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(Y₁)M(Y₂)⋯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.