An estimate on the maximum of a nice class of stochastic integrals

Let a sequence of iid. random variables ${\xi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\xi }_n$ be given on a space $(X,\mathcal{X})$ with distribution $\mu$ together with a nice class $\mathcal{F}$ of functions $f(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k)$ of $k$ variables on the product space $(X^k,{\cal X}^k)$\mathrm{.}Forall$f\in\cal F$weconsidertherandomintegral$J_{n,k}(f)$ 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 ${\xi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\xi }_n$ and investigate the probabilities $P({\sup }_{f\in \mathcal{F}}\mathrm{\mid }{J}_{n,k}(f)\mathrm{\mid }>x)$ for all $x>0$. We show that for nice classes of functions, for instance if $\mathcal{F}$ is a Vapnik-Cervonenkis 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.