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On the Submultiplicativity of Matrix Norms Induced by Random Vectors

Ludovick Bouthat

Abstract

In a recent article, Chávez, 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 $p=\max\{2+\varepsilon,d\}$.

On the Submultiplicativity of Matrix Norms Induced by Random Vectors

Abstract

In a recent article, Chávez, Garcia and Hurley introduced a new family of norms on the space of complex matrices which are induced by random vectors having finite -moments. Therein, the authors asked under which conditions the norms induced by a scalar multiple of are submultiplicative. In this paper, this question is completely answered by proving that this is always the case, as long as the entries of have finite -moments for .
Paper Structure (7 sections, 10 theorems, 57 equations)

This paper contains 7 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. An Extension of a Known Result
  3. Preliminary Estimations
  4. Bounding the Norm ||.||
  5. The Submultiplicativity of ||.||
  6. Example: Symmetric α-Stable Distributions
  7. Concluding remarks

Key Result

Proposition 1

aguilar2022norms Let $\mathcal{V}$ be a $\mathbb{C}$-vector space with conjugate-linear involution $*$ and suppose that the real-linear subspace $\mathcal{V}_{\mathbb{R}} = \{v \in \mathcal{V} : v = v^*\}$ of $*$-fixed points has the norm $\|\cdot\|$. Then for even $d\geq 2$, the following is a norm

Theorems & Definitions (19)

  • Proposition 1
  • Proposition 2
  • proof
  • Corollary 1
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 9 more