On the Submultiplicativity of Matrix Norms Induced by Random Vectors

In a recent article, Ch\'avez, Garcia and Hurley introduced a new family of norms $\|\cdot\|_{\mathbf{X},d}$ 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 $p=\max\{2+\varepsilon,d\}$.