The Berry-Esseen Bound for High-dimensional Self-normalized Sums

This manuscript studies the Gaussian approximation of the coordinate-wise maximum of self-normalized statistics in high-dimensional settings. We derive an explicit Berry-Esseen bound under weak assumptions on the absolute moments. When the third absolute moment is finite, our bound scales as ${\log }^{5\mathrm{/}4}(d)\mathrm{/}{n}^{1\mathrm{/}8}$ where $n$ is the sample size and $d$ is the dimension. Hence, our bound tends to zero as long as $\log (d)=o(n^{1\mathrm{/}10})$. Our results on self-normalized statistics represent substantial advancements, as such a bound has not been previously available in the high-dimensional central limit theorem (CLT) literature.