  We calculate the p-the moment of the sum of n independent random variables
with respect to symmetric norm in R^n. The order of growth for upper bound p/ln
p obtained in ths estimate is optimal. The result extends to generalized
Lorentz spaces l_{f,w} under mild assumptions on f. Indeed, the key
combinatorial estimate is obtained for the weak l_1 (l_{1,infinity})-norm.
Similar results have been obtained independently by Gordon, Litvak, Schuett and
Werner for Orlicz norms and by Montgomery-Smith using different techniques and
avoiding the combinatorial estimate.
