Gaussian approximations for high-dimensional non-degenerate [formula omitted]-statistics via exchangeable pairs

Abstract

In this paper, we obtain a non-asymptotic bound for Gaussian approximations for centered high-dimensional non-degenerate U-statistics over the class of hyperrectangles via exchangeable pairs and Stein’s method. We improve the upper bound of the convergence rate from n−1/6 in Chen (2018) to n−1/4 up to a polynomial factor of logd under the same conditions, where n is the sample size and d is the dimension of the U-statistic. Convergence to zero of the bound requires logd=o(n1/7) in Chen (2018), this requirement on d is weaken in this paper by allowing logd=o(n1/5).