On Random Operator-Valued Matrices: Operator-Valued Semicircular Mixtures and Central Limit Theorem

TL;DR

This paper introduces a new framework for random operator-valued matrices inspired by wireless communication models, establishing a central limit theorem and providing numerical methods for their analysis.

Contribution

It defines random operator-valued matrices within a non-commutative probability setting and proves a central limit theorem for their moments, along with a numerical algorithm for their Cauchy transform.

Findings

Established a CLT for operator-valued matrix moments.

Developed a numerical algorithm for the Cauchy transform.

Connected random matrix theory with wireless communication models.

Abstract

Motivated by a random matrix theory model from wireless communications, we define random operator-valued matrices as the elements of $L^{\infty-}(\Omega,{\mathcal F},{\mathbb P}) \otimes M_d({\mathcal A})$ where $(\Omega,{\mathcal F},{\mathbb P})$ is a classical probability space and $({\mathcal A},\varphi)$ is a non-commutative probability space. A central limit theorem for the mean $M_d(\mathbb{C})$-valued moments of these random operator-valued matrices is derived. Also a numerical algorithm to compute the mean $M_d({\mathbb C})$-valued Cauchy transform of operator-valued semicircular mixtures is analyzed.