Redundancy of the MTF scheme for stationary ergodic sources

In this paper, upper and lower bounds are derived for the redundancy of the move-to-front (MTF) scheme without the symbol extension for stationary ergodic sources and Markov sources. It is proved that for the stationary ergodic K-th order Markov sources, the MTF scheme can attain the entropy rate if and only if K=1 and the transition matrix of the source is a kind of doubly-stochastic matrix.