On short Edgeworth expansions for weighted sums of random vectors

The “typical” asymptotic behavior of the weighted sums of independent, identically distibuted random vectors in k-dimensional space is considered. It is shown that under finitnes of fifth absolute moment of an individual term the rate of convergence by Edgeworth correction in the multivariate central limit theorem is of order O(1/n^3/2 ). This extends the one-dimensional Bobkov(2020) result.