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Fluctuations of eigenvalues for random Toeplitz and related matrices


 
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1. Title Title of document Fluctuations of eigenvalues for random Toeplitz and related matrices
 
2. Creator Author's name, affiliation, country Dangzheng Liu; University of Science and Technology of China; China
 
2. Creator Author's name, affiliation, country Xin Sun; Massachusetts Institute of Technology
 
2. Creator Author's name, affiliation, country Zhengdong Wang; Peking University; China
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Toeplitz (band) matrix; Hankel matrix; Random matrices; Linear statistics of eigenvalues; Central limit theorem
 
3. Subject Subject classification 60B20;60F05
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2012-11-02
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2006
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-2006
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
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