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 \|·\|X,d on the space of n × n complex matrices which are induced by random vectors X having finite d-moments. Therein, the authors asked under which conditions the norms induced by a scalar multiple of 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 X have finite p-moments for p=max\2+ε,d\.