Multivariate asymptotic normality determined by high moments

TL;DR

This paper extends the moment method to multivariate cases, showing that high moments determine asymptotic normality in multiple dimensions, with applications to classical allocation schemes.

Contribution

It generalizes the moment method from one-dimensional to multivariate settings, providing new criteria for asymptotic normality based on high moments.

Findings

High moments determine multivariate asymptotic normality.

Application to joint distributions in allocation schemes.

Different from traditional moment convergence methods.

Abstract

We extend a general result showing that the asymptotic behavior of high moments, factorial or standard, of random variables, determines the asymptotically normality, from the one dimensional to the multidimensional setting. This approach differs from the usual moment method which requires that the moments of each fixed order converge. We illustrate our results by considering a joint distribution of the numbers of bins (having the same, finite, capacity) containing a prescribed number of balls in a classical allocation scheme.