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This is a follow up to this recent question: Berry Esseen type result for probability density functions

There exists a multidimensional version of the usual Berry-Esseen theorem (for cumulative distribution functions), see for example this paper (which I cannot find online) of 1945 by Bergström "On the central limit theorem in the space Rk, k> 1".

There also exists a local limit version of the Berry-Esseen theorem for probability density functions: see for instance chapter VII of the book "Sums of independent random variables" by Petrov.

I was not able, however, to find a multidimensional version of this local limit theorem. Does such a result appear somewhere in the literature?

Many thanks!

Edit (October 22): I've found a partial answer in this paper of Ričardas Zitikis: "A Berry-Esseen bound for multivariate l-estimates with explicit dependence on dimension"

In this paper, a bound is provided (Theorem 1.2) but still depends on a universal constant $c$. It would still be great to known an upper bound on $c$.

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I don't know of a precise reference, but mathscinet returns a large number of papers on this topic (many in russian, though). I don't have access to these journals, so I can't check, but looking at the first page of, it might contain something relevant... – Yvan Velenik Oct 13 '11 at 16:14
@Yvan Velenik Thanks a lot! I don't have access to mathscinet but I found the paper you mentioned: it can be downloaded here However, there are many notations which are obscure for me, and I'm wondering if there exists some more modern treatment of the question somewhere. – Anthony Leverrier Oct 18 '11 at 9:48

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