The Annals of Applied Probability

Moderate deviations for particle filtering

R. Douc, A. Guillin, and J. Najim

Full-text: Open access

Abstract

Consider the state space model (Xt,Yt), where (Xt) is a Markov chain, and (Yt) are the observations. In order to solve the so-called filtering problem, one has to compute ℒ(Xt|Y1,…,Yt), the law of Xt given the observations (Y1,…,Yt). The particle filtering method gives an approximation of the law ℒ(Xt|Y1,…,Yt) by an empirical measure $\frac{1}{n}$∑1nδxi,t. In this paper we establish the moderate deviation principle for the empirical mean $\frac{1}{n}$∑1nψ(xi,t) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 1B (2005), 587-614.

Dates
First available in Project Euclid: 1 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1107271661

Digital Object Identifier
doi:10.1214/105051604000000657

Mathematical Reviews number (MathSciNet)
MR2114983

Zentralblatt MATH identifier
1072.60018

Subjects
Primary: 60F10: Large deviations 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 93E11: Filtering [See also 60G35]

Keywords
Particle filters moderate deviation principle

Citation

Douc, R.; Guillin, A.; Najim, J. Moderate deviations for particle filtering. Ann. Appl. Probab. 15 (2005), no. 1B, 587--614. doi:10.1214/105051604000000657. https://projecteuclid.org/euclid.aoap/1107271661


Export citation

References

  • Bucklew, J. A. (1990). Large Deviations Techniques in Decision, Simulation and Estimation. Wiley, New York.
  • Chen, X. (1991). Moderate deviations of independent random vectors in Banach spaces. Chinese J. Appl. Probab. Statist. 7 24–32.
  • de Acosta, A. (1992). Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. Amer. Math. Soc. 329 357–375.
  • Del Moral, P. (1996). Nonlinear filtering: Interacting particle solution. Markov Process. Related Fields 2 555–579.
  • Del Moral, P. (1997). Nonlinear filtering: Interacting particle resolution. C. R. Acad. Sci. Paris Sér. I Math. 325 653–658.
  • Del Moral, P. and Guionnet, A. (1998). Large deviations for interacting particle systems: Applications to non-linear filtering. Stochastic Process. Appl. 78 69–95.
  • Del Moral, P. and Guionnet, A. (1999). Central limit theorem for nonlinear filtering and interacting particle systems. Ann. Appl. Probab. 9 275–297.
  • 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.
  • Del Moral, P. and Jacod, J. (2002). The Monte Carlo method for filtering with discrete-time observations: Central limit theorems. In Numerical Methods and Stochastics 29–53. Amer. Math. Soc., Providence, RI.
  • Del Moral, P., Jacod, J. and Protter, P. (2001). The Monte Carlo method for filtering with discrete-time observations. Probab. Theory Related Fields 120 346–368.
  • Del Moral, P. and Ledoux, M. (2000). Convergence of empirical processes for interacting particle systems with applications to nonlinear filtering. J. Theoret. Probab. 13 225–257.
  • Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering. Séminaire de Probabilités XXXIV. Lecture Notes in Math. 1729 1–145. Springer, Berlin.
  • Del Moral, P. and Miclo, L. (2003). Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups. ESAIM Probab. Statist. 7 171–208.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Deuschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston.
  • Djellout, H., Guillin, A. and Wu, L. (1999). Large and moderate deviations for quadratic empirical processes. Statist. Inference Stoch. Process. 2 195–225.
  • Doucet, A., de Freitas, N. and Gordon, N., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
  • Künsch, H. R. (2001). State space and hidden Markov models. In Complex Stochastic Systems 109–173. Chapman and Hall/CRC, Boca Raton, FL.
  • Künsch, H. R. (2002). Recursive Monte Carlo filters: Algorithms and theoretical analysis. Seminar für Statistik, ETH Zürich, Switzerland.
  • Ledoux, M. (1992). Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. H. Poincaré Probab. Statist. 28 267–280.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, New York.
  • Shephard, N. and Pitt, M. (1997). Likelihood analysis of non-Gaussian measurement time series. Biometrika 84 653–667.
  • Wu, L. (1994). Large deviations, moderate deviations and LIL for empirical processes. Ann. Probab. 22 17–27.
  • Wu, L. (1995). Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23 420–445.