Communications in Information & Systems

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

László Gerencser, György Michaletzky, and Gábor Molnár-Sáska

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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.

Article information

Source
Commun. Inf. Syst., Volume 7, Number 2 (2007), 133-152.

Dates
First available in Project Euclid: 20 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.cis/1184963915

Mathematical Reviews number (MathSciNet)
MR2344193

Zentralblatt MATH identifier
1263.62117

Keywords
hidden Markov models predictive filters random mappings Doeblin-condition risk processes L-mixing

Citation

Gerencser, László; Michaletzky, György; Molnár-Sáska, Gábor. An improved bound for the exponential stability of predictive filters of hidden Markov models. Commun. Inf. Syst. 7 (2007), no. 2, 133--152. https://projecteuclid.org/euclid.cis/1184963915


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