Electronic Journal of Probability

Fluctuations of eigenvalues for random Toeplitz and related matrices

Dangzheng Liu, Xin Sun, and Zhengdong Wang

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062417

Digital Object Identifier
doi:10.1214/EJP.v17-2006

Mathematical Reviews number (MathSciNet)
MR2994843

Zentralblatt MATH identifier
1284.60017

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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