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.

arXiv comment32 pages. Corrected some typos and streamlined proofs

References in corpus (4)

• Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition
• Stein's method and the zero bias transformation with application to simple random sampling
• A note on the exchangeability condition in Stein's method
• Existence of Stein Kernels under a Spectral Gap, and Discrepancy Bound

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