We extend a general result showing that the asymptotic behavior of high moments, factorial or standard, of random variables, determines 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.
Hitczenko et al. (2024) studied this question.