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

Abstract

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