A simple proof of Kaijser.s unique ergodicity result for hidden Markov .-chains

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 .-chain has a unique invariant limiting measure .. Here the .-chain {.n}={(.ni)} is given by .ni=P(Xn=i|Yn,Yn.1,.), where {(Xn,Yn)} is a finite state HMM with unobserved Markov chain component {Xn} and observed output component {Yn}. This defines {.n} as a stochastic process taking values in the probability simplex. It is not hard to see that {.n} is itself a Markov chain. The stepping matrices M(y)=(M(y)ij) give the probability that (Xn,Yn)=(j,y), conditional on Xn.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(Y1)M(Y2).M(Yn). In this paper we prove a slightly stronger form of Kaijser.s theorem with a simpler argument, exploiting the theory of e chains.