On the Submultiplicativity of Matrix Norms Induced by Random Vectors

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