On standard normal convergence of the multivariate Student tstatistic for symmetric random vectors

It is proved that if the multivariate Student t-statistic based on i.i.d. symmetric random vectors is asymptotically standard normal, then these random vectors are in the gener-alized domain of attraction of the normal law. Uniform integrability is also considered, even in the absence of symmetry. 1. Introduction. Let X, Xi, i ∈ N, be i.i.d. Rd valued random variables that we assume are full, that is, if for each u ∈ Sd−1, the random variable uTX is not almost surely constant. X is said to be in the generalized domain of attraction of the normal law (X ∈ GDOAN for short) if there exist (nonrandom) matrices An and constant vectors bn such tha