A new large deviation inequality for \(U\)-statistics of order \(2\). (English) Zbl 0957.60031

The second-order \(U\)-statistics \(U_n(f)=\sum_{1\leq i \neq j \leq n} f(\xi_i,\xi_j)\) are considered. Let \(U_n(f-E f)=2(n-1)U_n^{(1)}+U_n^{(2)}\) be the decomposition of \(U_n\) to the linear and second-order canonical projections. The upper bound for the moments \(E (U_n{(2)})^{2k}\) is proved. As a corollary the exponential inequality \[ P(U_n^{(2)}\geq C(f,F,n)x)\leq \exp (6-2\sqrt{x}) \] is derived. This inequality is compared with similar inequalities of M. A. Arcones and E. Giné [Ann. Probab. 21, No. 3, 1494-1542 (1993; Zbl 0789.60031)].


60F10 Large deviations


Zbl 0789.60031
Full Text: DOI EuDML


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