Universal prediction and universal coding

Although prediction schemes which are named \universal" are now abundant, very little has been addressed as to the de nition of universal prediction. This paper addresses for -nary ( 2) sequences the criteria of successful universal prediction and the prediction schemes which achieve the goals. We propose the following criteria: for any probability measures in a given measure class, the error probability of prediction (deterministic prediction) and the conditional probability of the next outcome given the past sequence (stochastic prediction) should converge to the optimal values in probability (weakly universal) and almost surely (strongly universal). We prove several properties with respect to the criteria in which a novel proof for Cover's open problem, which seems to be more simpli ed and intuitively appealing compared to the previous proofs, is presented. The proposed criteria are derived from an analogy with Davisson's universal coding, just like Feder, Merhav, and Gutman's scheme was inspired by Ziv and Lempel's universal coding scheme. At the same time, we explore the connection between universal prediction and universal coding. The measure class which we are considering is not con ned to the class of stationary ergodic measures: the criteria deal with general measure classes including the class of computable measures.