TY - JOUR

T1 - Spectral measure of large random Hankel, Markov and Toeplitz matrices

AU - Bryc, Włodzimierz

AU - Dembo, Amir

AU - Jiang, Tiefeng

PY - 2006/1

Y1 - 2006/1

N2 - 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 {X k} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {X ij} j>i of zero mean and unit variance, scaling the eigenvalues by √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 {X ij} 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 M n scaled by √2n log n converges almost surely to 1.

AB - 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 {X k} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {X ij} j>i of zero mean and unit variance, scaling the eigenvalues by √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 {X ij} 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 M n scaled by √2n log n converges almost surely to 1.

KW - Eulerian numbers

KW - Free convolution

KW - Random matrix theory

KW - Spectral measure

UR - http://www.scopus.com/inward/record.url?scp=33644950826&partnerID=8YFLogxK

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U2 - 10.1214/009117905000000495

DO - 10.1214/009117905000000495

M3 - Article

AN - SCOPUS:33644950826

SN - 0091-1798

VL - 34

SP - 1

EP - 38

JO - Annals of Probability

JF - Annals of Probability

IS - 1

ER -