Nearly optimal central limit theorem and bootstrap approximations in high dimensions

Abstract

In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors ${\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}}$ over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded ${\mathit{X}}_{\mathit{i}}$’s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form

$$\mathit{C}{({\mathit{B}}_{\mathit{n}}^2{log}^3\mathit{d}/\mathit{n})}^{1/2}log\mathit{n},$$

where d is the dimension of the vectors and ${\mathit{B}}_{\mathit{n}}$ is a uniform envelope constant on components of ${\mathit{X}}_{\mathit{i}}$’s. This bound is sharp in terms of d and ${\mathit{B}}_{\mathit{n}}$, and is nearly (up to $log\mathit{n}$) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded ${\mathit{X}}_{\mathit{i}}$’s, formulated solely in terms of moments of ${\mathit{X}}_{\mathit{i}}$’s. Finally, we demonstrate that the bounds can be further improved in some special smooth and moment-constrained cases.