Minimax noiseless universal coding for Markov sources

Upper and lower bounds are presented on the minimax redundancy for Markov source noiseless block-to-variable universal coding. The number of states, block size, and Markov order are arbitrary but finite. Unlike earlier results, the upper and lower bounds are absolute, that is, not merely asymptotic. The upper bound is established by combinatorial bounds. The lower bound is established by a bound on the average redundancy for an arbitrary distribution on the transition probabilities. The bound can be optimized across the choice of distribution.