Uniform test of algorithmic randomness over a general space
Theoretical Computer Science
An effective ergodic theorem and some applications
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Computability of probability measures and Martin-Löf randomness over metric spaces
Information and Computation
Applications of Effective Probability Theory to Martin-Löf Randomness
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
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
Ergodic-type characterizations of algorithmic randomness
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
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We prove the effective version of Birkhoff@?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, Nandakumar and Hoyrup, Rojas). The proof consists of two steps. First, we prove a generalization of Kucera@?s theorem, which is a particular case of effective ergodic theorem: a trajectory of a computable ergodic mapping that starts from a random point cannot remain inside an effectively open set of measure less than 1. Second, we show that the full statement of the effective ergodic theorem can be reduced to this special case. Both steps use the statement of classical ergodic theorem but not its usual classical proof. Therefore, we get a new simple proof of the effective ergodic theorem (with weaker assumptions than before). This result was recently obtained independently by Franklin, Greenberg, Miller and Ng.