Twice-Universal Coding

Assume that $A$ is a finite alphabet; $\Omega_i$ is a set of Markov sources of connectedness i that generate letters from $A$ ($i=1,2,\dots$) and $\Omega_0$ is a set of Bernoulli sources. A code is proposed whose redundancy as a function of the block length on each $\Omega_i$ is asymptotically as small as that of the universal code that is optimal on $\Omega_i$ ($i=0,1,2\dots)$. A generalization of this problem to the case of an arbitrary countable family of sets of stationary ergodic sources is considered.