# On the asymptotic variance in the central limit theorem for particle filters

• Volume: 16, page 151-164
• ISSN: 1292-8100

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## Abstract

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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.

## How to cite

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Favetto, Benjamin. "On the asymptotic variance in the central limit theorem for particle filters." ESAIM: Probability and Statistics 16 (2012): 151-164. <http://eudml.org/doc/276392>.

@article{Favetto2012,
abstract = {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.},
author = {Favetto, Benjamin},
journal = {ESAIM: Probability and Statistics},
keywords = {Hidden Markov Model; Particle filter; Central Limit Theorem; Asymptotic variance; Tightness; Sequential Monte-Carlo.; central limit theorem; asymptotic variance; particle filter; hidden Markov model; tightness; sequential Monte-Carlo},
language = {eng},
month = {7},
pages = {151-164},
publisher = {EDP Sciences},
title = {On the asymptotic variance in the central limit theorem for particle filters},
url = {http://eudml.org/doc/276392},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Favetto, Benjamin
TI - On the asymptotic variance in the central limit theorem for particle filters
JO - ESAIM: Probability and Statistics
DA - 2012/7//
PB - EDP Sciences
VL - 16
SP - 151
EP - 164
AB - 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.
LA - eng
KW - Hidden Markov Model; Particle filter; Central Limit Theorem; Asymptotic variance; Tightness; Sequential Monte-Carlo.; central limit theorem; asymptotic variance; particle filter; hidden Markov model; tightness; sequential Monte-Carlo
UR - http://eudml.org/doc/276392
ER -

## References

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17. R. Van Handel, Uniform time average consistency of Monte Carlo particle filters. Stoc. Proc. Appl.119 (2009) 3835–3861.  Zbl1176.93076

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