Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices

TL;DR

This paper investigates the $*$-convergence and spectral distribution limits of various structured random matrices, including Toeplitz, Hankel, and circulant types, revealing new limit theorems and connections.

Contribution

It establishes the $*$-convergence of symmetric Toeplitz and Hankel matrices to sums of Gaussian variables and provides new proofs for spectral distribution convergence.

Findings

Weak $*$-convergence of circulant and skew-circulant matrices to Gaussian distributions.

$*$-convergence of symmetric Toeplitz matrices to sums of Gaussian variables.

Convergence of Hankel matrices to sums involving symmetrized Rayleigh distribution.

Abstract

We establish the joint $*$-convergence of a random circulant matrix and a specific deterministic diagonal matrix. We also show that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge weakly a.s.~to complex Gaussian and symmetrized Rayleigh distributions, respectively. The $*$-convergence of symmetric Toeplitz and Hankel random matrices is well known. So is the weak convergence of their random spectrum. However, not much is known about the limits. We exploit the connections of circulant, reverse circulant, and left skew-circulants with the Hankel and Toeplitz matrices, to show the $*$-convergence of the random symmetric Toeplitz matrix to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. A similar result holds for the non-symmetric Toeplitz matrix, but the variables are not self-adjoint and have a complex Gaussian distribution. The random Hankel matrix is shown to converge in $*$-distribution to a sum of two self-adjoint variables, each of which has a symmetrized Rayleigh distribution. As a consequence of these results, we also obtain a different proof of the convergence of the empirical spectral distribution of symmetric Toeplitz and Hankel matrices, and a slightly different way of expressing the moments of the limit spectral distribution.