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The Bennett-Orlicz Norm.

Jon A Wellner¹

¹University of Washington, Seattle, USA.

Sankhya. Series A. (2008)

|August 21, 2018

Summary

This summary is machine-generated.

Researchers introduced a new Bennett-Orlicz norm, offering potentially tighter inequalities for expectations of maxima than the existing Bernstein-Orlicz norm. This work explores connections between these norms and various exponential inequalities.

Keywords:

Bennett’s inequality Exponential bound Maximal inequality Orlicz norm Poisson Primary: 60E15, 60F10 Prokhorov’s inequality Secondary: 60G50, 33E20

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Area of Science:

Probability theory
Mathematical analysis

Background:

The Bernstein-Orlicz norm, introduced by van de Geer and Lederer (2013), is linked to Bernstein-type inequalities.
Orlicz norms are fundamental in functional analysis and probability theory for bounding random variables.

Purpose of the Study:

Introduce a novel Bennett-Orlicz norm.
Establish connections between the new Bennett-Orlicz norm and Bennett-type inequalities.
Compare the efficacy of the Bennett-Orlicz norm with the Bernstein-Orlicz norm and other established inequalities.

Main Methods:

Definition and analysis of a new Orlicz norm, termed the Bennett-Orlicz norm.
Derivation of inequalities for expectations of maxima using the Bennett-Orlicz norm.
Comparative analysis with existing Bernstein-Orlicz norm and exponential inequalities (Bernstein, Bennett, Prokhorov).

Main Results:

The newly introduced Bennett-Orlicz norm provides inequalities for expectations of maxima.
These inequalities are potentially tighter than those derived from the Bernstein-Orlicz norm in applicable scenarios.
Cross-connections between the Bennett-Orlicz norm, Bernstein-Orlicz norm, and other exponential inequalities are discussed.

Conclusions:

The Bennett-Orlicz norm offers a valuable new tool in probability theory.
This norm provides potentially improved bounds for expectations of maxima.
The study contributes to the understanding of Orlicz norms and their applications in bounding random variables and comparing with existing theoretical frameworks.