an:05145416
Zbl 1128.62063
Major, P??ter
An estimate on the supremum of a nice class of stochastic integrals and U-statistics
EN
Probab. Theory Relat. Fields 134, No. 3, 489-537 (2006).
00122796
2006
j
62G20 60H05 60E15 62E20
degenerate U-statistics; supremum bounds
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 \textit{M. Arcones} and \textit{E. Gin??} [Stoch. Proc. Appl. 52, 17--38 (1994; Zbl 0807.62014)], where the kernels constitute a Vapnik-??ervonenkis class.
Neville Weber (Sydney)
Zbl 0807.62014