A note on a maximal Bernstein inequality

TL;DR

This paper demonstrates that if a Bernstein-type maximal inequality applies to partial sums of random variables, then a maximal form of this inequality also holds, revealing a fundamental connection between these inequalities.

Contribution

It establishes that maximal Bernstein inequalities automatically extend from their non-maximal counterparts under general conditions.

Findings

Maximal Bernstein inequalities are valid whenever Bernstein-type inequalities hold.

The result applies broadly to sequences of random variables.

This connection simplifies the analysis of maximal inequalities in probability theory.

Abstract

We show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums of a sequence of random variables, a maximal form of the inequality is also valid.