On empirical meaning of randomness with respect to a real parameter

Abstract

We study the empirical meaning of randomness with respect to a family of probability distributions Pθ, where θ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution Pθ an effectively strongly consistent estimate exists, we show that the Levin's a priory semicomputable semimeasure of the set of all Pθ-random sequences is positive if and only if the parameter θ is a computable real number. The different methods for generating "meaningful" Pθ-random sequences with noncomputable θ are discussed. © Springer-Verlag Berlin Heidelberg 2007.