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

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 $$$$\mathrm{Cov}\{(X_1,Y_1)\}=\begin{pmatrix} 1 & \sigma\\ \sigma & 1 \end{pmatrix}.$$$$ We assume the covariance matrix can be nearly degenerate, that is, $$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$$, such that $$$$\sup_{(x,y)\in\mathbb{R}^2}\left|P\left(\frac{1}{\sqrt{n}}\sum X_i\leq x,\frac{1}{\sqrt{n}}\sum Y_i\leq y\right)-P\left(Z_1\leq x,Z_2\leq y\right)\right|\leq\frac{c}{\sqrt{n}},$$$$ where $$(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.