  In this article, we are interested in the high dimensional normal
approximation of $T_n
=\Big(\sum_{i=1}^{n}X_{i1}/\Big(\sqrt{\sum_{i=1}^{n}X_{i1}^2}\Big),\dots,$
$\sum_{i=1}^{n}X_{ip}/\Big(\sqrt{\sum_{i=1}^{n}X_{ip}^2}\Big)\Big)$ 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. We assume that the components of
$X_i$ are independent and identically distributed (iid) and investigate the
optimal cut-off rate of $\log p$ 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 $\log p$ can not further be improved from
$o\big(n^{1/2}\big)$ as a power of $n$ even if $X_{ij}$'s are iid across
$(i,j)$ and $X_{11}$ is bounded. We also establish near$-n^{-\kappa/2}$
Berry-Esseen rate for $T_n$ in high dimension under the existence of
$(2+\kappa)$th absolute moments of $X_{ij}$ for $0< \kappa \leq 1$. When
$\kappa =1$, the obtained Berry-Esseen rate is also shown to be optimal. As an
application, we find respective versions for component-wise Student's
t-statistic, which may be useful in high dimensional statistical inference.
