Unconditional convergence of weakly sub gaussian series in Banach spaces

Characterizations of the class of Banach spaces isomorphing to the space $c_0$, as well as to the class of Banach spaces not containing $l_\infty^n$'s uniformly, are obtained in terms of almost surely unconditional convergence of weakly sub-Gaussian random series. Under almost surely unconditional convergence of random series, convergence of all permutations on the same set of full probability is understood. The questions of almost surely unconditional and weak absolute convergence in the spaces isomorphing to $c_0$ are investigated as well.