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October 2008 Limit theorems for weighted samples with applications to sequential Monte Carlo methods
Randal Douc, Eric Moulines
Ann. Statist. 36(5): 2344-2376 (October 2008). DOI: 10.1214/07-AOS514


In the last decade, sequential Monte Carlo methods (SMC) emerged as a key tool in computational statistics [see, e.g., Sequential Monte Carlo Methods in Practice (2001) Springer, New York, Monte Carlo Strategies in Scientific Computing (2001) Springer, New York, Complex Stochastic Systems (2001) 109–173]. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles, which are generated recursively.

Despite many theoretical advances [see, e.g., J. Roy. Statist. Soc. Ser. B 63 (2001) 127–146, Ann. Statist. 33 (2005) 1983–2021, Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications (2004) Springer, Ann. Statist. 32 (2004) 2385–2411], the large-sample theory of these approximations remains a question of central interest. In this paper we establish a law of large numbers and a central limit theorem as the number of particles gets large. We introduce the concepts of weighted sample consistency and asymptotic normality, and derive conditions under which the transformations of the weighted sample used in the SMC algorithm preserve these properties. To illustrate our findings, we analyze SMC algorithms to approximate the filtering distribution in state-space models. We show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies, including branching, resampling at randomly selected times, and so on.


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Randal Douc. Eric Moulines. "Limit theorems for weighted samples with applications to sequential Monte Carlo methods." Ann. Statist. 36 (5) 2344 - 2376, October 2008.


Published: October 2008
First available in Project Euclid: 13 October 2008

zbMATH: 1155.62056
MathSciNet: MR2458190
Digital Object Identifier: 10.1214/07-AOS514

Primary: 60F05 , 62L10 , 65C05
Secondary: 65C35 , 65C60

Keywords: branching , conditional central limit theorems , particle filtering , sequential importance sampling , sequential Monte Carlo

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 5 • October 2008
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