  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 $\alpha$-chain has a unique
invariant limiting measure $\lambda$. Here the $\alpha$-chain
$\{\alpha_n\}=\{(\alpha_{ni})\}$ is given by \[\alpha_{ni}=P(X_n=i|
Y_n,Y_{n-1},...),\] 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 $\{\alpha_n\}$ as a stochastic process taking values in the probability
simplex. It is not hard to see that $\{\alpha_n\}$ 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.
