A Limit Theorem for Partially Observed Markov Chains

Abstract

Let $\{X_n, n = 1, 2, \cdots\}$ be a Markov chain with finite state space $S = \{1, 2, \cdots, d\}$, transition probability matrix $P$ and initial distribution $p$. Let $g$ be a function with $S$ as domain and define $Y_n = g(X_n)$. Define \begin{align*}Z_n^i &= \operatorname{Pr}\lbrack X_n = i \mid Y_1, Y_2, \cdots, Y_n \rbrack, \\ Z_n &= (Z_n^1, Z_n^2, \cdots, Z_n^d),\end{align*} and let $\mu_n$ denote the probability distribution of the vector $Z_n$. In this paper we prove that if $\{X_n, n = 1, 2, \cdots\}$ is ergodic and if $P$ and $g$ satisfy a certain condition then $\mu_n$ converges to a limit and this limit is independent of the initial distribution $p$.