Electronic Journal of Probability

Fluctuations of eigenvalues for random Toeplitz and related matrices

Dangzheng Liu, Xin Sun, and Zhengdong Wang

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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.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 95, 22 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60F05: Central limit and other weak theorems

Toeplitz (band) matrix Hankel matrix Random matrices Linear statistics of eigenvalues Central limit theorem

This work is licensed under aCreative Commons Attribution 3.0 License.


Liu, Dangzheng; Sun, Xin; Wang, Zhengdong. Fluctuations of eigenvalues for random Toeplitz and related matrices. Electron. J. Probab. 17 (2012), paper no. 95, 22 pp. doi:10.1214/EJP.v17-2006. https://projecteuclid.org/euclid.ejp/1465062417

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