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 Electronic Communications in Probability > Vol. 14 (2009) > Paper 40 open journal systems 


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

Pages: 412-423

Published on: September 30, 2009
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Electronic Communications in Probability. ISSN: 1083-589X