Randomness Criteria in Terms of ${\alpha }$ -Divergences

Vovk´s randomness criterion characterizes sequences that are random relative to two distinct computable probability measures. The uniqueness of the criterion lies in the fact that, unlike the standard criterion based on the likelihood ratio test, it is expressed in terms of a geometrical quantity, the Hellinger distance, on the space of probability measures. In this paper, we generalize the randomness criterion to a wider class of geometrical quantities, the -divergences with . The nonextendibility of the criterion across the boundaries is investigated in connection with the likelihood ratio test and information geometry.