Symmetrization and decoupling of combinatorial random elements

Let Phi = (phi(if))(1 less than or equal to i,j less than or equal to n) be a random matrix whose components phi(ij) are independent stochastic processes on some index set J. Let S = Sigma(i=1)(n) phi(i Pi(i)), where Pi is a random permutation of {1,2,...,n}, independent from Phi. This random element is compared with its symmetrized version S-0 := Sigma(i=1)(n) xi(i)phi(i Pi(i)) and its decoupled version (S) over tilde := Sigma(i=1)(n) phi(<(i Pi)over tilde>(i)), where xi = (xi(i))(1 less than or equal to i less than or equal to n) is a Rademacher sequence and <(Pi)over tilde> is uniformly distributed on {1,2,...,n}(n) such that Phi, Pi, <(Pi)over tilde> and xi are independent. It is shown that for a broad class of convex functions Psi on R-J the following symmetrization and decoupling inequalities hold: [GRAPHICS] where kappa, gamma > 0 are universal constants. (C) 1998 Elsevier Science B.V. All rights reserved.