On the behaviour of the generalized Вayes estimates for Markov observations

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”.