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January 2006 Spectral measure of large random Hankel, Markov and Toeplitz matrices
Włodzimierz Bryc, Amir Dembo, Tiefeng Jiang
Ann. Probab. 34(1): 1-38 (January 2006). DOI: 10.1214/009117905000000495

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 $\sqrt{n}$ 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.

Citation

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Włodzimierz Bryc. Amir Dembo. Tiefeng Jiang. "Spectral measure of large random Hankel, Markov and Toeplitz matrices." Ann. Probab. 34 (1) 1 - 38, January 2006. https://doi.org/10.1214/009117905000000495

Information

Published: January 2006
First available in Project Euclid: 17 February 2006

zbMATH: 1094.15009
MathSciNet: MR2206341
Digital Object Identifier: 10.1214/009117905000000495

Subjects:
Primary: 15A52
Secondary: 60F10, 60F99, 62H10

Rights: Copyright © 2006 Institute of Mathematical Statistics

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Vol.34 • No. 1 • January 2006
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