  Central limit theorems (CLTs) for high-dimensional random vectors with
dimension possibly growing with the sample size have received a lot of
attention in the recent times. Chernozhukov et al. (2017) proved a
Berry--Esseen type result for high-dimensional averages for the class of
hyperrectangles and they proved that the rate of convergence can be upper
bounded by $n^{-1/6}$ upto a polynomial factor of $\log p$ (where $n$
represents the sample size and $p$ denotes the dimension). Convergence to zero
of the bound requires $\log^7p = o(n)$. We improve upon their result which only
requires $\log^4p = o(n)$ (in the best case). This improvement is made possible
by a sharper dimension-free anti-concentration inequality for Gaussian process
on a compact metric space. In addition, we prove two non-uniform variants of
the high-dimensional CLT based on the large deviation and non-uniform CLT
results for random variables in a Banach space by Bentkus, Ra{\v c}kauskas, and
Paulauskas. We apply our results in the context of post-selection inference in
linear regression and of empirical processes.
