On the asymptotic variance in the central limit theorem for particle filters
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 151-164.

Particle filter algorithms approximate a sequence of distributions by a sequence of empirical measures generated by a population of simulated particles. In the context of Hidden Markov Models (HMM), they provide approximations of the distribution of optimal filters associated to these models. For a given set of observations, the behaviour of particle filters, as the number of particles tends to infinity, is asymptotically Gaussian, and the asymptotic variance in the central limit theorem depends on the set of observations. In this paper we establish, under general assumptions on the hidden Markov model, the tightness of the sequence of asymptotic variances when considered as functions of the random observations as the number of observations tends to infinity. We discuss our assumptions on examples and provide numerical simulations.

DOI : 10.1051/ps/2010019
Classification : 60G35, 62M20, 60F05, 60J05
Mots-clés : hidden Markov model, particle filter, central limit theorem, asymptotic variance, tightness, sequential Monte-Carlo
@article{PS_2012__16__151_0,
     author = {Favetto, Benjamin},
     title = {On the asymptotic variance in the central limit theorem for particle filters},
     journal = {ESAIM: Probability and Statistics},
     pages = {151--164},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2010019},
     mrnumber = {2946125},
     zbl = {1273.60046},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2010019/}
}
TY  - JOUR
AU  - Favetto, Benjamin
TI  - On the asymptotic variance in the central limit theorem for particle filters
JO  - ESAIM: Probability and Statistics
PY  - 2012
SP  - 151
EP  - 164
VL  - 16
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2010019/
DO  - 10.1051/ps/2010019
LA  - en
ID  - PS_2012__16__151_0
ER  - 
%0 Journal Article
%A Favetto, Benjamin
%T On the asymptotic variance in the central limit theorem for particle filters
%J ESAIM: Probability and Statistics
%D 2012
%P 151-164
%V 16
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2010019/
%R 10.1051/ps/2010019
%G en
%F PS_2012__16__151_0
Favetto, Benjamin. On the asymptotic variance in the central limit theorem for particle filters. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 151-164. doi : 10.1051/ps/2010019. http://archive.numdam.org/articles/10.1051/ps/2010019/

[1] R. Atar and O. Zeitouni, Exponential stability for nonlinear filtering. Ann. Inst. Henri Poincaré 33 (1997) 697-725. | EuDML | Numdam | MR | Zbl

[2] O. Cappé, E. Moulines and T. Ryden, Inference in Hidden Markov Models, in Springer Series in Statistics. Springer-Verlag New York, Inc., Secaucus, NJ, USA (2005). | MR | Zbl

[3] M. Chaleyat-Maurel and V. Genon-Catalot, Computable infinite-dimensional filters with applications to discretized diffusion processes. Stoc. Proc. Appl. 116 (2006) 1447-1467. | MR | Zbl

[4] N. Chopin, Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 (2004) 2385-2411. | MR | Zbl

[5] E.B. Davies, Heat kernels and spectral theory, in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge 92 (1989). | MR | Zbl

[6] P. Del Moral, Feynman-Kac formulae, Genealogical and interacting particle systems with applications. Probab. Appl. Springer-Verlag, New York (2004). | MR | Zbl

[7] P. Del Moral and A. Guionnet, On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré 37 (2001) 155-194. | EuDML | Numdam | MR | Zbl

[8] P. Del Moral and J. Jacod, Interacting particle filtering with discrete observations, in Sequential Monte Carlo methods in practice, Springer, New York. Stat. Eng. Inf. Sci. (2001) 43-75. | MR | Zbl

[9] P. Del Moral and J. Jacod, Interacting particle filtering with discrete-time observations : asymptotic behaviour in the Gaussian case, in Stochastics in finite and infinite dimensions, Birkhäuser Boston, Boston, MA. Trends Math. (2001) 101-122. | MR | Zbl

[10] R. Douc, A. Guillin and J. Najim, Moderate deviations for particle filtering. Ann. Appl. Probab. 15 (2005) 587-614. | MR | Zbl

[11] R. Douc, G. Fort, E. Moulines and P. Priouret, Forgetting of the initial distribution for hidden Markov models. Stoc. Proc. Appl. 119 (2009) 1235-1256. | MR | Zbl

[12] A. Doucet, N. De Freitas and N. Gordon, Sequential Monte Carlo methods in practice, Stat. Eng. Inform. Sci. Springer-Verlag, New York (2001). | MR | Zbl

[13] H.R. Künsch, State space and hidden Markov models, in Complex Stochastic Systems. Eindhoven (1999); Chapman & Hall/CRC, Boca Raton, FL. Monogr. Statist. Appl. Probab. 87 (2001) 109-173. | MR | Zbl

[14] H.R. Künsch, Recursive Monte Carlo filters : algorithms and theoretical analysis. Ann. Statist. 33 (2005) 1983-2021. | MR | Zbl

[15] N. Oudjane and S. Rubenthaler, Stability and uniform particle approximation of nonlinear filters in case of non ergodic signals. Stoch. Anal. Appl. 23 (2005) 421-448. | MR | Zbl

[16] C.P. Robert and G. Casella, Monte Carlo statistical methods, 2nd edition, in Springer Texts in Statistics. Springer-Verlag, New York (2004). | MR | Zbl

[17] R. Van Handel, Uniform time average consistency of Monte Carlo particle filters. Stoc. Proc. Appl. 119 (2009) 3835-3861. | MR | Zbl

Cité par Sources :