Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

We provide a synthetic yet comprehensive review of the so-called fourth moment criterion, and of universal limit theorems, for multilinear homogeneous sums, in both the classical and the free probability settings. In addition to such a general picture, we also prove a novel multidimensional transfer principle for Central Limit Theorems involving homogeneous sums with leptokurtic or mesokurtic entries. The key step will be to prove that joint and component-wise convergence are indeed equivalent for these random objects, encompassing well-known results concerning Wiener and Wigner Chaoses.