Berry Esseen inequality for multidimensional distributions

Question

The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary $$ \sup_{t \in \mathbb{R}} |F(t) - G(t)| \ll \frac{1}{T} + \int_{-T}^{T} \bigg | \frac{f(t) - g(t)}{t} \bigg | dt $$ provided that one of the $F$ or $G$ is in Schwartz class (say). Is there a generalization of this inequality for distribution functions in $\mathbb{R}^k$, with $k = 2$ specifically? Precisely, I'm looking for a bound for $$ \sup_{\mathcal{R}} |\mathbb{P}(X \in \mathcal{R}) - \mathbb{P}(Y \in \mathcal{R}) | $$ in terms of the characteristic functions of $X$ and $Y$, with $X,Y$ random variables in $\mathbb{R}^2$, and $\mathcal{R}$ rectangles in $\mathbb{R}^2$.

Answer 1

The sharpest multidimensional Berry--Esseen Theorem I know is due to Bentkus and appears in the paper "A Lyapunov type bound in ${\mathbb R}^d$".

http://epubs.siam.org/doi/abs/10.1137/S0040585X97981123

It does not use the characteristic function, though.

Answer 2

There are many results along those lines in Bhattacharya and Rao, Normal Approximation and Asymptotic Expansions.

Answer 3

I was also looking for a higher dimensional analogue of the so-called Berry-Esseen inequality and ran into this unanswered question.

For a two dimensional analogue look at Theorem 1 and its Corollary in the paper On two-dimensional analogues of an inequality of Esseen and their application to the central limit theorem by S.M. Sadikova. It is mentioned here that the proof of the result generalizes to higher dimensions.

For an explicit statement of the general higher dimensional analogue look at Theorem 2 and Corollary 2.2 of the paper Higher dimensional quasi-power theorem and Berry–Esseen inequality by Clemens Heuberger and Sara Kropf.