Convergence of U-Statistics for Interacting Particle Systems

  • P. Del Moral
  • F. Patras
  • S. RubenthalerEmail author


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.


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

Mathematics Subject Classification (2000)

82C22 60K35 65C35 60F05 


  1. 1.
    Del Moral, P.: Feynman-Kac formulae. Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications. Springer, New York (2004) zbMATHGoogle Scholar
  2. 2.
    Del Moral, P., Miclo, L.: Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. In: Séminaire de Probabilités, XXXIV. Lecture Notes in Mathematics, vol. 1729, pp. 1–145. Springer, Berlin (2000) CrossRefGoogle Scholar
  3. 3.
    Del Moral, P., Miclo, L.: Strong propagations of chaos in Moran’s type particle interpretations of Feynman-Kac measures. Stoch. Anal. Appl. 25(3), 519–575 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Del Moral, P., Patras, F., Rubenthaler, S.: Coalescent tree based functional representations for some Feynman-Kac particle models. Ann. Appl. Probab., 19(2), 778–825 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dynkin, E.B., Mandelbaum, A.: Symmetric statistics, Poisson point processes, and multiple Wiener integrals. Ann. Stat. 11(3), 739–745 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Itzykson, C., Zuber, J.-B.: Quantum Field Theory. International Series in Pure and Applied Physics. McGraw-Hill, New York (1980) Google Scholar
  7. 7.
    Lee, A.J.: Theory and practice, U-statistics. Statistics: Textbooks and Monographs, vol. 110. Dekker, New York (1990) Google Scholar
  8. 8.
    de la Peña, V.H., Giné, E.: From dependence to independence, Randomly stopped processes. U-statistics and processes. Martingales and beyond. In: Decoupling, Probability and Its Applications. Springer, New York (1999) Google Scholar
  9. 9.
    Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Addison-Wesley, Reading (1995). Edited and with a foreword by David Pines Google Scholar
  10. 10.
    Rubenthaler, S.: Expansion of the propagation of chaos for Bird and Nanbu systems, Tech. report, HAL (2009).
  11. 11.
    Rota, G.-C., Wallstrom, T.C.: Stochastic integrals: a combinatorial approach. Ann. Probab. 25(3), 1257–1283 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Rousset, M.: On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38(3), 824–844 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Wick, G.C.: The evaluation of the collision matrix. Phys. Rev. (2) 80, 268–272 (1950) MathSciNetzbMATHCrossRefGoogle Scholar

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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