Asymptotically minimax regret by Bayes mixtures

We study the problem of data compression, gambling and prediction of a sequence x/sup n/ = x/sub 1/x/sub 2/...x/sub n/ from a certain alphabet X, in terms of regret (Shtarkov 1988) and redundancy with respect to a general exponential family, a general smooth family, and also Markov sources. In particular, we show that variants of Jeffreys mixture asymptotically achieve their minimax values.