Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

Abstract

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulae of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit ${\gamma }_{{}_T}^{(m)}$ $({\gamma }_{{}_H}^{(m)})$ of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where $m$ is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that ${\gamma }_{{}_T}^{(m)}$ $({\gamma }_{{}_H}^{(m)})$ converges weakly to the semicircle law as $m\ra\iy$.

26 pages, 1 figure, accepted by Journal of Theoretical Probability

References in corpus (2)

• Toeplitz Block Matrices in Compressed Sensing
• Notes on Free Probability Theory