Using systematic sampling for approximating Feynman-Kac solutions by Monte Carlo methods Ivan Gentil ∗ and Bruno R´ emillard † Abstract While convergence properties of many sampling selection methods can be proven to hold in a context of approximation of Feynman-Kac solutions using sequential Monte Carlo simulations, there is one particular sampling selection method introduced by Baker (1987), closely related with “systematic sampling” in statistics, that has been exclusively treated on an empirical basis. The main motivation of the paper is to start to study formally its convergence properties, since in practice it is by far the fastest selection method available. One will show that convergence results for the systematic sampling selection method are related to properties of peculiar Markov chains. Keywords: Feynman-Kac formulæ, sequential Monte Carlo, genetic algorithms, systematic sampling, Markov chains. MSC (2000): 65C35, 60G35, 60J10. 1 Introduction Let (X k ) k≥0 be a non-homogeneous Markov chain on a locally compact metric space E, with transition kernels (K n ) n≥1 and initial law η 0 defined on the Borel σ-field B(E). Further let B b (E) be the set of bounded B(E)-measurable functions. Given a sequence (g n ) n≥1 of positive functions in B b (E), suppose that one wants to calculate recur- sively the following Feynman-Kac formulæ (η n ) n≥1 : (1) η n (f )= γ n (f ) γ n (1) ,f ∈B b (E), where (2) γ n (f )= E  f (X n ) n−1  k=1 g k (X k−1 )  . Note that most nonlinear filtering problems are particular cases of Feynman-Kac formulæ. Following Crisan et al. (1999) and Del Moral and Miclo (2000), let M 1 (E) denotes the set of probability measures on (E, B(E)). If µ ∈ M 1 (E) and n ≥ 0, let µK n be the probability measure defined on B b (E) by µK n (f )= µ(K n f )=  E  E f (z )K n (x,dz )µ(dx). In order to understand the relation between the η n s, for any n ≥ 1, let ψ n : M 1 (E) → M 1 (E) be defined by ψ n (η)f = η(g n f ) η(g n ) ,η ∈ M (E),f ∈B b (E), and let Φ n denotes the mapping from M 1 (E) to M 1 (E) defined by Φ n (η)= ψ n (η)K n . * CEREMADE, Universit´ e Paris-Dauphine, Place du Mar´ echal de Lattre de Tassigny, F-75775 Paris cedex 16, France, gentil@ceremade.dauphine.fr † HEC Montr´ eal, Service de l’enseignement des m´ ethodes quantitatives de gestion, 3000, chemin de la cˆ ote-Sainte- Catherine, Canada H3T 2A7, Bruno.Remillard@hec.ca 1