Comparison of weak and strong moments for vectors with independent coordinates

TL;DR

This paper compares the p-th moments of suprema of linear combinations of independent centered random variables with their weak moments, establishing conditions under which these moments are comparable, especially in the i.i.d. case.

Contribution

It introduces a necessary and sufficient condition involving integral moments for the comparability of moments of suprema in the i.i.d. setting.

Findings

Moment comparability holds under specific integral moment conditions.

The condition involving 2q-th and q-th moments is necessary for i.i.d. variables.

Provides a characterization of when weak and strong moments are comparable.

Abstract

We show that for $p\ge 1$, the $p$-th moment of suprema of linear combinations of independent centered random variables are comparable with the sum of the first moment and the weak $p$-th moment provided that $2q$-th and $q$-th integral moments of these variables are comparable for all $ q \ge 2$. The latest condition turns out to be necessary in the i.i.d. case.