## The Annals of Probability

### Spectral measure of large random Hankel, Markov and Toeplitz matrices

#### Abstract

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 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 converges almost surely to 1.

#### Article information

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

Dates
First available in Project Euclid: 17 February 2006

https://projecteuclid.org/euclid.aop/1140191531

Digital Object Identifier
doi:10.1214/009117905000000495

Mathematical Reviews number (MathSciNet)
MR2206341

Zentralblatt MATH identifier
1094.15009

#### Citation

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. https://projecteuclid.org/euclid.aop/1140191531

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