The Annals of Applied Probability

Concentration inequalities for mean field particle models

Pierre Del Moral and Emmanuel Rio

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Abstract

This article is concerned with the fluctuations and the concentration properties of a general class of discrete generation and mean field particle interpretations of nonlinear measure valued processes. We combine an original stochastic perturbation analysis with a concentration analysis for triangular arrays of conditionally independent random sequences, which may be of independent interest. Under some additional stability properties of the limiting measure valued processes, uniform concentration properties, with respect to the time parameter, are also derived. The concentration inequalities presented here generalize the classical Hoeffding, Bernstein and Bennett inequalities for independent random sequences to interacting particle systems, yielding very new results for this class of models.

We illustrate these results in the context of McKean–Vlasov-type diffusion models, McKean collision-type models of gases and of a class of Feynman–Kac distribution flows arising in stochastic engineering sciences and in molecular chemistry.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 3 (2011), 1017-1052.

Dates
First available in Project Euclid: 2 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1307020390

Digital Object Identifier
doi:10.1214/10-AAP716

Mathematical Reviews number (MathSciNet)
MR2830611

Zentralblatt MATH identifier
1234.60019

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F99: None of the above, but in this section 60F10: Large deviations 82C22: Interacting particle systems [See also 60K35]

Keywords
Concentration inequalities mean field particle models measure valued processes Feynman–Kac semigroups McKean–Vlasov models

Citation

Del Moral, Pierre; Rio, Emmanuel. Concentration inequalities for mean field particle models. Ann. Appl. Probab. 21 (2011), no. 3, 1017--1052. doi:10.1214/10-AAP716. https://projecteuclid.org/euclid.aoap/1307020390


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