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.