A simple proof of Kaijser's unique ergodicity result for hidden Markov $α$-chains

Abstract

According to a 1975 result of T. Kaijser, if some nonvanishing product of hidden Markov model (HMM) stepping matrices is subrectangular, and the underlying chain is aperiodic, the corresponding $α$-chain has a unique invariant limiting measure $λ$. Here the $α$-chain $\{α_n\}=\{(α_{ni})\}$ is given by \[α_{ni}=P(X_n=i| Y_n,Y_{n-1},...),\] where $\{(X_n,Y_n)\}$ is a finite state HMM with unobserved Markov chain component $\{X_n\}$ and observed output component $\{Y_n\}$. This defines $\{α_n\}$ as a stochastic process taking values in the probability simplex. It is not hard to see that $\{α_n\}$ is itself a Markov chain. The stepping matrices $M(y)=(M(y)_{ij})$ give the probability that $(X_n,Y_n)=(j,y)$, conditional on $X_{n-1}=i$. A matrix is said to be subrectangular if the locations of its nonzero entries forms a cartesian product of a set of row indices and a set of column indices. Kaijser's result is based on an application of the Furstenberg--Kesten theory to the random matrix products $M(Y_1)M(Y_2)... M(Y_n)$. In this paper we prove a slightly stronger form of Kaijser's theorem with a simpler argument, exploiting the theory of e chains.