125 research outputs found
Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness
We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr
randomness (which we will call uniformly relative Schnorr randomness). An
immediate corollary is one direction of van Lambalgen's theorem for Schnorr
randomness. It has been claimed in the literature that this corollary (and the
analogous result for computable randomness) is a "straightforward modification
of the proof of van Lambalgen's Theorem." This is not so, and we point out why.
We also point out an error in Miyabe's proof of van Lambalgen's Theorem for
truth-table reducible randomness (which we will call uniformly relative
computable randomness). While we do not fix the error, we do prove a weaker
version of van Lambalgen's Theorem where each half is computably random
uniformly relative to the other
Randomness on computable probability spaces - A dynamical point of view
We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff’s pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications
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