A Simple Proof of Kaijser ’ S Unique Ergodicity Result for Hidden Markov Α - Chains

According to a 1975 result of T. Kaijser, if some nonvanishing product of hidden Markov model (HMM) stepping matrices is sub-rectangular, and the underlying chain is aperiodic, the corresponding α-chain has a unique invariant limiting measure λ. Here the α-chain {αn} = {(αni)} is given by where {(Xn, Yn)} is a finite state HMM with unobserved Markov chain component {Xn} and observed output component {Yn}. This defines {αn} as a stochastic process taking values in the probability simplex. It is not hard to see that {αn} 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. 1. Introduction. In 1975 Kaijser [9] gave a simple sufficient condition for the uniqueness of the invariant measure for the so-called α-chain, a certain weak Feller chain with compact state space arising in the study of an arbitrary finite state hidden Markov model. This provided an elegant partial answer to a question posed by David Blackwell in 1957 [3]. (We follow Blackwell in using α n to denote the state of the α-chain at time n. Kaijser calls it Z n .) The transition behavior of a finite state hidden Markov model (HMM) and of its associated α-chain can be specified by a finite collection