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

Abstract

Motivated by a random matrix theory model from wireless communications, we define random operator-valued matrices as the elements of $L^{\infty-}(Ω,{\mathcal F},{\mathbb P}) \otimes M_d({\mathcal A})$ where $(Ω,{\mathcal F},{\mathbb P})$ is a classical probability space and $({\mathcal A},φ)$ 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.

17 pages, 1 figure