Spectral measure of large random Hankel, Markov and Toeplitz matrices
| dc.creator | Bryc, Włodzimierz | |
| dc.creator | Dembo, Amir | |
| dc.creator | Jiang, Tiefeng | |
| dc.date | 2003-07-25 | |
| dc.date | 2006-02-27 | |
| dc.date.accessioned | 2026-07-07T08:06:09Z | |
| dc.date.available | 2026-07-07T08:06:09Z | |
| dc.description | 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 $γ_H$, $γ_M$ and $γ_T$ of unbounded support. The moments of $γ_H$ and $γ_T$ are the sum of volumes of solids related to Eulerian numbers, whereas $γ_M$ 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 $\mathbf {M}_n$ scaled by $\sqrt{2n\log n}$ converges almost surely to 1. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117905000000495 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0307330 | |
| dc.identifier | http://arxiv.org/abs/math/0307330 | |
| dc.identifier | Annals of Probability 2006, Vol. 34, No. 1, 1-38 | |
| dc.identifier | doi:10.1214/009117905000000495 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130511 | |
| dc.subject | Probability | |
| dc.subject | Combinatorics | |
| dc.subject | Statistics Theory | |
| dc.subject | 15A52 (Primary) 60F99, 62H10, 60F10 (Secondary) | |
| dc.title | Spectral measure of large random Hankel, Markov and Toeplitz matrices | |
| dc.type | text |