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