Taylor series expansions for the entropy rate of Hidden Markov Processes

Finding the entropy rate of Hidden Markov Processes is an active research topic, of both theoretical and practical importance. A recently used approach is studying the asymptotic behavior of the entropy rate in various regimes. In this paper we generalize and prove a previous conjecture relating the entropy rate to entropies of finite systems. We use the proof to establish series expansions for the entropy rate in two different regimes. We also study the radius of convergence of the two series expansions.