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A hidden Markov model (HMM) is said to have path-mergeable states if for anytwo states i,j there exists a word w and state k such that it is possible totransition from both i and j to k while emitting w. We show that for a finiteHMM with path-mergeable states the block estimates of the entropy rate convergeexponentially fast. We also show that the path-mergeability property isasymptotically typical in the space of HMM topolgies and easily testable.

Exponential Bounds for Convergence of Entropy Rate Approximations in Hidden Markov Models Satisfying a Path-Mergeability Condition