The Annals of Applied Probability

Forgetting of the initial distribution for nonergodic Hidden Markov Chains

Randal Douc, Elisabeth Gassiat, Benoit Landelle, and Eric Moulines

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In this paper, the forgetting of the initial distribution for a nonergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter. Both a pathwise and mean convergence of the total variation distance of the filter started from two different initial distributions are obtained. The results are illustrated using a generic nonergodic state-space model for which both pathwise and mean exponential stability is established.

Article information

Ann. Appl. Probab. Volume 20, Number 5 (2010), 1638-1662.

First available in Project Euclid: 25 August 2010

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Zentralblatt MATH identifier

Primary: 93E11: Filtering [See also 60G35] 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

Nonlinear filtering forgetting of the initial distribution nonergodic Hidden Markov Chains Feynman–Kac semigroup


Douc, Randal; Gassiat, Elisabeth; Landelle, Benoit; Moulines, Eric. Forgetting of the initial distribution for nonergodic Hidden Markov Chains. Ann. Appl. Probab. 20 (2010), no. 5, 1638--1662. doi:10.1214/09-AAP632.

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  • [1] Atar, R. and Zeitouni, O. (1997). Exponential stability for nonlinear filtering. Ann. Inst. H. Poincaré Probab. Statist. 33 697–725.
  • [2] Budhiraja, A. and Ocone, D. (1997). Exponential stability of discrete-time filters for bounded observation noise. Systems Control Lett. 30 185–193.
  • [3] Budhiraja, A. and Ocone, D. (1999). Exponential stability in discrete-time filtering for non-ergodic signals. Stochastic Process. Appl. 82 245–257.
  • [4] Chigansky, P., Liptser, R. and van Handel, R. (2009). Intrinsic methods in filter stability. In Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford.
  • [5] Crisan, D. and Heine, K. (2008). Stability of the discrete time filter in terms of the tails of noise distributions. J. Lond. Math. Soc. (2) 78 441–458.
  • [6] Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems With Applications. Springer, New York.
  • [7] Douc, R., Fort, G., Moulines, E. and Priouret, P. (2009). Forgetting the initial distribution for hidden Markov models. Stochastic Process. Appl. 119 1235–1256.
  • [8] Douc, R., Moulines, É. and Rydén, T. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 2254–2304.
  • [9] Doucet, A., De Freitas, N. and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice. Information Science and Statistics. Springer, New York.
  • [10] Hu, Z.-C. and Sun, W. (2006). A note on exponential stability of the nonlinear filter for denumerable Markov chains. Systems Control Lett. 55 955–960.
  • [11] Kleptsyna, M. L. and Veretennikov, A. Y. (2008). On discrete time ergodic filters with wrong initial data. Probab. Theory Related Fields 141 411–444.
  • [12] Le Gland, F. and Oudjane, N. (2004). Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. Ann. Appl. Probab. 14 144–187.
  • [13] LeGland, F. and Oudjane, N. (2003). A robustification approach to stability and to uniform particle approximation of nonlinear filters: The example of pseudo-mixing signals. Stochastic Process. Appl. 106 279–316.
  • [14] Ocone, D. and Pardoux, E. (1996). Asymptotic stability of the optimal filter with respect to its initial condition. SIAM J. Control Optim. 34 226–243.
  • [15] Oudjane, N. and Rubenthaler, S. (2005). Stability and uniform particle approximation of nonlinear filters in case of nonergodic signals. Stoch. Anal. Appl. 23 421–448.
  • [16] Ristic, B., Arulampalam, M. and Gordon, A. (2004). Beyond Kalman Filters: Particle Filters for Tracking Applications. Artech House, Boston.
  • [17] Van Handel, R. (2008). Discrete time nonlinear filters with informative observations are stable. Electron. Commun. Probab. 13 562–575.
  • [18] Van Handel, R. (2009). Uniform observability of hidden Markov models and filter stability for unstable signals. Ann. Appl. Probab. 19 1172–1199.
  • [19] Veretennikov, A. Y. (2002). Coupling method for Markov chains under integral Doeblin type condition. Theory Stoch. Process. 8 383–390.