There are quite a few results in the literature that provide Berry-Esseen type bounds for the convergence of a standardized sum $S_n:=\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i$ of iid random vectors in $\mathbb{R}^d$, $X_1, \dots, X_n$ with $\mathbb{E}=0$ and $Cov(X_i)= I$, towards a standard normal vector $Y=\mathcal{N}(0, I)$. For example, this paper by Bentkus, where it is proven that, for the class of convex subsets of $\mathbb{R}^d$ denoted by $\mathcal{C}$ it holds $$ \sup_{C \in \mathcal{C}}| P(S_n \in C) - P(Y \in C)| \leq 400 d^{1/2}\beta/ \sqrt{n}. $$ There exist other results of that kind, for example by Goetze or Bhattacharya and Rao. The authors also speak like a generalization to random vectors $X'$ with general Covariance $Cov(X')=\Sigma$ is rather straightforward. However, I do not see how this is done. The commenter, Did, in this related post Rate of Convergence in CLT for IID random vectors with dependent entries has seemingly provided an answer. He says: "You might want to use the fact that $L\Sigma L^T=I$ for some $L$". As far as I understand this means I could generalize the result in the following way:
Define $S'_n= \frac{1}{\sqrt{n}} \sum_{i=1}^nX'_i$, and $Y'= \mathcal{N}(0, \Sigma)$. Above result then can be rewritten as $$ \sup_{C \in \mathcal{C}}| P(LS'_n \in C) - P(LY' \in C)| \leq 400 d^{1/2}\beta/ \sqrt{n}. $$ and hence $$ \sup_{C \in \mathcal{C}}| P(S'_n \in L^{-1}C) - P(Y' \in L^{-1}C)| \leq 400 d^{1/2}\beta/ \sqrt{n}. $$ and since $L^{-1}C$ is still a convex set, $$ \sup_{C \in \mathcal{C}}| P(S'_n \in C) - P(Y' \in C)| \leq 400 d^{1/2}\beta/ \sqrt{n}. $$
Is this correct? It seems weird to me that the bound does not change actually.