On the Complexity of Real Functions
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Uniform test of algorithmic randomness over a general space
Theoretical Computer Science
Applications of Effective Probability Theory to Martin-Löf Randomness
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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
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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.