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 convergence rates for the multi-dimensional central limit theorem in terms of Wasserstein distances of any order $$p \ge 2$$ under simple moment assumptions.