Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the   Invariant Measure of a Diffusion
  We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure $\nu$ of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f -- $\nu$(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr{\"o}benius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem. 
Igor Honor\'e (LaMME), Stephane Menozzi (LaMME), Gilles Pag\`es (LPMA)
math.PR