The classical BerryEsseen 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 $\mathbb{R}^2$.

$\begingroup$ There are many ways to bound that quantity. Why does it have to be in term of characteristic functions in particular? $\endgroup$ – an12 Oct 29 '12 at 6:22
The sharpest multidimensional BerryEsseen 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.

1$\begingroup$ There's now also the followup work in arxiv.org/pdf/1802.06475 which has explicit constants. $\endgroup$ – Thomas Dybdahl Ahle Oct 10 '19 at 13:53
There are many results along those lines in Bhattacharya and Rao, Normal Approximation and Asymptotic Expansions.
I was also looking for a higher dimensional analogue of the socalled BerryEsseen inequality and ran into this unanswered question.
For a two dimensional analogue look at Theorem 1 and its Corollary in the paper On twodimensional 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 quasipower theorem and Berry–Esseen inequality by Clemens Heuberger and Sara Kropf.