An introduction to Kolmogorov complexity and its applications
An introduction to Kolmogorov complexity and its applications
Computable analysis: an introduction
Computable analysis: an introduction
Random elements in effective topological spaces with measure
Information and Computation
Mathematical Structures in Computer Science
Uniform test of algorithmic randomness over a general space
Theoretical Computer Science
Admissible Representations of Probability Measures
Electronic Notes in Theoretical Computer Science (ENTCS)
Dynamics of a generic Brownian motion: Recursive aspects
Theoretical Computer Science
An effective ergodic theorem and some applications
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The Law of the Iterated Logarithm for Algorithmically Random Brownian Motion
LFCS '07 Proceedings of the international symposium on Logical Foundations of Computer Science
A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties
Theoretical Computer Science
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
An Application of Martin-Löf Randomness to Effective Probability Theory
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Effective symbolic dynamics, random points, statistical behavior, complexity and entropy
Information and Computation
Randomness and the ergodic decomposition
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
A constructive version of Birkhoff's ergodic theorem for Martin-Löf random points
Information and Computation
Von Neumann's Biased Coin Revisited
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
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We pursue the study of the framework of layerwise computability introduced in a preceding paper and give three applications. (i) We prove a general version of Birkhoff's ergodic theorem for random points, where the transformation and the observable are supposed to be effectively measurable instead of computable . This result significantly improves V'yugin and Nandakumar's ones. (ii) We provide a general framework for deriving sharper theorems for random points, sensitive to the speed of convergence. This offers a systematic approach to obtain results in the spirit of Davie's ones. (iii) Proving an effective version of Prokhorov theorem, we positively answer a question recently raised by Fouché: can random Brownian paths reach any random number? All this shows that layerwise computability is a powerful framework to study Martin-Löf randomness, with a wide range of applications.