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Convergence of U-Statistics for Interacting Particle Systems

Author

Listed:
  • P. Moral

    (Université Bordeaux I)

  • F. Patras

    (Université de Nice-Sophia Antipolis)

  • S. Rubenthaler

    (Université de Nice-Sophia Antipolis)

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.

Suggested Citation

  • P. Moral & F. Patras & S. Rubenthaler, 2011. "Convergence of U-Statistics for Interacting Particle Systems," Journal of Theoretical Probability, Springer, vol. 24(4), pages 1002-1027, December.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-011-0355-6
    DOI: 10.1007/s10959-011-0355-6
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    Cited by:

    1. Radosław Adamczak & Piotr Miłoś, 2014. "$$U$$ U -Statistics of Ornstein–Uhlenbeck Branching Particle System," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1071-1111, December.
    2. Cloez, Bertrand & Corujo, Josué, 2022. "Uniform in time propagation of chaos for a Moran model," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 251-285.

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