Open Access
February 2014 Loss of memory of hidden Markov models and Lyapunov exponents
Pierre Collet, Florencia Leonardi
Ann. Appl. Probab. 24(1): 422-446 (February 2014). DOI: 10.1214/13-AAP929

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.

Citation

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Pierre Collet. Florencia Leonardi. "Loss of memory of hidden Markov models and Lyapunov exponents." Ann. Appl. Probab. 24 (1) 422 - 446, February 2014. https://doi.org/10.1214/13-AAP929

Information

Published: February 2014
First available in Project Euclid: 9 January 2014

zbMATH: 1293.62173
MathSciNet: MR3161652
Digital Object Identifier: 10.1214/13-AAP929

Subjects:
Primary: 60J2
Secondary: 62M09

Keywords: Oseledec’s theorem , perturbed processes , Random functions of Markov chains

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 1 • February 2014
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