A multivariate CLT for <> weighted sums with rate of convergence of order O(1/n)

Abstract

The "typical" asymptotic behavior of the weighted sums of independent random vectors in -dimensional space is considered. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order (1/ ). This extends the one-dimensional Klartag and Sodin (2011) result.