A Note on Sequence Prediction over Large Alphabets

Building on results from data compression, we prove nearly tight bounds on how well sequences of length n can be predicted in terms of the size σ of the alphabet and the length k of the context considered when making predictions. We compare the performance achievable by an adaptive predictor with no advance knowledge of the sequence, to the performance achievable by the optimal static predictor using a table listing the frequency of each (k + 1)-tuple in the sequence. We show that, if the elements of the sequence are chosen uniformly at random, then an adaptive predictor can compete in the expected case if k ≤ logσ n− 3− , for a constant > 0, but not if k ≥ logσ n.