Convergence of joint moments for independent random patterned matrices

TL;DR

This paper investigates the joint convergence behaviors of various patterned matrices, revealing different types of independence in their limits, including classical, half independence, and complex dependencies.

Contribution

It introduces a framework for understanding joint limit laws of patterned matrices beyond classical notions like independence and freeness.

Findings

Symmetric circulants exhibit classical independence in the limit.

Reverse circulants demonstrate half independence asymptotically.

Toeplitz and Hankel matrices do not conform to standard independence notions.

Abstract

It is known that the joint limit distribution of independent Wigner matrices satisfies a very special asymptotic independence, called freeness. We study the joint convergence of a few other patterned matrices, providing a framework to accommodate other joint laws. In particular, the matricial limits of symmetric circulants and reverse circulants satisfy, respectively, the classical independence and the half independence. The matricial limits of Toeplitz and Hankel matrices do not seem to submit to any easy or explicit independence/dependence notions. Their limits are not independent, free or half independent.