TY - JOUR

T1 - Extreme eigenvalues of sparse, heavy tailed random matrices

AU - Auffinger, Antonio

AU - Tang, Si

N1 - Funding Information:
The authors would like to thank an anonymous referee. His valuable comments greatly improved the presentation of the paper. The research of A.A. is supported by National Science Foundation grant DMS-1517864 .
Publisher Copyright:
© 2016 Elsevier B.V.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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.

AB - 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.

KW - Eigenvalue distribution

KW - Heavy tail

KW - Random matrices

KW - Sparse

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U2 - 10.1016/j.spa.2016.04.029

DO - 10.1016/j.spa.2016.04.029

M3 - Article

AN - SCOPUS:84979641041

VL - 126

SP - 3310

EP - 3330

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 11

ER -