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Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

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  • Deng Zhang

    (Shanghai Jiao Tong University)

Abstract

This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the $$\beta $$ β -Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.

Suggested Citation

  • Deng Zhang, 2017. "Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1076-1103, September.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-016-0683-7
    DOI: 10.1007/s10959-016-0683-7
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    References listed on IDEAS

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    1. Hanna Döring & Peter Eichelsbacher, 2013. "Moderate Deviations via Cumulants," Journal of Theoretical Probability, Springer, vol. 26(2), pages 360-385, June.
    2. Florence Merlevède & Magda Peligrad, 2010. "Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables," Journal of Theoretical Probability, Springer, vol. 23(1), pages 277-300, March.
    3. Arup Bose & Sanchayan Sen, 2013. "Finite Diagonal Random Matrices," Journal of Theoretical Probability, Springer, vol. 26(3), pages 819-835, September.
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    Cited by:

    1. Deng Zhang, 2019. "Gaussian Fluctuations and Moderate Deviations of Eigenvalues in Unitary Invariant Ensembles," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1647-1687, December.

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