Central limit theorem and near classical Berry-Esseen rate for self normalized sums in high dimensions

Abstract

In this article, we are interested in the normal approximation of

$$T_n=(\sum _{i=1}^n{X}_{i1}∕(\sqrt{\sum _{i=1}^n{X}_{i1}^2}),\dots ,\sum _{i=1}^n{X}_{ip}∕(\sqrt{\sum _{i=1}^n{X}_{ip}^2}\left)\right)$$

in ${\mathcal{R}}^p$ uniformly over the class of hyper-rectangles ${\mathcal{A}}^{re}=\{{\prod }_{j=1}^p[a_j,b_j]\cap \mathcal{R}:-\mathrm{\infty }\le a_j\le b_j\le \mathrm{\infty },j=1,\dots ,p\}$, where $X_1,\dots ,X_n$ are non-degenerate independent p-dimensional random vectors. We assume that the components of $X_i$ are independent and identically distributed (iid) and investigate the optimal cut-off rate of $logp$ in the uniform central limit theorem (UCLT) for $T_n$ over ${\mathcal{A}}^{re}$. The aim is to reduce the exponential moment conditions, generally assumed for exponential growth of the dimension with respect to the sample size in high dimensional CLT, to some polynomial moment conditions. Indeed, we establish that only the existence of some polynomial moment of order $\in [2,4]$ is sufficient for exponential growth of p. However the rate of growth of $logp$ cannot further be improved from $o\left(n^{1∕2}\right)$ as a power of n even if $X_{ij}$’s are iid across $(i,j)$ and $X_{11}$ is bounded. We also establish near$-n^{-\mathrm{\kappa }∕2}$ Berry-Esseen rate for $T_n$ in high dimension under the existence of $(2+\mathrm{\kappa })$th absolute moments of $X_{ij}$ for $0<\mathrm{\kappa }\le 1$. When $\mathrm{\kappa }=1$, the obtained Berry-Esseen rate is also shown to be optimal. As an application, we find respective versions for componentwise Student’s t-statistic, which may be useful in high dimensional statistical inference.