Notre Dame Journal of Formal Logic

Uncomputably Noisy Ergodic Limits

Jeremy Avigad


V’yugin has shown that there are a computable shift-invariant measure on 2N and a simple function f such that there is no computable bound on the rate of convergence of the ergodic averages Anf. Here it is shown that in fact one can construct an example with the property that there is no computable bound on the complexity of the limit; that is, there is no computable bound on how complex a simple function needs to be to approximate the limit to within a given ε.

Article information

Notre Dame J. Formal Logic, Volume 53, Number 3 (2012), 347-350.

First available in Project Euclid: 25 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]
Secondary: 37A25: Ergodicity, mixing, rates of mixing

ergodic theorems computable analysis


Avigad, Jeremy. Uncomputably Noisy Ergodic Limits. Notre Dame J. Formal Logic 53 (2012), no. 3, 347--350. doi:10.1215/00294527-1716757.

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  • [1] Hoyrup, M., and C. Rojas, “Computability of probability measures and Martin-Löf randomness over metric spaces,” Information and Computation, vol. 207 (2009), pp. 830–847.
  • [2] V’yugin, V. V., “Ergodic convergence in probability, and an ergodic theorem for individual random sequences,” Rossiĭskaya Akademiya Nauk: Teoriya Veroyatnosteĭ i ee Primeneniya, vol. 42 (1997), pp. 39–50.
  • [3] V’yugin, V. V., “Ergodic theorems for individual random sequences,” Theoretical Computer Science, vol. 207 (1998), pp. 343–361.
  • [4] Weihrauch, K., “Computability on the probability measures on the Borel sets of the unit interval,” Theoretical Computer Science, vol. 219 (1999), pp. 421–437.