Approximating a sequence of observations by a simple process

Given a sequence (s0; s1,..., sN) of observations from a finite set S, we construct a process (sn)n_<N that satisfies the following properties: (i) (Sn)n_< ·N is a piecewise Markov chain, (ii) the conditional distribution of sn given S0,...,Sn-1 is close to the empirical transition given by the observed sequence, for most n's, (iii) under (Sn)n_<N, with high probability the empirical frequency of the realized sequence is close to the one given by the observed sequence. We generalize this result to the case that the conditional distribution of Sn given S0,...,Sn-1 is required to be in some polyhedron Vs n-1.