Random rearrangements and operators

Consider an n by n matrix x_ij, and consider the quantity || x_{i,pi(i)} ||_X where X is a symmetric sequence space as a random variable where the permutation pi is chosen randomly. This was considered by Kwapien and Schutt, and we extend their results to rearrangement invariant spaces. We also consider the property of D and D^* convexity for rearrangement invariant spaces.