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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:
60E15 Inequalities; stochastic orderings
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