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Order of magnitude bounds for expectations of \(\Delta_2\)-functions of nonnegative random bilinear forms and generalized \(U\)-statistics. (English) Zbl 0895.60018
Summary: Let \(X_1,Y_1,X_2,Y_2, \dots, X_n,Y_n\) 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
\(E\Phi (\sum_{1\leq i,j \leq n} b_{ij} X_iY_j)\), 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 \(E\Phi (|\sum_{1\leq i\neq j\leq n} g_{ij} (X_i,X_j)|)\) whenever decoupling results and Khinchin-type inequalities apply, such as \(\Phi\) is convex, \({\mathcal L}(g_{ij} (X_i, X_j))={\mathcal L}(g_{ji} (X_j,X_i))\) and \(Eg_{ij} (X_i,x) \equiv 0\) for all \(x\) in the range of \(X_j\).

MSC:
60E15 Inequalities; stochastic orderings
60F25 \(L^p\)-limit theorems
60G50 Sums of independent random variables; random walks
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