Rates in the Central Limit Theorem and diffusion approximation via Stein's Method

Thomas Bonis (1)

1 DATASHAPE - Understanding the Shape of Data

We present a way to apply Stein's method in order to bound the Wasserstein distance between a, possibly discrete, measure and another measure assumed to be the invariant measure of a diffusion operator. We apply this construction to obtain convergence rates, in terms of p-Wasserstein distance for p ≥ 2, in the Central Limit Theorem in dimension 1 under precise moment conditions. We also establish a similar result for the Wasserstein distance of order 2 in the multidimensional setting. In a second time, we study the convergence of stationary distributions of Markov chains in the context of diffusion approximation, with applications to density estimation from geometric random graphs and to sampling using the Langevin Monte Carlo algorithm.

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Thomas Bonis. Rates in the Central Limit Theorem and diffusion approximation via Stein's Method. 2016. ⟨hal-01167372v2⟩