On Empirical Meaning of Randomness with Respect to a Real Parameter

摘要

We study the empirical meaning of randomness with respect to a family of probability distributions P_θ, where θ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution P_θ an effectively strongly consistent estimate exists, we show that the Levin's a priory semicomputable semimeasure of the set of all P_θ-random sequences is positive if and only if the parameter θ is a computable real number. The different methods for generating "meaningful" P_θ-random sequences with noncomputable θ are discussed.