Loss of memory of hidden Markov models and Lyapunov exponents

TL;DR

This paper establishes a connection between the exponential loss of memory in finite state hidden Markov models and the difference of their Lyapunov exponents, providing bounds and almost sure attainability of this rate.

Contribution

It proves that the asymptotic loss of memory is bounded by the difference of Lyapunov exponents and that this bound is almost surely achieved for typical observed sequences.

Findings

Loss of memory rate bounded by Lyapunov exponents difference

Bound is tight and attained for almost all realizations

Results apply to observed process and filter in total variation

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.