https://cnrs.hal.science/hal-03890805Pagès, GillesGillesPagèsLPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche ScientifiqueRey, ClémentClémentReyFirst and second order Central Limit Theorems for the recursive computation of the invariant distribution of a Feller processHAL CCSD2022[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Pagès, Gilles2022-12-08 17:52:422023-02-11 04:05:082022-12-08 17:52:42enPreprints, Working Papers, ...1This paper studies the convergence of empirical measures of a stochastic approximation toward the invariant distribution of a Feller process. In particular, we provide a general and abstract approach to establish Central Limit Theorems (CLT) with given rate . Moreover, considering weighted empirical measures of a weak order two stochastic approximation, we show its second order convergence while the CLT for standard empirical measures has order one. We also propose various applications: First order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from literature) and second order CLT for the approximation of Brownian diffusion stationary regimes using Talay scheme (1990) of weak order two.