Revisiting Toeplitz and Hankel random matrices via *-convergence of circulant-type matrices
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.
Related
- Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
- Generalization of Toeplitz + Hankel operators in the polydisc
- The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space
- Algebraic properties of Truncated Toeplitz + Truncated Hankel operators
- Toeplitz and Hankel operators on Bergman spaces with doubling weights
BibTeX
@misc{bose2026revisiting,
title = {{Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices}},
author = {Arup Bose and Pradeep Vishwakarma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.16160},
eprint = {2605.16160},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.16160},
}
RIS
TY - UNPB
TI - Revisiting Toeplitz and Hankel random matrices via *-convergence of circulant-type matrices
AU - Arup Bose
AU - Pradeep Vishwakarma
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2605.16160
UR - https://arxiv.org/abs/2605.16160
AB - 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.
ER -