A lower bound on the performance of sequential prediction

We consider the problem of sequential linear prediction of real-valued sequences under the square-error loss function. For this problem, a prediction algorithm has been demonstrated whose accumulated squared prediction error, for every bounded sequenee, is asymptotically as small as the best fixed linear predictor for that sequence, taken from the class of all linear predictors of a given order p. The redundancy, or excess prediction error above that of the best predictor for that sequence, is upper bounded by A²p ln(n)/n, where n is the data length and the sequence is assumed to be bounded by some A. In this paper, we show that this predictor is optimal in a min-max sense, by deriving a corresponding lower bound, such that no sequential predictor can ever do better than a redundancy of A²p ln(n)/n.