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

The “typical” asymptotic behavior of the weighted sums of independent random vectors in $k$-dimensional space is considered. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order $O(1\mathrm{/}n)$. This extends the one-dimensional Klartag and Sodin (2011) result.