An estimate on the supremum of a nice class of stochastic integrals and U-statistics |
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Authors: | Péter Major |
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Institution: | (1) Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127 1364, Hungary |
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Abstract: | 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.
Supported by the OTKA foundation Nr. 037886 |
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Keywords: | |
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