New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems

2020/04/05 by Xiao Fang, Fang, Xiao, Yuta Koike +1 · 2 citations
Computer Science · Mathematics · #60F05 #62E17 #Complexity and Algorithms in Graphs #FOS: Mathematics #Mathematical Approximation and Integration #Point processes and geometric inequalities #Probability (math.PR) #math.PR #msc:60F05 #msc:62E17

paper · pdf · doi:10.48550/arxiv.2004.02101

35 pages. Major update. Added applications to degenerate U-statistics and homogeneous sums. Obtained the optimal Wasserstein bound for the Wishart matrix example

openalex publication_date 2020/04/05 · arxiv created 2020/09/19 · arxiv updated 2020/09/22 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28

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 √(n³/d) under only moment assumptions, and to quadratic forms and Poisson functionals, where we strengthen a few of the fourth moment bounds in the literature on the Wasserstein distance.

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