Spectral Measure of Large Random Hankel, Markov and Toeplitz Matrices

Spectral Measure of Large Random Hankel, Markov and Toeplitz Matrices

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}...

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Veröffentlicht in:The Annals of Probability. - Institute of Mathematical Statistics. - 34(2006), 1, Seite 1-38
1. Verfasser: Bryc, Włodzimierz (VerfasserIn)
Weitere Verfasser: Dembo, Amir, Jiang, Tiefeng
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 2006
Zugriff auf das übergeordnete Werk:The Annals of Probability
Schlagworte:Random matrix theory Spectral measure Free convolution Eulerian numbers Mathematics Physical sciences
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520 |a 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. 
540 |a Copyright 2006 The Institute of Mathematical Statistics 
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650 4 |a Mathematics  |x Pure mathematics  |x Probability theory  |x Random variables 
650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Applied statistics  |x Descriptive statistics  |x Statistical distributions  |x Distribution functions  |x Probability distributions  |x Mathematical moments 
650 4 |a Mathematics  |x Pure mathematics  |x Geometry  |x Euclidean geometry  |x Plane geometry  |x Vertices 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Matrix theory  |x Eigenvalues 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Matrix theory  |x Matrices 
650 4 |a Mathematics  |x Pure mathematics  |x Geometry  |x Geometric shapes  |x Semicircles 
650 4 |a Physical sciences  |x Physics  |x Mechanics  |x Density 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Matrix theory  |x Matrices  |x Covariance matrices 
650 4 |a Mathematics  |x Pure mathematics  |x Discrete mathematics  |x Number theory  |x Numbers  |x Real numbers  |x Rational numbers  |x Integers  |x Zero 
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