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2010 Balanced random and Toeplitz matrices
Aniran Basak, Arup Bose
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Electron. Commun. Probab. 15: 134-148 (2010). DOI: 10.1214/ECP.v15-1537


Except for the Toeplitz and Hankel matrices, the common patterned matrices for which the limiting spectral distribution (LSD) are known to exist share a common property–the number of times each random variable appears in the matrix is (more or less) the same across the variables. Thus it seems natural to ask what happens to the spectrum of the Toeplitz and Hankel matrices when each entry is scaled by the square root of the number of times that entry appears in the matrix instead of the uniform scaling by $n^{−1/2}$. We show that the LSD of these balanced matrices exist and derive integral formulae for the moments of the limit distribution. Curiously, it is not clear if these moments define a unique distribution


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Aniran Basak. Arup Bose. "Balanced random and Toeplitz matrices." Electron. Commun. Probab. 15 134 - 148, 2010.


Accepted: 27 April 2010; Published: 2010
First available in Project Euclid: 6 June 2016

zbMATH: 1225.60014
MathSciNet: MR2643593
Digital Object Identifier: 10.1214/ECP.v15-1537

Primary: 60B20
Secondary: 60B10 , 60F05 , 60G57 , 62E20

Keywords: Almost sure convergence , balanced matrix , Bounded Lipschitz metric , Carleman condition , Convergence in distribution , Eigenvalues , Moment method , Random matrix , uniform integrability

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