Let $S_n$ be a sequence of partial sums of mean zero purely $d$-dimensional i.i.d. random vectors. Necessary and sufficient conditions are given for the existence of matrices $A_n$ such that the transform of $S_n$ by $A_n$ is asymptotically multivariate normal with identity covariance matrix. This is more general than previous $d$-dimensional results. Examples are given to illustrate the need for the present approach. The matrices $A_n$ take a particularly simple form because of a degree of uncorrelatedness between certain pairs of 1-dimensional random variables obtained by projection.
"Matrix Normalization of Sums of Random Vectors in the Domain of Attraction of the Multivariate Normal." Ann. Probab. 8 (2) 262 - 280, April, 1980. https://doi.org/10.1214/aop/1176994776