Let us say have a sequence of $n$ 2-$D$ random variables $X\_i=(\varepsilon\_i/\sqrt{n},i\varepsilon\_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon\_{i}$ are independent random variables such that $\mathbb{P}(\varepsilon\_i =\pm 1)=1/2$. Denote by $S\_n$ their sum and take a $2$-dimensional Gaussian random variable $Z$ with the same covariance matrix as $S\_n$. Any standard Berry-Esseen theorem (say by Gotze or Bhachattarya or Bentkus) gives us that for any convex set $C$ we have
\begin{equation}
|\mathbb{P}(S\_n\in C)-\mathbb{P}(Z\in C)|\leq c\gamma,
\end{equation}
where gamma is the sum of third absolute moments of $X\_i$, which in this case behaves like $n^{-1/2}$.


Question: is there a standard way to pass from the distance between distribution functions to expectations of Lipschitz functions? That is, suppose $f$ is Lipschitz, can we still bound 


\begin{equation}
|\mathbb{E}f(S\_n)-\mathbb{E}f(Z)|
\end{equation}
in therms of $\gamma$? If so, does the same bound of magnitude $\gamma$ still apply?



Berry-Esseen bound in 2 dimensions for linear combinations
I'm fairly confident the answer is yes, but unfortunately I can't specify an exact reference just now.


If you just want to see that one can get a similar bound (but with worse dependence on $\gamma$), then it's not too hard to prove multidimensional Berry--Esseen theorems via the "Lindeberg replacement method"; see, e.g., Exercise 11.46(a) (cf. Corollary 11.59) in <http://get.analysisofbooleanfunctions.org>


Probably to get the correct dependence on $\gamma$ in the Lipschitz case, the best strategy would be a proof via Stein's method. Unfortunately, I didn't see the exact result you want in, say, the Chen-Goldstein-Shao text, and unfortunately this Valiant-Valiant paper's Theorem 2 (<http://eccc.hpi-web.de/report/2010/179/>) is not quantitatively optimal (probably). 


But I feel pretty confident the result you want should be known. Perhaps try searching for 'multidimensional Stein's method'. 


(If you find an appropriate theorem, please post here!)


