The Bernstein-Orlicz norm and deviation inequalities

TL;DR

This paper introduces the Bernstein-Orlicz norm and simplified chaining methods to derive deviation inequalities for empirical processes, bridging sub-Gaussian and sub-exponential behaviors and extending to unbounded cases.

Contribution

It presents a new Bernstein-Orlicz norm and simplified chaining techniques, enhancing the analysis of empirical processes and deviation inequalities.

Findings

Introduces Bernstein-Orlicz norm for tail behavior interpolation

Simplifies chaining and generic chaining concepts

Establishes deviation inequality for unbounded empirical processes

Abstract

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein-Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case.