Concentration inequalities for mean field particle models

TL;DR

This paper develops new concentration inequalities for mean field particle models, extending classical inequalities to interacting systems and demonstrating their application in various scientific models.

Contribution

It introduces a novel stochastic perturbation and concentration analysis for mean field particle systems, generalizing classical inequalities to dependent, interacting particles.

Findings

Derived uniform concentration inequalities for mean field models.

Extended classical Hoeffding, Bernstein, and Bennett inequalities to interacting particles.

Applied results to McKean-Vlasov, gas collision models, and Feynman-Kac flows.

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.