LJKProbability & Statistics Seminar

On Thursday January 12 2017 at 14h00 in Room 106  IMAG Building

Seminary of Stéphane MENOZZI (Université d'Evry)

NonAsymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion

Summary

We obtain nonasymptotic Gaussian concentration bounds for the difference between the invariant measure ν 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 − ν(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fröbenius norm of the diffusion coefficient lies in this class. In some cases, we can even find the expected asymptotic optimal variance.
We apply these bounds to design computable nonasymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive nonasymptotic deviation bounds for the almost sure Central Limit Theorem.
