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Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem

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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.

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Acknowledgements

The author would like to thank Michel Ledoux for his many comments and advice regarding the redaction of this paper as well as Jérôme Dedecker, Yvik Swan, Frédéric Chazal and anonymous reviewers for their multiple remarks.

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Correspondence to Thomas Bonis.

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The author was supported by the French Délégation Générale de l’Armement (DGA) and by ANR project TopData ANR-13-BS01-0008.

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Bonis, T. Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem. Probab. Theory Relat. Fields 178, 827–860 (2020). https://doi.org/10.1007/s00440-020-00989-4

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