Convergence of U-Statistics for Interacting Particle Systems

Article

Abstract

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 (Lee in Statistics: Textbooks and Monographs, vol. 10, Dekker, New York, 1990; de la Peña and Giné in Decoupling. Probability and Its Application, Springer, New York, 1999). 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.

Keywords

Interacting particle systems Feynman–Kac models U-statistics Fluctuations Limit theorems 

Mathematics Subject Classification (2000)

82C22 60K35 65C35 60F05 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centre INRIA Bordeaux Sud-Ouest & Institut de Mathématiques de BordeauxUniversité Bordeaux ITalence cedexFrance
  2. 2.Laboratoire de Mathématiques J.A. Dieudonné, CNRS UMR 6621Université de Nice-Sophia AntipolisNice cedex 2France

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