How much randomness is needed for statistics?

Abstract

In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure $\lambda$, a choice needs to be made. One approach is to allow randomness tests to access the measure $\lambda$ as an oracle (which we call the "classical approach"). The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in $\lambda$ (we call this approach "Hippocratic"). While the Hippocratic approach is in general much more restrictive, there are cases where the two coincide. The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure $\lambda$ is a Bernoulli measure, classical randomness and Hippocratic randomness coincide. In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.

Preliminary version in: Computability in Europe, Lecture Notes in Computer Science 7318, Springer, Berlin, 2012, 395--404