Uniform Central Limit Theorem for self normalized sums in high dimensions

In this article, we are interested in the normal approximation of the self-normalized random vector $\Big(\frac{\sum_{i=1}^{n}X_{i1}}{\sqrt{\sum_{i=1}^{n}X_{i1}^2}},\dots,\frac{\sum_{i=1}^{n}X_{ip}}{\sqrt{\sum_{i=1}^{n}X_{ip}^2}}\Big)$ in $\mathcal{R}^p$ uniformly over the class of hyper-rectangles $\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leq b_j\leq \infty, j=1,\ldots,p\}$, where $X_1,\dots,X_n$ are non-degenerate independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of $\log p$ in the uniform central limit theorem (UCLT) under variety of moment conditions. When $X_{ij}$'s have $(2+\delta)$th absolute moment for some $0< \delta\leq 1$, the optimal rate of $\log p$ is $o\big(n^{\delta/(2+\delta)}\big)$. When $X_{ij}$'s are independent and identically distributed (iid) across $(i,j)$, even $(2+\delta)$th absolute moment of $X_{11}$ is not needed. Only under the condition that $X_{11}$ is in the domain of attraction of the normal distribution, the growth rate of $\log p$ can be made to be $o(\eta_n)$ for some $\eta_n\rightarrow 0$ as $n\rightarrow \infty$. We also establish that the rate of $\log p$ can be pushed to $\log p =o(n^{1/2})$ if we assume the existence of fourth moment of $X_{ij}$'s. By an example, it is shown however that the rate of growth of $\log p$ can not further be improved from $n^{1/2}$ as a power of $n$. As an application, we found respective versions of the high dimensional UCLT for component-wise Student's t-statistic. An important aspect of the these UCLTs is that it does not require the existence of some exponential moments even when dimension $p$ grows exponentially with some power of $n$, as opposed to the UCLT of normalized sums. Only the existence of some absolute moment of order $\in [2,4]$ is sufficient.