The Multidimensional Central Limit Theorem for Arrays Normed by Affine Transformations

Abstract

Let $X_{n1}, \cdots, X_{nk_n}$ be independent random vectors in $\mathbb{R}^d$. Necessary and sufficient conditions are found for the existence of linear operators $A_n$ on $\mathbb{R}^d$ such that $\mathscr{L}(A_n(\sum^{k_n}_{j = 1} X_{nj})) \rightarrow N(\overset{\rightarrow}{0}, I)$, where $I$ is the $d \times d$ identity covariance matrix. These results extend the authors' previous work on sums of i.i.d. random vectors. The proof of the main theorem is constructive, yielding explicit centering vectors and norming linear operators.