Randomness and imprecision: from supermartingales to randomness tests

TL;DR

This paper extends classical randomness tests to interval-valued forecasts, establishing their equivalence with martingale-based notions and characterizing them via universal supermartingales and tests.

Contribution

It generalizes randomness test definitions to interval forecasts and links them to Levin's uniform randomness, unifying different approaches under computability conditions.

Findings

Interval-valued forecasts can be incorporated into randomness tests.

The generalized tests are equivalent to martingale-based notions.

Universal supermartingales characterize the new randomness notions.

Abstract

We generalise the randomness test definitions in the literature for both the Martin-L\"of and Schnorr randomness of a series of binary outcomes, in order to allow for interval-valued rather than merely precise forecasts for these outcomes, and prove that under some computability conditions on the forecasts, our definition of Martin-L\"of test randomness can be seen as a special case of Levin's uniform randomness. We show that the resulting randomness notions are, under some computability and non-degeneracy conditions on the forecasts, equivalent to the martingale-theoretic versions we introduced in earlier papers. In addition, we prove that our generalised notion of Martin-L\"of randomness can be characterised by universal supermartingales and universal randomness tests.