On the Submultiplicativity of Matrix Norms Induced by Random Vectors

In a recent article, Chávez, Garcia and Hurley introduced a new family of norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathbf{X},d}$ on the space of $n\times n$ complex matrices which are induced by random vectors $\mathbf{X}$ having finite $d$-moments. Therein, the authors asked under which conditions the norms induced by a scalar multiple of $\mathbf{X}$ are submultiplicative. In this paper, this question is completely answered by proving that this is always the case, as long as the entries of $\mathbf{X}$ have finite $p$-moments for $p=\max \{2+\epsilon ,d\}$.