Uniform bounds for norms of sums of independent random functions

In this paper, we develop a general machinery for finding explicit uni- form probability and moment bounds on sub-additive positive functionals of random processes. Using the developed general technique, we derive uniform bounds on the _s-norms of empirical and regression-type processes. Use-fulness of the obtained results is illustrated by application to the processes appearing in kernel density estimation and in nonparametric estimation of regression functions.