Imprecision in martingale- and test-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 on this prequential approach, based on ideas borrowed from our earlier imprecise-probabilistic, standard 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, both in a martingale-theoretic and a test-theoretic sense. We prove that these two versions of prequential randomness coincide, we compare the resulting prequential randomness notions with the more standard ones, and we investigate where the prequential and standard randomness notions coincide.