A simple proof of Kaijser’s unique ergodicity result for hidden Markov α-chains
Kochman, Fred ; Reeds, Jim
Ann. Appl. Probab., Tome 16 (2006) no. 1, p. 1805-1815 / Harvested from Project Euclid
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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.
Publié le : 2006-11-14
Classification:  Hidden Markov models,  uniform mean stability,  e-chain,  ergodicity,  60J10,  60J05,  60F99 
@article{1169065208,
     author = {Kochman, Fred and Reeds, Jim},
     title = {A simple proof of Kaijser's unique ergodicity result for hidden Markov $\alpha$-chains},
     journal = {Ann. Appl. Probab.},
     volume = {16},
     number = {1},
     year = {2006},
     pages = { 1805-1815},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1169065208}
}

Kochman, Fred; Reeds, Jim. A simple proof of Kaijser’s unique ergodicity result for hidden Markov α-chains. Ann. Appl. Probab., Tome 16 (2006) no. 1, pp.  1805-1815. http://gdmltest.u-ga.fr/item/1169065208/