Abstract
We study the a priori semimeasure of sets of P θ -random infinite sequences, where P θ is a family of probability distributions depending on a real parameter θ. In the case when for a computable probability distribution P θ an effectively strictly consistent estimator exists, we show that Levin’s a priory semimeasure of the set of all P θ -random sequences is positive if and only if the parameter θ is a computable real number. We show that the a priory semimeasure of the set \(\bigcup_{\theta}I_{\theta}\), where I θ is the set of all P θ -random sequences and the union is taken over all algorithmically non-random θ, is positive.
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V’yugin, V. On Empirical Meaning of Randomness with Respect to Parametric Families of Probability Distributions. Theory Comput Syst 50, 296–312 (2012). https://doi.org/10.1007/s00224-010-9300-9
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DOI: https://doi.org/10.1007/s00224-010-9300-9