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Martin-Löf randomness and Galton–Watson processes

David Diamondstone & Bjørn Kjos-Hanssen

Annals of Pure and Applied Logic 163 (5):519-529 (2012)

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Effectively closed sets of measures and randomness.Jan Reimann - 2008 - Annals of Pure and Applied Logic 156 (1):170-182.details We show that if a real x2ω is strongly Hausdorff -random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective continuous transformations and a basis theorem for -classes applied to closed sets of probability measures. We use the main result to derive a (...) Download Export citation Bookmark 9 citations
Lowness for Kurtz randomness.Noam Greenberg & Joseph S. Miller - 2009 - Journal of Symbolic Logic 74 (2):665-678.details We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity. We also consider Low(M, Kurtz), the class of degrees a such that every element of M is a-Kurtz random. These are characterised when M is the class of Martin-Löf random, computably random, or Schnorr random reals. (...) Download Export citation Bookmark 9 citations

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