Spectral measure of large random Hankel, Markov and Toeplitz matrices

• 成果类型: Article
• 署名作者: Bryc, W; Dembo, A; Jiang, TF
• 署名单位: University System of Ohio; University of Cincinnati; Stanford University; Stanford University; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000495
• 发表日期: 2006
• 页码: 1-38
• 关键词: eigenvalues

摘要

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 eigen-values by root 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 M-n scaled by root 2n log n converges almost surely to 1.