On the (dis)similarities between stationary imprecise and non-stationary precise uncertainty models in algorithmic randomness

Abstract

The field of algorithmic randomness studies what it means for infinite binary sequences to be random for some given uncertainty model. Classically, martingale-theoretic notions of such randomness involve precise uncertainty models, and it is only recently that imprecision has been introduced into this context. As a consequence, the investigation into how imprecision alters our view on martingale-theoretic random sequences has only just begun. In this contribution, where we allow for non-computable uncertainty models, we establish a close and surprising connection between precise and imprecise uncertainty models in this randomness context. In particular, we show that there are stationary imprecise models and non-computable non-stationary precise models that have the exact same set of random sequences. We also give a preliminary discussion of the possible implications of our result for a statistics based on imprecise probabilities, and shed some light on the practical (ir)relevance of both imprecise and non-computable precise uncertainty models in that context.