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

¹ CMAP, École Polytechnique, Palaiseau, France
² Laboratoire Traitement et Communication de l'Information, CNRS / GET Télécom Paris, France

Abstract

In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics (see for instance [Doucet, De Freitas and Gordon,Sequential Monte Carlo Methods in Practice. Springer, New York, 2001], [Liu. Monte Carlo Strategies in Scientific Computing. Springer, New York, 2001], [Künsch. Complex Stochastic Systems, 109-173, CRC Publisher, Boca raton, 2001]). 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 for instance [Gilks and Berzuini. J. Roy. Statist. Soc. Ser. B, 63(1), 127-146, 2001], [Künsch. Ann. Statist., 33(5), 1983-2021, 2005], [Del Moral. Feynman-Kac Formulae. Genealogical and Interacting Particle Systems with Applications, Springer, 2004], [Chopin. Ann. Statist., 32(6), 2385-2411, 2004]), the asymptotic property of these approximations remains a question of central interest. In this paper, we analyze sequential Monte Carlo (SMC) 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 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, selection at randomly selected times, etc..