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Computability of probability measures and Martin-Löf randomness over metric spaces
TLDR
In this paper, we investigate algorithmic randomness on more general spaces than the Cantor space in a computable and measure-theoretic sense. Expand
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Randomness on Computable Probability Spaces—A Dynamical Point of View
TLDR
We extend Schnorr randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Expand
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Applications of Effective Probability Theory to Martin-Löf Randomness
TLDR
We pursue the study of the framework of layerwise computability introduced in a preceding paper and give three applications to Martin-Lof randomness. Expand
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Algorithmic tests and randomness with respect to a class of measures
TLDR
This paper offers some new results on randomness with respect to classes of measures, along with a didactic exposition of their context based on results that appeared elsewhere. Expand
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An Application of Martin-Löf Randomness to Effective Probability Theory
TLDR
We introduce and study the framework of layerwise computability which lies on Martin-Lof randomness and the existence of a universal randomness test. Expand
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A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points
TLDR
We prove the effective version of [email protected]s ergodic theorem for Martin-Lof random points and effectively open sets, improving the results previously obtained in this direction (in particular those of Vyugin and Hoyrup, Rojas). Expand
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Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
TLDR
A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system. Expand
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A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties
TLDR
In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets A" i with effectively summable measures, there are computable points which are not contained in infinitely many A"i. Expand
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The dimension of ergodic random sequences
  • M. Hoyrup
  • Mathematics, Computer Science
  • STACS
  • 6 July 2011
TLDR
We prove the strong law of large numbers extends to Birkhoff's ergodic theorem for Martin-Löf random sequences. Expand
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Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
TLDR
We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sinai entropy of the system. Expand
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