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](http://www.jstor.org/stable/2244370) 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.



Multivariate Berry-Esseen Theorem for possibly co-linear random vector
By Theorem 1.3 in the [article of Götze you linked to](https://projecteuclid.org/download/pdf_1/euclid.aop/1176990448), 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.


