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An estimate on the supremum of a nice class of stochastic integrals and U-statistics. (English) Zbl 1128.62063
The paper aims to bound the upper tail probability of the supremum of appropriate classes of multiple integrals with respect to a normalized empirical measure. This problem is closely related to the study of the supremum of classes of degenerate \(U\)-statistics. Let \(\xi_1, \ldots, \xi_n\) be a sequence of independent and identically distributed random variables on a given space \((X, \chi)\) with distribution \(\mu\). Let \({\mathcal{F}}\) denote a class of functions of \(k\) variables on the product space \((X^k, \chi^k)\). For all \(f \in {\mathcal{F}}\) we consider the random integral \(J_{n,k}(f)\) of the function \(f\) with respect to the \(k\)-fold product of the normalized signed measure \(\sqrt{n}(\mu_n - \mu),\) where \(\mu_n\) denotes the empirical measure defined by the random variables \(\xi_1, \ldots, \xi_n.\) The paper gives bounds for the probabilities \(P( \sup_{f\in {\mathcal F}}| J_{n,k}(f)| \geq x)\) for all \(x>0.\) The results provide an improvement of similar bounds for degenerate \(U\)-statistics found by M. Arcones and E. Giné [Stoch. Proc. Appl. 52, 17–38 (1994; Zbl 0807.62014)], where the kernels constitute a Vapnik-Červonenkis class.

MSC:
62G20 Asymptotic properties of nonparametric inference
60H05 Stochastic integrals
60E15 Inequalities; stochastic orderings
62E20 Asymptotic distribution theory in statistics
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