The Annals of Probability

Spectral measure of large random Hankel, Markov and Toeplitz matrices

Włodzimierz Bryc, Amir Dembo, and Tiefeng Jiang

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We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure.

For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xij}j>i of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γH, γM and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM has a bounded smooth density given by the free convolution of the semicircle and normal densities.

For symmetric Markov matrices generated by i.i.d. random variables {Xij}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m=0, and the fourth moment is finite, we prove that the spectral norm of Mn scaled by $\sqrt{2n\log n}$ converges almost surely to 1.

Article information

Ann. Probab., Volume 34, Number 1 (2006), 1-38.

First available in Project Euclid: 17 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 60F99: None of the above, but in this section 62H10: Distribution of statistics 60F10: Large deviations

Random matrix theory spectral measure free convolution Eulerian numbers


Bryc, Włodzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1--38. doi:10.1214/009117905000000495.

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