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 $P_{\theta }$, where $\theta$ is a real parameter, using algorithmic randomness theory. In the case when for a computable probability distribution $P_{\theta }$ an effectively strongly consistent estimate exists, we show that the Levin's a priory semicomputable semimeasure of the set of all $P_{\theta }$-random sequences is positive if and only if the parameter $\theta$ is a computable real number. The different methods for generating ``meaningful'' $P_{\theta }$-random sequences with noncomputable $\theta$ are discussed.

14 pages