On Short Chebyshev--Edgeworth Expansions for Weighted Sums of Random Vectors

We consider “typical” asymptotic behavior of weighted sums of independent identically distributed random vectors in the $k$-dimensional space. We show that if the fifth absolute moment of a separate term is finite, then in the multidimensional central limit theorem, the convergence rate is $O(1/n^{3/2})$ under the Chebyshev--Edgeworth correction. This result generalizes a result of Bobkov [Edgeworth corrections in randomized central limit theorems, in Geometric Aspects of Functional Analysis, Springer, 2020, pp. 71--97] to the multidimensional case.