Geodesic
On the behaviour of the generalized Вayes estimates for Markov observations
Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 340-354

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We consider the behaviour of the generalized Bayes estimates $\widehat\theta_n$ of an unknown parameter $\theta$ assuming that the observations $X_1,\dots,X_n,\dots$ form a transient Markov chain. It is shown that, under some conditions, the limit distribution of the sequence $\sqrt n(\widehat\theta_n-\theta)$ is a weighted normal law. The main restriction is that there must exist a.s. nonzero limit of the $I(X_n)$, where $I(X)$ is the (conditioned) Fiser information. Some examples show that, if this restriction is not satisfied, the estimation problem becomes “irregular”. 
@article{TVP_1974_19_2_a7,
     author = {B. Ya. Levit},
     title = {On the behaviour of the generalized {{\CYRV}ayes} estimates for {Markov} observations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {340--354},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1974_19_2_a7/}
}
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B. Ya. Levit. On the behaviour of the generalized Вayes estimates for Markov observations. Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 340-354. http://geodesic.mathdoc.fr/item/TVP_1974_19_2_a7/