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Fluctuations of eigenvalues for random Toeplitz and related matrices

Author(s)
Liu, Dangzheng; Sun, Xin; Wang, Zhengdong
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Abstract
Consider random symmetric Toeplitz matrices Tn=(ai−j)[superscript n]i,j=1 with matrix entries aj,j=0,1,2,⋯, being independent real random variables such that E[aj]=0, E[|aj|2]=1 for j=0,1,2,⋯, (homogeneity of 4-th moments) κ=E[|aj|4], and further (uniform boundedness) supj≥0E[|aj|k]=Ck<∞ for k≥3. Under the assumption of a0≡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 aj'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.
Date issued
2012-11
URI
http://hdl.handle.net/1721.1/89526
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Electronic Journal of Probability
Publisher
Institute of Mathematical Statistics
Citation
Liu, Dangzheng, Xin Sun, and Zhengdong Wang. “Fluctuations of Eigenvalues for Random Toeplitz and Related Matrices.” Electronic Journal of Probability 17, no. 0 (January 1, 2012).
Version: Final published version
ISSN
1083-6489

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