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For n independent random variables f1, . . . ,fn and a symmetric norm || ||X on ℝn, we show that for 1≤ p < ∞     Here      is the disjoint sum of the fi's and h* is the non-increasing rearrangement. Similar results (where Lp is replaced by a more general rearrangement invariant function space) were obtained first by Litvak, Gordon, Schtt and Werner for Orlicz spaces X and independently by S. Montgomery-Smith [22] for general X but without an explicit analysis of the order of growth for the constant in the upper estimate. The order  is optimal and obtained from combinatorial estimates for doubly stochastic matrices. The result extends to Lorentz-norms lf, q on ℝn under mild assumptions on f. We give applications to the theory of noncommutative Lp spaces.

The Optimal Order for the p-th Moment of Sums of Independent Random Variables with Respect to Symmetric Norms and Related Combinatorial Estimates