Sharp non-asymptotic concentration inequalities for the approximation of the invariant distribution of a diffusion

摘要

Let (Y-t)(t)(>= 0) be an ergodic diffusion with invariant distribution nu. Consider the empirical measure nu(n) := (Sigma(n)(k=1) gamma k)(-1) Sigma(n)(k=1)gamma k delta(Xk-1) where (X-k)(k >= 0) is an Euler scheme with decreasing steps (gamma(k))(k >= 0 )which approximates (Y-t)(t)(>= 0). Given a test function f, we obtain sharp concentration inequalities for nu(n)(f) - nu(f) which improve the results in Honore et al. (2019). Our hypotheses on the test function f cover many real applications: either f is supposed to be a coboundary of the infinitesimal generator of the diffusion, or f is supposed to be Lipschitz. (C) 2019 Elsevier B.V. All rights reserved.