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An estimate on the supremum of a nice class of stochastic integrals and U-statistics

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Let a sequence of iid. random variables ξ 1, . . . ,ξ n be given on a space with distribution μ together with a nice class of functions f(x 1, . . . ,x k ) of k variables on the product space For all 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 where μ n denotes the empirical measure defined by the random variables ξ 1, . . . ,ξ n and investigate the probabilities for all x>0. We show that for nice classes of functions, for instance if is a Vapnik–Červonenkis class, an almost as good bound can be given for these probabilities as in the case when only the random integral of one function is considered. A similar result holds for degenerate U-statistics, too.

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Correspondence to Péter Major.

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Supported by the OTKA foundation Nr. 037886

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Major, P. An estimate on the supremum of a nice class of stochastic integrals and U-statistics. Probab. Theory Relat. Fields 134, 489–537 (2006). https://doi.org/10.1007/s00440-005-0440-9

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  • Probability Measure
  • Kernel Function
  • Measurable Space
  • Product Space
  • Separable Banach Space