Imprecision in Martingale-Theoretic Prequential Randomness

Abstract: In a prequential approach to algorithmic randomness, probabilities for the next outcome can be forecast `on the fly' without the need for fully specifying a probability measure on all possible sequences of outcomes, as is the case in the more standard approach. We take the first steps in allowing for probability intervals instead of precise probabilities in this prequential approach, based on ideas from our earlier imprecise-probabilistic and martingale-theoretic account of algorithmic randomness. We define what it means for an infinite sequence $(I_1,x_1,I_2,x_2,\dots)$ of successive interval forecasts $I_k$ and subsequent binary outcomes $x_k$ to be random. We compare the resulting prequential randomness notion with the more standard one, and investigate where both randomness notions coincide, as well as where their properties correspond.