On multiplier processes under weak moment assumptions

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.