Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion

2016/05/27 by Igor Honoré, Honoré, Igor, Stéphane Menozzi +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1605.08525

openalex publication_date 2016/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain non-asymptotic 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. 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.

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