Loss of memory of hidden Markov models and Lyapunov exponents
Abstract
In this paper we prove that the asymptotic rate of exponential loss of memory of a finite state hidden Markov model is bounded above by the difference of the first two Lyapunov exponents of a certain product of matrices. We also show that this bound is in fact realized, namely for almost all realizations of the observed process we can find symbols where the asymptotic exponential rate of loss of memory attains the difference of the first two Lyapunov exponents. These results are derived in particular for the observed process and for the filter; that is, for the distribution of the hidden state conditioned on the observed sequence. We also prove similar results in total variation.
 Publication:

arXiv eprints
 Pub Date:
 August 2009
 arXiv:
 arXiv:0908.0077
 Bibcode:
 2009arXiv0908.0077C
 Keywords:

 Mathematics  Probability
 EPrint:
 Published in at http://dx.doi.org/10.1214/13AAP929 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)