Notre Dame Journal of Formal Logic

Uncomputably Noisy Ergodic Limits

Abstract

V’yugin has shown that there are a computable shift-invariant measure on $2^{\mathbb{N}}$ and a simple function $f$ such that there is no computable bound on the rate of convergence of the ergodic averages $A_{n}f$. 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 $\varepsilon$.

Article information

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

Dates
First available in Project Euclid: 25 September 2012

https://projecteuclid.org/euclid.ndjfl/1348577704

Digital Object Identifier
doi:10.1215/00294527-1716757

Mathematical Reviews number (MathSciNet)
MR2981012

Zentralblatt MATH identifier
1258.03084

Citation

Avigad, Jeremy. Uncomputably Noisy Ergodic Limits. Notre Dame J. Formal Logic 53 (2012), no. 3, 347--350. doi:10.1215/00294527-1716757. https://projecteuclid.org/euclid.ndjfl/1348577704

References

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• [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.