Decoupling Inequalities for Polynomial Chaos

Abstract

Let $X, X_1,\ldots, X_d$ be a sequence of independent, symmetric, identically distributed random vectors with independent components. The main subject of this paper is the so-called decoupling inequalities, i.e., inequalities of the form \begin{align*}E\phi (cQ(X, X,\ldots, X)) &\leq E\phi (Q(X_1, X_2,\ldots, X_d)) \\ &\leq E\phi(CQ(X, X,\ldots, X)), \\ \end{align*} where $Q$ is a symmetric multilinear form with values in a vector space $F$ with all "diagonal" terms equal to zero and $\phi$ is a convex function on $F$.