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