Let $X_i$ be a sequence of i.i.d. $\mathbb{R}^d$-valued, continuous (i.e. with density) random variables. We assume that $E X_i =0$ and $Cov(X_i)=Id$. Let
$S_n:=\frac{1}{\sqrt{n}}\sum_{i=1}^n X_i.$
For $d=1$, under assumption of existence of the $k$-th moment, it is known that
$p_n(x)=g(x) + f_k(x) +o(n^{\frac{k-2}{2}}),$
where $g(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$, $p_n$ is the density of $S_n$ and $f_k$ is some (known) correction term. (This can be found in "Limit distributions..." - Gendenko, Kolmogorov, p.228).
Does anthing like that is known for $d>1$? Does existence of exponential moments of $X_i$ implies exponentailly small error?
