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

TL;DR

This paper explores the relationship between stationary imprecise and non-stationary precise uncertainty models in algorithmic randomness, revealing surprising equivalences and discussing implications for statistical applications.

Contribution

It establishes a novel connection showing that certain stationary imprecise models and non-computable non-stationary precise models share the same set of random sequences.

Findings

Stationary imprecise models and non-computable non-stationary precise models have identical random sequences.

The study highlights the potential practical relevance of imprecise and non-computable models in statistical contexts.

It opens new avenues for understanding the impact of imprecision and computability in randomness theory.

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.