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

Laboratoire MAP5, Université Paris Descartes, U.F.R. de Mathématique et Informatique, CNRS UMR 8145, 45, rue des Saints-Pères, 75270 Paris Cedex 06, France
Benjamin.Favetto@mi.parisdescartes.fr

Received: 23 November 2009
Revised: 4 May 2010

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.