Analyticity of Entropy Rate in Families of Hidden Markov Chains (II)

We give relaxed sufficient conditions (compared to D. Blackwell (1957)) for analyticity of the entropy rate of a hidden Markov chain. Several special cases of the relaxed conditions are discussed. A general principle to calculate the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. Finally, we prove a "stabilizing" property of "black hole" case, which suggests that one can explicitly compute the derivatives and obtain an explicit Taylor series in certain cases, generalizing the results in O. Zuk et al. (2004)