Spectral measure of large random Hankel, Markov and Toeplitz matrices

By Wlodzimierz Bryc, Amir Dembo, Tiefeng Jiang. 2003

269 citations recorded by the source corpus.

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 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 iy } y>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.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cf47a700da36a6e1383df12373c8edc9f1a5c748

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Citations in the corpus

On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries (2011, result): “Similar results have been also obtained for structured random matrices (like Toeplitz or Hankel Matrices, see [12]).”
Fluctuations of eigenvalues and second order Poincaré inequalities (2007, unknown): “In a recent preprint, Jiang [32] proves the CLT for the Jacobi ensemble.”
Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices (2005, methodology): “spectral measure (see Deﬁnitions 1.1 and 1.3) of the normalized eigenvalues. The fourth moment is 22 3 , close to but not equal to the standard Gaussian density’s fourth moment of 3. Bryc-Dembo-Jiang [BDJ] (calculating the moments using uniform variables and interpreting the results as volumes of so”
Random matrix theory in statistics: A review (2014, methodology): “Bryc et al. (2006) used combinatorial methods to prove the existence of limiting ESDs for random matrices of Hankel, Toeplitz and Markov types.”
On the spectral norm of a random Toeplitz matrix (2007, background): “bgaussian estimate for the Xj. Since kJnk = n, (7) and the triangle inequality imply a strong law of large numbers: (8) lim n→∞ kTnk n = |m| almost surely. In [3], (8) was proved using estimates from [5] under the assumption that the Xj are identically distributed and have ﬁnite variance. We emphasi”
Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, background): “They are proved in Jiang[44].”