022
  • (2022)Ergodic Theorems and Converses for PSPACE FunctionsTheory of Computing Systems10.1007/s00224-022-10094-967:3(491-520)Online publication date: 30-Aug-2022
  • (2019)Martin-Löf random quantum statesJournal of Mathematical Physics10.1063/1.509466060:9(092201)Online publication date: Sep-2019
  • (2016)On Stability of Probability Laws with Respect to Small Violations of Algorithmic RandomnessTheory of Computing Systems10.1007/s00224-015-9632-658:3(403-423)Online publication date: 1-Apr-2016
  • (2011)Randomness and the ergodic decompositionProceedings of the 7th conference on Models of computation in context: computability in Europe10.5555/2040947.2040960(122-131)Online publication date: 27-Jun-2011
  • (2010)Effective symbolic dynamics, random points, statistical behavior, complexity and entropyInformation and Computation10.1016/j.ic.2009.05.001208:1(23-41)Online publication date: 1-Jan-2010

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