Decoupling Inequalities for Multilinear Forms in Independent Symmetric Random Variables

Abstract

Let $X^1, X^2,\ldots$ be independent copies of a sequence $X = (X_1, X_2, \ldots)$ of independent symmetric random variables. Let $M$ be a symmetric multilinear form of rank $s$ on $\mathbb{R}^\mathbb{N}$ whose components $a_{i_1,\ldots, i_s}$ relative to the standard basis of $\mathbb{R}^\mathbb{N}$ satisfy $a_{i_1,\ldots, i_s} = 0$ for all but finitely many multi-indices and whenever two indices agree. If $\phi$ is nondecreasing, convex, $\phi(0) = 0$ and $\phi$ satisfies a $\Delta_2$ growth condition then $E\phi(|M(X,\ldots, X)|) \leq cE\phi(|M(X^1,\ldots, X^s)|),$ where $c$ depends only on $\phi$ and $s$.