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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 of 1945 by Bergström "On the central limit theorem in the space $R_k$, $k > 1$". Skand. Aktuarie Tidskr. 28, 106–127. Zbl 0060.28708

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 – Esséen bound for multivariate $L$-estimates with explicit dependence on dimension". In: Kalashnikov, V.V., Zolatarev, V.M. (eds) Stability Problems for Stochastic Models. Lecture Notes in Mathematics, vol 1546. Springer, Berlin, Heidelberg. Zbl 0804.60023

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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    $\begingroup$ 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 springerlink.com/content/v7071t24g455837h, it might contain something relevant... $\endgroup$ Oct 13, 2011 at 16:14
  • $\begingroup$ @Yvan Velenik Thanks a lot! I don't have access to mathscinet but I found the paper you mentioned: it can be downloaded here perso.telecom-paristech.fr/~leverrie/Rozowski.pdf 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. $\endgroup$ Oct 18, 2011 at 9:48
  • $\begingroup$ The link to springerlink.com in a comment above seems to be broken. The alternative link at perso.telecom-paristech.fr in another comment above also seems to be broken. I'm unable to find any snapshots saved on the Wayback Machine. $\endgroup$ Feb 24, 2023 at 12:45

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