  For a Banach space X we define RUMD_n(X) to be the infimum of all c>0 such
that (AVE_{\epsilon_k =\pm 1} || \sum_1^n epsilon_k (M_k - M_{k-1}
)||_{L_2^X}^2 )^{1/2} <= c || M_n ||_{L_2^X} holds for all Walsh-Paley
martingales {M_k}_0^n subset L_2^X with M_0 =0. We relate the asymptotic
behaviour of the sequence {RUMD(X)}_{n=1}^{infinity} to geometrical properties
of the Banach space X such as K-convexity and superreflexivity.
