Symmetrization and decoupling of combinatorial random elements

Abstract

Let Φ = ( φ ij ) 1 ⩽ ij⩽ n be a random matrix whose components φ ij are independent stochastic processes on some index set T . Let S = ∑ i=1 n φiπ(i) , where Π is a random permutation of {1,2, …, n}, independent from Φ. This random element is compared with its symmetrized version S 0 := ∑ i=1 n ξ iφiπ(i) and its decoupled version S := ∑ i=1 n φ iπ(i) , where ξ = ( ξ i ) 1 ⩽ i⩽ n is a Rademacher sequence and Π is uniformly distributed on {1,2,…,n} n such that Φ, Π, Π and ξ are independent. It is shown that for a broad class of convex functions Ψ on R T the following symmetrization and decoupling inequalities hold: EΨ(S−ES) ⩽ EΨ(kS 0) EΨ(γ( S−ES)) where κ, γ > 0 are universal constants.