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?



Multivariate CLT, convergence of densities
The answer to your first question is yes. See Chapter 19 of [\*Normal Approximation and Asymptotic Expansions](http://books.google.com/books?id=H1lOIVHcRDEC&lpg=PP1&dq=bhattacharya%2520and%2520rao&pg=PP1#v=onepage&q&f=false) by Bhattacharya and Rao.


I believe the answer to your second question is no, but don't know where to find details offhand.


In general, the kinds of results you're looking for are called *local limit theorems*.


