On the generalized domain of attraction of the multivariate normal law and asymptotic normality of the multivariate Student tt-statistic

It is well-known that if a random vector XX is in the generalized domain of attraction of the multivariate normal law (GDAN), then all its components are in the domain of attraction of the normal law (DAN) and, moreover, the Euclidean inner products of XX with all the nonrandom vectors of unit Euclidean norm are also in DAN. However, these two implications are known to be nonreversible in general. In this paper, a condition is given under which these implications are proved to become reversible, and thus characterizations of GDAN. Large enough classes and an example of random vectors satisfying this condition are provided. Also, the multivariate Student tt-statistic that is based on independent copies of a random vector XX satisfying this condition is proved to be asymptotically standard normal only if XX is in GDAN. A corollary to the thus established result parallels a previous resolution of this problem for a spherically symmetric XX in the literature.