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2012 Fluctuations of eigenvalues for random Toeplitz and related matrices
Dangzheng Liu, Xin Sun, Zhengdong Wang
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Electron. J. Probab. 17: 1-22 (2012). DOI: 10.1214/EJP.v17-2006


Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$ with matrix entries $a_{j}, j=0,1,2,\cdots,$ being independent real random variables such that $$ \mathbb{E}[a_{j}]=0, \ \ \mathbb{E} [|a_{j}|^{2}]=1 \ \mathrm{for}\,\ \ j=0,1,2,\cdots,$$ (homogeneity of 4-th moments) $$\kappa=\mathbb{E} [|a_{j}|^{4}],$$ and further (uniform boundedness) $$\sup\limits_{j\geq 0} \mathbb{E} [|a_{j}|^{k}]=C_{k}<\infty\ \ \mathrm{for} \ \ \ k\geq 3.$$ Under the assumption of $a_{0}\equiv 0$, we prove a central limit theorem for linear statistics of eigenvalues for a fixed polynomial with degree at least 2. Without this assumption, the CLT can be easily modified to a possibly non-normal limit law. In a special case where $a_{j}$'s are Gaussian, the result has been obtained by Chatterjee for some test functions. Our derivation is based on a simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our method can apply to other related random matrix models, including Hermitian Toeplitz and symmetric Hankel matrices. Since Toeplitz matrices are quite different from Wigner and Wishart matrices, our results enrich this topic.


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Dangzheng Liu. Xin Sun. Zhengdong Wang. "Fluctuations of eigenvalues for random Toeplitz and related matrices." Electron. J. Probab. 17 1 - 22, 2012.


Accepted: 2 November 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1284.60017
MathSciNet: MR2994843
Digital Object Identifier: 10.1214/EJP.v17-2006

Primary: 60B20
Secondary: 60F05

Keywords: central limit theorem , Hankel matrix , Linear statistics of eigenvalues , random matrices , Toeplitz (band) matrix

Vol.17 • 2012
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