Stein's method for normal approximation in Wasserstein distances with application to the multivariate Central Limit Theorem

Abstract

We use Stein's method to bound the Wasserstein distance of order $2$ between a measure $\nu$ and the Gaussian measure using a stochastic process $(X_t{)}_{t\ge 0}$ such that $X_t$ is drawn from $\nu$ for any $t>0$. If the stochastic process $(X_t{)}_{t\ge 0}$ satisfies an additional exchangeability assumption, we show it can also be used to obtain bounds on Wasserstein distances of any order $p\ge 1$. Using our results, we provide optimal convergence rates for the multi-dimensional Central Limit Theorem in terms of Wasserstein distances of any order $p\ge 2$ under simple moment assumptions.

32 pages. Corrected some typos and streamlined proofs