Order-of-Magnitude Bounds for Expectations Involving Quadratic Forms

Abstract

Let $X_1, X_2,\ldots,X_n$ be independent mean-zero random variables and let $a_{ij}, 1 \leq i, j \leq n$, be an array of constants with $a_{ii} \equiv 0$. We present a method of obtaining the order of magnitude of $E\Phi(\sum_{1\leq i,j\leq n}a_{ij}X_iX_j)$ for any such $\{X_i\}$ and $\{a_{ij}\}$ and any nonnegative symmetric (convex) function $\Phi$ with $\Phi(0) = 0$ such that, for some integer $k \geq 0, \Phi(x^{2-k})$ is convex and simultaneously $\Phi(x^{2^{-k-1}})$ is concave on $\lbrack 0, \infty)$. The approximation is based on decoupling inequalities valid for all such mean-zero $\{X_i\}$ and reals $\{a_{ij}\}$ and a certain further "independentization" procedure.