Spectral Measure of Large Random Hankel, Markov and Toeplitz Matrices

• Published in: The Annals of probability 2006-01, Vol.34 (1), p.1-38
• Main Authors: Bryc, Włodzimierz, Dembo, Amir, Jiang, Tiefeng
• Format: Article
• Language: English
• Subjects: 15A52; 60F10; 60F99; 62H10; Covariance matrices; Density; Distribution theory; Eigenvalues; Eulerian numbers; Exact sciences and technology; free convolution; General topics; Linear inference, regression; Markov analysis; Mathematical models; Mathematical moments; Mathematics; Matrices; Probability; Probability and statistics; Probability theory and stochastic processes; Random matrix theory; Random variables; Sciences and techniques of general use; Semicircles; spectral measure; Statistics; Stochastic processes; Studies; Vertices; Zero
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/009117905000000495

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$\sqrt{n}$we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions$\gamma _{H}$,$\gamma _{M}$and$\gamma _{T}$of unbounded support. The moments of$\gamma _{H}$and$\gamma _{T}$are the sum of volumes of solids related to Eulerian numbers, whereas$\gamma _{M}$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${\bf M}_{n}$scaled by$\sqrt{2n\,{\rm log}\,n}$log n converges almost surely to 1.