An improved bound for the exponential stability of predictive filters of hidden Markov models

Abstract

We consider hidden Markov processes in discrete time with a finite state space $X$ and a general observation or read-out space $Y$, which is assumed to be a Polish space. It is well-known that in the statistical analysis of HMMs the so-called predictive filter plays a fundamental role. A useful result establishing the exponential stability of the predictive filter with respect to perturbations of its initial condition was given in "Exponential forgetting and geometric ergodicity in hidden Markov models" (F. LeGland, L. Mevel, Mathematics of control, signals and systems, 13(2000), pp. 63-93) in the case, when the assumed transition probability matrix was primitive. The main technical result of the present paper is the extension of the cited result by showing that the random constant and the deterministic positive exponent showing up in the inequality stating exponential stability can be chosen so that for any prescribed ${s\geq 0}$ the $s$-th exponential moment of the random constant is finite. An application of this result to the estimation of HMMs with primitive transition probabilities will be also briefly presented.