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Decoupling and Khintchine’s inequalities for $$U$$-statistics. (English) Zbl 0761.60014
Summary: We introduce a fairly general decoupling inequality for $$U$$-statistics. Let $$\{X_ i\}$$ be a sequence of independent random variables in a measurable space $$(S,{\mathcal S})$$, and let $$\{\widetilde X_ i\}$$ 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 $\max_{1\leq i\neq j\leq n}E\Phi\left(\sup_{f_{ij}\in\Pi_{ij}}\| f_{ij}(X_ i,X_ j)\|\right)<\infty,$ then, under measurability conditions, $E\Phi\left(\sup_{{\mathbf f}\in{\pmb\Pi}}\left\|\sum_{1\leq i\neq j\leq n}f_{ij}(X_ i,X_ j)\right\|\right)\leq E\Phi\left(8\sup_{{\mathbf f}\in{\pmb\Pi}}\left\|\sum_{1\leq i\neq j\leq n}f_{ij}(X_ i,\widetilde X_ j)\right\|\right),$ where $${\mathbf f}=(f_{ij},\;1\leq i\neq j\leq n)$$ and $$\Pi=(\Pi_{ij},\;1\leq i\neq j\leq n)$$. In the case where $$\Pi$$ 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.

##### MSC:
 6e+16 Inequalities; stochastic orderings
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