  We study multivariate generalizations of the $q$-central limit theorem, a
generalization of the classical central limit theorem consistent with
nonextensive statistical mechanics. Two types of generalizations are addressed,
more precisely the {\it direct} and {\it sequential} $q$-central limit theorems
are proved. Their relevance to the asymptotic scale invariance of some
specially correlated systems is studied. A $q$-analog of the classic weak
convergence is introduced and its equivalence to the $q$-convergence is proved
for $q>1$.
