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 $c$.
springerlink.com
in a comment above seems to be broken. The alternative link atperso.telecom-paristech.fr
in another comment above also seems to be broken. I'm unable to find any snapshots saved on the Wayback Machine. $\endgroup$