On Empirical Meaning of Randomness with Respect to a Real Parameter

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.

• Type: Conference
• Year: 2007
• Where: CSR
• Authors: Vladimir V. V'yugin
• Keywords: Algorithmic Randomness Theory, Computable Probability Distribution, CSR 2007, Probability Distribution, Theoretical Computer Science