Uniform bounds for norms of sums of independent random functions

TL;DR

This paper introduces a general method for deriving explicit uniform bounds on the norms of sums of independent random functions, with applications to kernel density estimation and nonparametric regression.

Contribution

It develops a new machinery for obtaining uniform probability and moment bounds on sub-additive functionals of random processes, applicable to empirical and regression processes.

Findings

Derived uniform bounds for ${\\mathbb{L}}_s$-norms of empirical processes

Applied bounds to kernel density estimation processes

Applied bounds to nonparametric regression estimation

Abstract

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