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Extreme eigenvalues of sparse, heavy tailed random matrices

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  • Auffinger, Antonio
  • Tang, Si

Abstract

We study the statistics of the largest eigenvalues of p×p sample covariance matrices Σp,n=Mp,nMp,n∗ when the entries of the p×n matrix Mp,n are sparse and have a distribution with tail t−α, α>0. On average the number of nonzero entries of Mp,n is of order nμ+1, 0≤μ≤1. We prove that in the large n limit, the largest eigenvalues are Poissonian if α<2(1+μ−1) and converge to a constant in the case α>2(1+μ−1). We also extend the results of Benaych-Georges and Péché (2014) in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.

Suggested Citation

  • Auffinger, Antonio & Tang, Si, 2016. "Extreme eigenvalues of sparse, heavy tailed random matrices," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3310-3330.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:11:p:3310-3330
    DOI: 10.1016/j.spa.2016.04.029
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    References listed on IDEAS

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    1. Bai, Z. D. & Silverstein, Jack W. & Yin, Y. Q., 1988. "A note on the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 166-168, August.
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    Cited by:

    1. Elizaveta Rebrova, 2020. "Constructive Regularization of the Random Matrix Norm," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1768-1790, September.

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