High-dimensional CLT for Sums of Non-degenerate Random Vectors: $n^{-1/2}$-rate

Understand

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.