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.

https://www.esaim-proc.org/articles/proc/pdf/2007/05/proc071913.pdf

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

In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles. These particles and weights are generated recursively according to elementary transformations: mutation and selection. Examples of applications include the sequential Monte-Carlo techniques to solve optimal non-linear filtering problems in state-space models, molecular simulation, genetic optimization, etc. 
Despite many theoretical advances the asymptotic property of these approximations remains of course a question of central interest. In this paper, we analyze sequential Monte Carlo methods from an asymptotic perspective, that is, we establish law of large numbers and invariance principle as the number of particles gets large. We introduce the concepts of "weighted sample" consistency and asymptotic normality, and derive conditions under which the mutation and the selection procedure used in the sequential Monte-Carlo build-up 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.

Limit theorems for weighted samples with applications to Sequential Monte Carlo Methods

Some examples and results in the theory of mixing and random-sum central limit theorems

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Convergence faible avec indices aléatoires

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M. Csörgö

Papers

On mixing and the central limit theorem