## Electronic Communications in Probability

### Spectrum of random Toeplitz matrices with band structure

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 40, 412-423.

Dates
Accepted: 30 September 2009
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465234749

Digital Object Identifier
doi:10.1214/ECP.v14-1492

Mathematical Reviews number (MathSciNet)
MR2551851

Zentralblatt MATH identifier
1188.15036

Subjects
Primary: 15A52

Keywords
random matrices

Rights

#### Citation

Kargin, Vladislav. Spectrum of random Toeplitz matrices with band structure. Electron. Commun. Probab. 14 (2009), paper no. 40, 412--423. doi:10.1214/ECP.v14-1492. https://projecteuclid.org/euclid.ecp/1465234749

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