The Annals of Applied Probability

Central limit theorem for nonlinear filtering and interacting particle systems

P. Del Moral and A. Guionnet

Full-text: Open access

Abstract

Several random particle systems approaches were recently suggested to solve nonlinear filtering problems numerically. The present analysis is concerned with genetic-type interacting particle systems. Our aim is to study the fluctuations on path space of such particle-approximating models.

Article information

Source
Ann. Appl. Probab. Volume 9, Number 2 (1999), 275-297.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1029962742

Digital Object Identifier
doi:10.1214/aoap/1029962742

Mathematical Reviews number (MathSciNet)
MR1687359

Zentralblatt MATH identifier
0938.60022

Subjects
Primary: 60F05: Central limit and other weak theorems 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 93E11: Filtering [See also 60G35] 62L20: Stochastic approximation

Keywords
Central limit interacting random processes filtering stochastic approximation

Citation

Del Moral, P.; Guionnet, A. Central limit theorem for nonlinear filtering and interacting particle systems. Ann. Appl. Probab. 9 (1999), no. 2, 275--297. doi:10.1214/aoap/1029962742. http://projecteuclid.org/euclid.aoap/1029962742.


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References

  • [1] Ben Arous, G. and Brunaud, M. (1990). Methode de Laplace: Etude variationnelle des fluctuations de diffusions de ty pe "champ moy en." Stochastics 31-32 79-144.
  • [2] Benes, B. E. (1981). Exact finite-dimensional filters for certain diffusions with nonlinear drift. Stochastics 5 65-92.
  • [3] Bolthausen, E. (1986). Laplace approximation for sums of independant random vectors I. Probab. Theory Related Fields 72 305-318.
  • [4] Bucy, R. S. (1994). Lectures on discrete time filtering. In Signal Processing and Digital Filtering. Springer, New York.
  • [5] Carvalho, H. (1995). Filtrage optimal non lin´eaire du signal GPS NAVSTAR en racalage de centrales de navigation. Th ese de L'Ecole Nationale Sup´erieure de l'A´eronautique et de l'Espace, Toulouse.
  • [6] Carvalho, H., Del Moral, P., Monin, A. and Salut, G. (1997). Optimal non-linear filtering in GPS/INS integration. IEEE Trans. Aerospace Electron. Sy stems 33 835-850.
  • [7] Chaley at-Maurel, M. and Michel, D. (1983). Des r´esultats de non existence de filtres de dimension finie. C. R. Acad. Sci. Paris S´er. I Math. 296.
  • [8] Crisan, D., Del Moral, P. and Ly ons, T. J. (1997). Non linear filtering using branching and interacting particle sy stems. Publications du Laboratoire de Statistiques et Probabilit´es, Universit´e Paul Sabatier 01-98.
  • [9] Crisan, D., Gaines, J. and Ly ons, T. J. (1998). A particle approximation of the solution of the Kushner-Stratonovitch equation. SIAM J. Appl. Math. To appear.
  • [10] Crisan, D. and Ly ons, T. J. (1996). Convergence of a branching particle method to the solution of the Zakai equation. Imperial College, London. Preprint.
  • [11] Crisan, D. and Ly ons, T. J. (1997). Nonlinear filtering and measure valued processes. Probab. Theory Related Fields 109 217-244.
  • [12] Del Moral, P. (1996). Non linear filtering using random particles. Theory Probab. Appl. 40 690-701.
  • [13] Del Moral, P. (1996). Non-linear filtering: interacting particle resolution. Markov Processes Related Fields 2 555-580.
  • [14] Del Moral, P. (1996). Measure valued processes and interacting particle sy stems. Application to non linear filtering problems. Unpublished manuscript.
  • [15] Del Moral, P. (1996). A uniform theorem for the numerical solving of non linear filtering problems. Publications du Laboratoire de Statistiques et Probabilit´es, Universit´e Paul Sabatier 14-96. J. Appl. Probab. To appear.
  • [16] Del Moral, P. (1997). Filtrage non lin´eaire par sy st emes de particules en int´eraction. C.R. Acad. Sci. Paris S´er. I Math. 325 653-658.
  • [17] Del Moral, P. and Guionnet, A. (1997). Large deviations for interacting particle sy stems. Applications to non linear filtering problems. Stochastic Process. Appl. To appear.
  • [18] Del Moral, P., Noy er, J. C. and Salut, G. (1995). R´esolution particulaire et traitement nonlin´eaire du signal: application radar/sonar. Traitement du signal 12 287-301.
  • [19] Del Moral P., Rigal, G., Noy er, J. C. and Salut, G. (1993). Traitement non-lin´eaire du signal par reseau particulaire: application radar. In 14th colloque GRETSI sur le Traitement du Signal et des Images 399-402.
  • [20] Dy nkin, E. B. and Mandelbaum, A. (1983). Sy mmetric statistics, Poisson processes and multiple Wiener integrals. Ann. Statist. 11 739-745.
  • [21] Guionnet, A. (1997). About precise Laplace's method; Applications to fluctuations for mean field interacting particles. Preprint.
  • [22] Goldberg, D. E. (1985), Genetic algorithms and rule learning in dy namic control sy stems. In Proceedings of the First International Conference on Genetic Algorithms 8-15. Erlbaum, Hillsdale, NJ.
  • [23] Goldberg, D. E. (1987). Simple genetic algorithms and the minimal deceptive problem. In Genetic Algorithms and Simulated Annealing (Lawrence Davis, ed.) Pitman, New York.
  • [24] Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA.
  • [25] Gordon, N. J., Salmon, D. J. and Smith, A. F. M. (1993). Novel Approach to non-linear/nonGaussian Bayesian state estimation. IEE Proceedings 140 107-133.
  • [26] Holland, J. H. (1975). Adaptation in Natural and Artificial Sy stems. Univ. Michigan Press.
  • [27] Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York.
  • [28] Kunita, H. (1971). Asy mptotic behavior of nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 365-393.
  • [29] Kusuoka, S. and Tamura, Y. (1984). Gibbs measures for mean field potentials. J. Fac. Sci. Univ. Toky o Sect. IA Math 31.
  • [30] Liang, D. and McMillan, J. (1989). Development of a marine integrated navigation sy stem. Kalman Filter Integration of Modern Guidance and Navigation Sy stems, AGARD-LS166, OTAN.
  • [31] Simon, B. (1977). Trace Ideals and Their Applications. Cambridge Univ. Press.
  • [32] Stettner, L. (1989). On invariant measures of filtering processes. Stochastic Differential Sy stems. Lecture Notes in Control and Inform. Sci. 126. Springer, New York.
  • [33] Shiga, T. and Tanaka, H. (1985). Central limit theorem for a sy stem of markovian paricles with mean field interaction. Z. Wahrsch. Verw. Gebiete 69 439-459.
  • [34] Shiry aev A. N. (1996). Probability, 2nd ed. Springer, New York.
  • [35] Tanaka, H. (1984). Limit theorems for certain diffusion processes. In Proceedings of the Taniguchi Sy mposium Katata 1982 469-488. Kinokuniy a, Toky o.
  • [36] Van Dootingh, M., Viel, F., Rakotopara, D. and Gauthier, J. P. (1991). Coupling of nonlinear control with a stochastic filter for state estimation: application on a free radical polimerization reactor. I.F.A.C. International Sy mposium ADCHEM'91. Toulouse, France, October 14-15.
  • [37] Vose, M. D. (1993). Modelling simple genetic algorithms. In Foundations of Genetic Algorithms. Morgan Kaufmann.
  • [38] Vose, M. D. (1995). Modelling simple genetic algorithms. Elementary Computations 3 453- 472.
  • [39] Williams, D. (1992). Probability with Martingales. Cambridge Univ. Press.