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 $mathcal{A}^{re}={prod_{j=1}^{p}[a_j,b_j]capmathcal{R}:-inftyleq a_jleq b_jleq infty, j=1,ldots,p}$, 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 $0< deltaleq 1$, the optimal rate of $log p$ is $obig(n^{delta/(2+delta)}big)$. 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 $eta_nrightarrow 0$ as $nrightarrow infty$. 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 respecti