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The central limit theorem is, with the strong law of large numbers, one ofthe two fundamental limit theorems in probability theory. Benjamin Jourdain andAlvin Tse have extended to non-linear functionals of the empirical measure ofindependent and identically distributed random vectors the central limittheorem which is well known for linear functionals. The main tool permittingthis extension is the linear functional derivative, one of the notions ofderivation on the Wasserstein space of probability measures that have recentlybeen developed. The purpose of this work is to generalize what has been done byJourdain and Tse: provide a Central Limit Theorem for non-linear functionals ofindependent and non equidistributed random vectors such as the successivevalues of an ergodic Markov chain.

Central limit theorem over non-linear functionals of empirical measures: beyond the iid setting