Randomness and imprecision: from supermartingales to randomness tests

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.