Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying $$ \sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A), $$ for any Borel set $A\subset X\setminus \{0\}$.
Please find sufficient and necessary conditions on $X$ and $1\leq p<\infty$ so that there exists a constant $C_{p, X}$ satisfying $$ \mathbb E \bigl\|\sum_n\xi_n\bigr\|^p \leq C_{p, X} \mathbb E \bigl\|\sum_n\eta_n\bigr\|^p. $$
This problem stated for a general Banach space was originally posed here. It turned out that $X$ can not be $\ell^{\infty}$ (see this answer) and that $X$ should have a finite cotype due to the Maurey–Pisier theorem (see this answer).
