On multiplier processes under weak moment assumptions

TL;DR

This paper demonstrates that under certain symmetry and moment conditions, empirical and multiplier processes behave similarly to subgaussian processes, even with weak moment assumptions on the random vectors.

Contribution

It establishes conditions under which empirical and multiplier processes exhibit subgaussian-like behavior despite weak moment assumptions.

Findings

Processes behave as if $X$ were $L$-subgaussian

Symmetry condition is crucial for the results

Applicable for vectors with moments up to $ ot hickapprox rac{1}{2} \

Abstract

We show that if $V \subset \R^n$ satisfies a certain symmetry condition (closely related to unconditionaity) and if $X$ is an isotropic random vector for which $\|\inr{X,t}\|_{L_p} \leq L \sqrt{p}$ for every $t \in S^{n-1}$ and $p \lesssim \log n$, then the corresponding empirical and multiplier processes indexed by $V$ behave as if $X$ were $L$-subgaussian.