Matricially free random variables

TL;DR

This paper extends Voiculescu's operatorial framework to arrays of random variables with matricial multiplication, introducing matricial freeness as a generalization of existing independence notions, and explores their limit distributions related to Gaussian matrices.

Contribution

It introduces the concept of matricial freeness, generalizing freeness and monotone independence, and establishes central limit theorems for associated random pseudomatrices.

Findings

Matricial freeness generalizes freeness and monotone independence.

Central limit theorems for random pseudomatrices are proved.

Limit distributions are matricial analogs of semicircle laws.

Abstract

We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a generalization of both freeness and monotone independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to Gaussian random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial generalizations of semicirle laws.