The optimal order for the p-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates

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We calculate the p-the moment of the sum of n independent random variables with respect to symmetric norm in Rⁿ. The order of growth for upper bound p/ln p obtained in ths estimate is optimal. The result extends to generalized Lorentz spaces lf,w under mild assumptions on f. Indeed, the key combinatorial estimate is obtained for the weak l₁ (l1,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.