A simple proof of Kaijser's unique ergodicity result for hidden Markov α-chains
成果类型：
Article
署名作者：
Kochman, Fred; Reeds, Jim
署名单位：
Center for Communications & Computing
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/105051606000000367
发表日期：
2006
页码：
1805-1815
关键词：
摘要：
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 gimel. Here the alpha-chain {alpha(n)} = {(alpha(ni))} is given by alpha(ni) = P(X-n = i vertical bar Y-n, Yn-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 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 (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.
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