The Annals of Statistics

Limit theorems for weighted samples with applications to sequential Monte Carlo methods

Randal Douc and Eric Moulines

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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.

Article information

Ann. Statist., Volume 36, Number 5 (2008), 2344-2376.

First available in Project Euclid: 13 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62L10: Sequential analysis 65C05: Monte Carlo methods
Secondary: 65C35: Stochastic particle methods [See also 82C80] 65C60: Computational problems in statistics

Branching conditional central limit theorems particle filtering sequential importance sampling sequential Monte Carlo


Douc, Randal; Moulines, Eric. Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36 (2008), no. 5, 2344--2376. doi:10.1214/07-AOS514.

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