Universal Coding and Prediction on Martin-Löf Random Points
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We perform an effectivization of classical results concerning universal coding and prediction for stationary ergodic processes over an arbitrary finite alphabet. That is, we lift the well-known almost sure statements to statements about Martin-Löf random sequences. Most of this work is quite mechanical but, by the way, we complete a result of Ryabko from 2008 by showing that each universal probability measure in the sense of universal coding induces a universal predictor in the prequential sense. Surprisingly, the effectivization of this implication holds true provided the universal measure does not ascribe too low conditional probabilities to individual symbols. As an example, we show that the Prediction by Partial Matching (PPM) measure satisfies this requirement. In the almost sure setting, the requirement is superfluous.
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