Order of magnitude bounds for expectations of $\Delta_2$-functions of nonnegative random bilinear forms and generalized $U$-statistics

Abstract

Let $X_1, Y_1, Y_2, \dots, X_n, Y_n$ be independent nonnegative rv’s and let $\{b_{ij}\}_{1 \leq i, j \leq n}$ be an array of nonnegative constants. We present a method of obtaining the order of magnitude of

$$E \Phi (\sum_{1 \leq i, j \leq n} b_{ij} X_i Y_j),$$

for any such ${X_i}, {Y_j}$ and ${b_{ij}}$ and any nondecreasing function $\Phi$ on $[0, \infty)$ with $\Phi (0) = 0$ and satisfying a $\Delta_2$ growth condition. Furthermore, this technique is extended to provide the order of magnitude of

$$E \Phi (\sum_{1 \leq i, j \leq n} f_{ij} (X_i Y_j)),$$

where ${f_{ij} (x, y)}_{1 \leq i, j \leq n}$ is any array of nonnegative functions.

For arbitrary functions ${g_{ij} (x, y)}_{1 \leq i \neq j \leq n}$, the aforementioned approximation enables us to identify the order of magnitude of

$$E \Phi (|\sum_{1 \leq i \not= j \leq n} g_{ij} (X_i Y_j)|),$$

whenever decoupling results and Khintchine-type inequalities apply, such as $\Phi$ is convex, $\mathscr{L} (g_{ij}(X_i, X_j)) = \mathscr{L}(g_{ij}(X_j, X_i))$ and $Eg_{ij}(X_i, x) \equiv 0$ for all $x$ in the range of $X_j$.