2006/11/01 by Fred Kochman, Jim Reeds
Mathematics · #Graph theory and applications #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #math.PR #msc:60F99 #msc:60J05 #msc:60J10
paper · pdf · doi:10.1214/105051606000000367
published as Annals of Applied Probability 2006, Vol. 16, No. 4, 1805-1815 · Published at http://dx.doi.org/10.1214/105051606000000367 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2006/11/01 · arxiv created 2007/02/09 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 αn=(αni) is given by αni=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 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.