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In this work, a generalised version of the central limit theorem is proposedfor nonlinear functionals of the empirical measure of i.i.d. random variables,provided that the functional satisfies some regularity assumptions for theassociated linear functional derivative. This generalisation can be applied toMonte-Carlo methods, even when there is a nonlinear dependence on the measurecomponent. We use this result to deal with the contribution of theinitialisation in the convergence of the fluctuations between the empiricalmeasure of interacting diffusion and their mean-field limiting measure (as thenumber of particles goes to infinity), when the dependence on measure isnonlinear. A complementary contribution related to the time evolution istreated using the master equation, a parabolic PDE involving L-derivatives withrespect to the measure component, which is a stronger notion of derivative thatis nonetheless related to the linear functional derivative.

electronic journal of probability

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