Original article at: http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=2129

Spectrum of random Toeplitz matrices with band structure

Vladislav Kargin, Stanford University

Abstract

This paper considers the eigenvalues of symmetric Toeplitz matrices with independent random entries and band structure. We assume that the entries of the matrices have zero mean and a uniformly bounded 4th moment, and we study the limit of the eigenvalue distribution when both the size of the matrix and the width of the band with non-zero entries grow to infinity. It is shown that if the bandwidth/size ratio converges to zero, then the limit of the eigenvalue distributions is Gaussian. If the ratio converges to a positive limit, then the distributions converge to a non-Gaussian distribution, which depends only on the limit ratio. A formula for the fourth moment of this distribution is derived.

Full text: PDF | PostScript




Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings. The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article. Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to Philippe Carmona Laboratoire Jean Leray UMR 6629 Universite de Nantes, 2, Rue de la Houssinière BP 92208 F-44322 Nantes Cédex 03 France You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr. If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.

Original article at: http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=2129