In this paper we prove rates of uniform strong convergence, convergence rates of the mean square error and the asymptotic normality of the kernel estimator for the transition density of a geometrically ergodic Markov chain. The assumptions on the Markov chain are closely related to absolute regularity. We allow the initial distribution of the Markov chain to be arbitrary.

We consider Hidden Markov Chains obtained by passing a Markov Chain with rare transitions through a noisy memoryless channel. We obtain asymptotic estimates for the entropy of the resulting Hidden Markov Chain as the transition rate is reduced to zero. Let (Xn) be a Markov chain with finite state space S and transition matrix P (p) and let (Yn) be the Hidden Markov chain observed by passing (Xn...