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August, 1975 A Limit Theorem for Partially Observed Markov Chains
Thomas Kaijser
Ann. Probab. 3(4): 677-696 (August, 1975). DOI: 10.1214/aop/1176996308


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 $p$.


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Thomas Kaijser. "A Limit Theorem for Partially Observed Markov Chains." Ann. Probab. 3 (4) 677 - 696, August, 1975.


Published: August, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0315.60038
MathSciNet: MR383536
Digital Object Identifier: 10.1214/aop/1176996308

Primary: 60J10
Secondary: 60F99 , 60J05

Keywords: Partially observed Markov chains , Products of random matrices , random systems with complete connection

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 4 • August, 1975
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