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In this article, we are interested in the high dimensional normalapproximation 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$\mathcal{A}^{re}=\{\prod_{j=1}^{p}[a_j,b_j]\cap\mathcal{R}:-\infty\leq a_j\leqb_j \leq \infty, j=1,\ldots,p\}$, where $X_1,\dots,X_n$ are non-degenerateindependent $p-$dimensional random vectors. We assume that the components of$X_i$ are independent and identically distributed (iid) and investigate theoptimal 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 momentconditions, generally assumed for exponential growth of the dimension withrespect to the sample size in high dimensional CLT, to some polynomial momentconditions. Indeed, we establish that only the existence of some polynomialmoment 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 anapplication, we find respective versions for component-wise Student'st-statistic, which may be useful in high dimensional statistical inference.

Central Limit Theorem and Near classical Berry-Esseen rate for self normalized sums in high dimensions