Decoupling and Khintchine's Inequalities for $U$-Statistics

Abstract

In this paper we introduce a fairly general decoupling inequality for $U$-statistics. Let $\{X_i\}$ be a sequence of independent random variables in a measurable space $(S, \mathscr{J})$, and let $\{\tilde{X}_i\}$ be an independent copy of $\{X_i\}$. Let $\Phi(x)$ be any convex increasing function for $x \geq 0$. Let $\Pi_{ij}$ be families of functions of two variables taking $(S \times S)$ into a Banach space $(D, \|\cdot\|)$. If the $f_{ij} \in \Pi_{ij}$ are Bochner integrable and $\max_{1\leq i\neq j\leq n} E\Phi\big(\sup_{f_{ij}\in\Pi_{ij}}\|f_{ij}(X_i, X_j)\|\big) < \infty,$ then, under measurability conditions, $E\Phi\big(\sup_{\mathbf{f}\in\mathbf{\Pi}}\big\|\sum_{1\leq i\neq j\leq n} f_{ij}(X_i, X_j)\big\|\big) \leq E\Phi\big(8 \sup_{\mathbf{f}\in\mathbf{\Pi}}\big\|\sum_{1\leq i\neq j\leq n} f_{ij}(X_i, \tilde{X}_j)\big\|\big),$ where $\mathbf{f} = (f_{ij}, 1 \leq i \neq j \leq n)$ and $\mathbf{\Pi} = (\Pi_{ij}, 1 \leq i \neq j \leq n)$. In the case where $\mathbf{\Pi}$ is a family of functions of two variables satisfying $f_{ij} = f_{ji}$ and $f_{ij}(X_i, X_j) = f_{ij}(X_j, X_i)$, the reverse inequality holds (with a different constant). As a corollary, we extend Khintchine's inequality for quadratic forms to the case of degenerate $U$-statistics. A new maximal inequality for degenerate $U$-statistics is also obtained. The multivariate extension is provided.