  In this note, we provide a Berry--Esseen bounds for rectangles in
high-dimensions when the random vectors have non-singular covariance matrices.
Under this assumption of non-singularity, we prove an $n^{-1/2}$ scaling for
the Berry--Esseen bound for sums of mean independent random vectors with a
finite third moment. The proof is essentially the method of compositions proof
of multivariate Berry--Esseen bound from Senatov (2011). Similar to other
existing works (Kuchibhotla et al. 2018, Fang and Koike 2020a), this note
considers the applicability and effectiveness of classical CLT proof techniques
for the high-dimensional case.
