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

TL;DR

This paper develops Stein's method to bound Wasserstein distances between probability measures, enabling convergence rate analysis in the Central Limit Theorem, diffusion approximation, and applications in data analysis.

Contribution

It introduces a novel approach using Stein's method to quantify convergence rates for measures related to diffusion processes and Gaussian measures, with applications in data analysis.

Findings

Bounded Wasserstein distance of order 2 between measures supported on .

Established convergence rates for the multi-dimensional Central Limit Theorem.

Provided bounds for diffusion approximation and Monte Carlo algorithms.

Abstract

We present a way to use Stein's method in order to bound the Wasserstein distance of order $2$ between two measures $\nu$ and $\mu$ supported on $\mathbb{R}^d$ such that $\mu$ is the reversible measure of a diffusion process. In order to apply our result, we only require to have access to a stochastic process $(X_t)_{t \geq 0}$ such that $X_t$ is drawn from $\nu$ for any $t > 0$. We then show that, whenever $\mu$ is the Gaussian measure $\gamma$, one can use a slightly different approach to bound the Wasserstein distances of order $p \geq 1$ between $\nu$ and $\gamma$ under an additional exchangeability assumption on the stochastic process $(X_t)_{t \geq 0}$. Using our results, we are able to obtain convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of order $p \geq 2$. Our results can also provide bounds for steady-state diffusion approximation, allowing us to tackle two problems appearing in the field of data analysis by giving a quantitative convergence result for invariant measures of random walks on random geometric graphs and by providing quantitative guarantees for a Monte Carlo sampling algorithm.