Berry-Esseen bound in 2 dimensions for linear combinations

Question

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?

Answer 1

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!)