Unconditional convergence of Gaussian random series in Banach spaces

A sufficient condition is given for the a.s. unconditional convergence of Gaussian series in Banach spaces with unconditional bases not containing $l^n_\infty$'s uniformly. By the a.s. unconditional convergence of random series we understand the convergence of all rearrangements of the series on the same set of total probability.