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

TL;DR

This paper establishes that the weighted sums of independent random vectors in multivariate space converge to a normal distribution at a rate of O(1/n), extending previous one-dimensional results.

Contribution

It extends the multivariate central limit theorem to show a convergence rate of O(1/n) for weighted sums, generalizing prior one-dimensional findings.

Findings

Convergence rate of O(1/n) for multivariate weighted sums

Extension of Klartag and Sodin's 2011 one-dimensional result

Applicable to independent random vectors in k-dimensional space

Abstract

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 $O(1/n)$. This extends the one-dimensional Klartag and Sodin (2011) result.