Electronic Communications in Probability

Balanced random and Toeplitz matrices

Aniran Basak and Arup Bose

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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

Article information

Electron. Commun. Probab., Volume 15 (2010), paper no. 14, 134-148.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory 60G57: Random measures 60B10: Convergence of probability measures

Random matrix eigenvalues balanced matrix moment method bounded Lipschitz metric Carleman condition almost sure convergence convergence in distribution uniform integrability

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Basak, Aniran; Bose, Arup. Balanced random and Toeplitz matrices. Electron. Commun. Probab. 15 (2010), paper no. 14, 134--148. doi:10.1214/ECP.v15-1537. https://projecteuclid.org/euclid.ecp/1465243957

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