Forgetting of the initial distribution for Hidden Markov Models

Randal Douc (1) , Gersende Fort (2) , Éric Moulines (2) , Pierre Priouret (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LTCI - Laboratoire Traitement et Communication de l'Information
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the non-linear state space model and the stochastic volatility model.

Mots clés: Nonlinear filtering; Hidden Markov Models; Asymptotic stability; Total Variation Norm.; Total Variation Norm

Domaines: Statistiques [math.ST]

Randal Douc, Gersende Fort, Éric Moulines, Pierre Priouret. Forgetting of the initial distribution for Hidden Markov Models. Stochastic Processes and their Applications, 2009, 119 (4), pp.1235--1256. ⟨hal-00138902⟩