We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and Tweedie, and from estimates using coupling, although we start from essentially the same assumptions of a drift condition toward a "small set." The estimates show a noticeable improvement on existing results if the Markov chain is reversible with respect to its stationary distribution, and especially so if the chain is also positive. The method of proof uses the first-entrancelast-exit decomposition, together with new quantitative versions of a result of Kendall from discrete renewal theory.
Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the non-ergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure. S = a 1 1 , . . . , a 1 n1 S1 , . . . , a m 1 , . . . , a m nm Sm , where the subalphabets S 1 , . . . , S m are noncommunicating in the sense that for any i = j and t ≥ sSo, unless m = 1, X t is a compound Markov chain with the transition intensities *
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