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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
##### Keywords:
degenerate U-statistics; supremum bounds
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##### References:
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