Multivariate Berry-Esseen Theorem for possibly co-linear random vector

Question

I was considering the following problem. Let $\{(X_i,Y_i)\}_{i=1}^n$ be i.i.d. zero-mean random vectors with covariance matrix \begin{equation} \mathrm{Cov}\{(X_1,Y_1)\}=\begin{pmatrix} 1 & \sigma\\ \sigma & 1 \end{pmatrix}. \end{equation} We assume the covariance matrix can be nearly degenerate, that is, $X$ and $Y$ can be highly Co-linear (but not exactly, since otherwise it reduces to univariate case). Now I want to know, if the multivariate Berry-Esseen Theorem still holds true. Mathematically speaking, does there exists a uniform constant $c>0$, such that \begin{equation} \sup_{(x,y)\in\mathbb{R}^2}\left|P\left(\frac{1}{\sqrt{n}}\sum X_i\leq x,\frac{1}{\sqrt{n}}\sum Y_i\leq y\right)-P\left(Z_1\leq x,Z_2\leq y\right)\right|\leq\frac{c}{\sqrt{n}}, \end{equation} where $(Z_1,Z_2)$ admits multivariate normal distribution with mean zero and covariance matrix $\mathrm{Cov}\{(X_1,Y_1)\}$, holds true for all $-1\leq\sigma\leq1$? I have found some articles about multivariate Berry-Esseen theorem, usually they assume the covariance matrix to be identity matrix in order to bound the variance of each component, but I'm not sure whether Co-linearity affect the result.

Answer 1

By Theorem 1.3 in the article of Götze you linked to, the answer is yes, because the result holds for all convex sets and by a linear transformation you can make the covariance an identity. In fact a result of Sazonov (1968) that Götze refers to already contains this.