Applications of Effective Probability Theory to Martin-Löf Randomness
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.
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