Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting particle systems.

Summary In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivatives of various orders. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. As a consequence of this result, we also analyse the convergence of fluctuation between the empirical measure of particles in an interacting particle system and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear.