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" http://www.springerlink.com/content/916020285j808858/

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 $c$.