NEW ERROR BOUNDS IN MULTIVARIATE NORMAL APPROXIMATIONS VIA EXCHANGEABLE PAIRS WITH APPLICATIONS TO WISHART MATRICES AND FOURTH MOMENT THEOREMS

• 成果类型: Article
• 署名作者: Fang, Xiao; Koike, Yuta
• 署名单位: Chinese University of Hong Kong; University of Tokyo
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1690
• 发表日期: 2022
• 页码: 602-631
• 关键词: steins method; nonnormal approximation; homogeneous sums; CONVERGENCE; clt; rates

摘要

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 root n(3)/d 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.