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.
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