A simple proof of Kaijser's unique ergodicity result for hidden Markov $α$-chains
| dc.creator | Kochman, Fred | |
| dc.creator | Reeds, Jim | |
| dc.date | 2007-02-09 | |
| dc.date.accessioned | 2026-07-07T07:45:46Z | |
| dc.date.available | 2026-07-07T07:45:46Z | |
| dc.description | 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 $M(Y_1)M(Y_2)... M(Y_n)$. In this paper we prove a slightly stronger form of Kaijser's theorem with a simpler argument, exploiting the theory of e chains. | |
| dc.description | 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) | |
| dc.identifier | https://arxiv.org/abs/math/0702248 | |
| dc.identifier | http://arxiv.org/abs/math/0702248 | |
| dc.identifier | Annals of Applied Probability 2006, Vol. 16, No. 4, 1805-1815 | |
| dc.identifier | doi:10.1214/105051606000000367 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123643 | |
| dc.subject | Probability | |
| dc.subject | 60J10 (Primary) 60J05, 60F99 (Secondary) | |
| dc.title | A simple proof of Kaijser's unique ergodicity result for hidden Markov $α$-chains | |
| dc.type | text |