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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?

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up vote 2 down vote accepted

The answer to your first question is yes. See Chapter 19 of *Normal Approximation and Asymptotic Expansions 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.

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