Abstract
We extend Stein’s celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs to obtain an error bound for smooth test functions. We also obtain a continuous version of the multi-dimensional Wasserstein bound in terms of fourth moments. We apply the main results to multivariate normal approximations to Wishart matrices of size n and degree d, where we obtain the optimal convergence rate under only moment assumptions, and to degenerate U-statistics and Poisson functionals, where we strengthen a few of the fourth moment bounds in the literature on the Wasserstein distance.
Funding Statement
Fang X. was partially supported by Hong Kong RGC ECS 24301617 and GRF 14302418 and 14304917, a CUHK direct grant and a CUHK start-up grant. Koike Y. was partially supported by JST CREST Grant Number JPMJCR14D7 and JSPS KAKENHI Grant Numbers JP17H01100, JP18H00836, JP19K13668.
Acknowledgments
We thank the two anonymous referees for their careful reading of the manuscript and for their valuable suggestions which led to many improvements.
Citation
Xiao Fang. Yuta Koike. "New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems." Ann. Appl. Probab. 32 (1) 602 - 631, February 2022. https://doi.org/10.1214/21-AAP1690
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