Convergence of U-statistics for interacting particle systems

The convergence of U -statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, dlPG99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U -statistics in this framework. Key-words: Interacting particle systems, Feynman-Kac models, U -statistics, fluctuations, limit theorems. ∗ Centre INRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux , Université de Bordeaux I, 351 cours de la Libération 33405 Talence cedex, France, [email protected] † CNRS UMR 6621, Université de Nice, Laboratoire de Mathématiques J.-A. Dieudonné, Parc Valrose, 06108 Nice Cedex 2, France ‡ CNRS UMR 6621, Université de Nice, Laboratoire de Mathématiques J.-A. Dieudonné, Parc Valrose, 06108 Nice Cedex 2, France in ria -0 03 97 36 6, v er si on 1 21 J un 2 00 9 Convergence de U-statistiques pour des systèmes de particules en interaction Résumé : L’analyse de la convergence de U -statistiques a été abondamment étudiée sous des angles différents pour des familles de variables aléatoires indépendantes ou différentes variantes. Dans la plupart des études, l’hypothèse d’indépendance est cruciale [Lee90, dlPG99]. Pour des modèles de systèmes de particules en interaction de type FeynmanKac ou autres, nous sommes confrontés à un nouveau type de problème. Pour être plus précis, dans de tels algorithmes les échantillons de N particules sont formés de variables aléatoires corrélées; bien que tout bloc de taille finie soit formé de variables asymptotiquement indépendantes lorsque la taille des populations tend vers l’infini. Dans cette étude, nous analysons finement la convergence de ces modèles et nous démontrons des théorèmes limites pour des U -statistiques associées aux mesures d’occupation de systèmes de particules en interaction. Mots-clés : Systèmes de particules en interaction, formules de Feynman-Kac, U -statistiques, fluctuations, théorèmes limites. in ria -0 03 97 36 6, v er si on 1 21 J un 2 00 9 Convergence of U -statistics for interacting particle systems 3 Introduction The convergence of U -statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial. When dealing with Feynman-Kac and other interacting particle systems of MonteCarlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated although any finite number of them is asymptotically independent with respect to the total number N of particles. It happens so (and this is the main contribution of the present article to show) that this asymptotic independence is enough in practice to insure the convergence of U -statistics based on interacting particle systems. In the following, we prove therefore the convergence of U -statistics for different particle systems under mild assumption that are satisfied by Feynmann-Kac particle systems. The case of Bird and Nanbu systems also fits in this framework and will be treated elsewhere by the third Author [Rub09]. To study the asymptotics of Feynman-Kac systems, whose properties are crucial to ensure the convergence of the statistics, we will use a functional representation, as introduced in [DPR09] in the framework of discrete interacting particle systems. The article is organized as follows. To fix the notation and the general framework of interacting particle systems, we first recall the (Feynman-Kac, continuous) interacting particle model. The model first appeared in quantum physics, in the work of Feynman and Kac in the 1940-50’s, as a way to encode the motion of a quantum particle evolving in a potential (e.g. the interaction potential of a quantum field theory, viewed as a perturbation of the free Hamiltonian) in terms of path-integral formulas. It was realized progressively that interacting particle systems could be used in incredibly many different settings in probability and statistics. A detailed list of the (still expanding) application areas of these models is contained in [Del04], to which we refer for further informations. Recall simply, since we focus here on its statistical features, that the model is mainly used, in applied statistics, as a Bayesian nonlinear filtering model: the motion of the particles is driven by a diffusion process and the potential encodes the likewood of the states with respect to observations or to some reference path. We study then the associated empirical joint distributions of a finite number k of particles and study the convergence of the distributions in terms of the total number N of particles of the system. This is closely related to our previous joint work [DPR09] on discrete FeynmanKac models -although the continuous hypothesis we use in the present article leads to some simplification of the tricky combinatorics that showed up in the discrete framework. We turn then to U -statistics for interacting particle systems and prove that under mild asumptions on the behavior of the system (satisfied e.g. by Feynman-Kac and Boltzmann systems) several asymptotic normality properties holds. 1 Feynman-Kac particle systems Let us consider a E-valued Markov process Xt, where E = R (or an arbitrary metric space) with a time-inhomogeneous infinitesimal generator Lt, continuous trajectories, and a positive bounded potential function Vt, 0 ≤ Vt(x) ≤ V∞. We assume that the distribution of X0 is γ0 = η0. Notice that these hypothesis are meaningful for most applications, but could be accomodated to more general ones, see [Del04]. We are interested in the unnormalized (resp. normalized) distribution flows γt and ηt that are solutions, for sufficiently regular test functions f and under appropriate regularity conditions, of the nonlinear equations: d dt γt(f) = γt(Lt(f))− γt(fVt) and d dt ηt(f) = ηt(Lt(f)) + ηt(f(ηt(Vt)− Vt)). In terms of Xt, we have: γt(f) = E ( f(Xt) exp ( − ∫ t 0 Vs(Xs)ds )) , ηt(f) = γt(f) γt(1) . RR n° 6966 in ria -0 03 97 36 6, v er si on 1 21 J un 2 00 9 4 Del Moral & Patras & Rubenthaler 1.1 Definitions and notation Let us fix first of all some notation. For q ∈ N∗, we write [q] := {1, . . . , q}. For q,N ∈ N∗, we set 〈q,N〉 := {s ∈ [N ], s injective}. For q even, we write Iq for the set of partitions of [q] in pairs. We have #Iq = q! 2q/2 ( q 2 ) ! . The set of smooth bounded (resp. smooth bounded symmetric) functions on E is written Bb(E) (resp. B b (E)). We also write B sym 0 (E ) for the set of symmetric functions: B 0 (E) := { F ∈ B b (E ) : ∫ E F (x1, . . . , xq)γt(dxq) = 0 } . Notice that the set of functions B 0 (E) depends on t, so that a better notation would be B 0,t (E). However, since in practice the abbreviated notation should not lead to confusion, we decided not emphasize this dependency for notational simplicity. We write simply B0(E) for the centered functions: F ∈ Bb(E) : ∫ E F (x)γt(dx) = 0 The empirical (possibly random) measure associated to a (possibly random) vector x = (x1, . . . , xN ) ∈ E is given by