The Annals of Applied Probability

Stability of nonlinear filters in nonmixing case

Pavel Chigansky and Robert Liptser

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The nonlinear filtering equation is said to be stable if it “forgets” the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure.

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Ann. Appl. Probab., Volume 14, Number 4 (2004), 2038-2056.

First available in Project Euclid: 5 November 2004

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Zentralblatt MATH identifier

Primary: 93E11: Filtering [See also 60G35] 60J57: Multiplicative functionals

Filtering stability geometric ergodicity


Chigansky, Pavel; Liptser, Robert. Stability of nonlinear filters in nonmixing case. Ann. Appl. Probab. 14 (2004), no. 4, 2038--2056. doi:10.1214/105051604000000873.

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  • Atar, R. and Zeitouni, O. (1997). Exponential stability for nonlinear filtering. Ann. Inst. H. Poincaré Probab. Statist. 33 697–725.
  • Baxendale, P., Chigansky, P. and Liptser, R. (2002). Asymptotic stability of the Wonham filter: Ergodic and nonergodic signals. Preprint. Available at math.PR/0205230.
  • Beneš, V. E. and Karatzas, I. (1983). Estimation and control for linear, partially observable systems with non-Gaussian initial distribution. Stochastic Process. Appl. 14 233–248.
  • Budhiraja, A. and Ocone, D. (1997). Exponential stability of discrete-time filters for bounded observation noise. Systems Control Lett. 30 185–193.
  • Budhiraja, A. and Ocone, D. (1999). Exponential stability of discrete time filters without signal ergodicity. Stochastic Process. Appl. 82 245–257.
  • Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Probab. Statist. 37 155–194.
  • Delyon, B. and Zeitouni, O. (1991). Lyapunov exponents for filtering problem. In Applied Stochastic Analysis (M. H. A. Davis and R. J. Elliot, eds.) 411–521. Gordon and Breach, London.
  • Kaijser, T. (1975). A limit theorem for partially observed Markov chains. Ann. Probab. 3 677–696.
  • Le Gland, F. and Mevel, L. (2000). Exponential forgetting and geometric ergodicity in hidden Markov models. Math. Control Signals Systems 13 63–93.
  • Le Gland, F. and Oudjane, N. (2004). Stability and uniform approximation of nonlinear filters using the Hilbert metric and applications to particle filters. Ann. Appl. Probab. 14 144–187.
  • Liptser, R. Sh. and Shiryaev, A. N. (2000). Statistics of Random Processes II, 2nd ed. Springer, Berlin.
  • Makowski, A. M. (1986). Filtering formulae for partially observed linear systems with non-Gaussian initial conditions. Stochastics 16 1–24.
  • Makowski, A. M. and Sowers, R. B. (1992). Discrete-time filtering for linear systems with non-Gaussian initial conditions: Asymptotic behaviors of the difference between the MMSE and the LMSE estimates. IEEE Trans. Automat. Control 37 114–121.
  • Norris, J. R. (1998). Markov Chains. Cambridge Univ. Press.
  • Ocone, D. and Pardoux, E. (1986). Asymptotic stability of the optimal filter with respect to its initial condition. SIAM J. Control Optim. 34 226–243.
  • Oudjane, N. and Rubenthaler, S. (2003). Stability and uniform particle approximation of nonlinear filters in case of non ergodic signal. Prepublication PMA-786, Laboratoire de Probabilites et Modeles Aleatoires, Univ. de Paris VI. Available at mathdoc/textes/PMA-786.pdf.
  • Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, Berlin.