Extreme eigenvalues of sparse, heavy tailed random matrices

Abstract

We study the statistics of the largest eigenvalues of $p\times p$ sample covariance matrices ${\Sigma }_{p,n}=M_{p,n}{M}_{p,n}^{\ast }$ when the entries of the $p\times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-\alpha }$, $\alpha >0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{\mu +1}$, $0\le \mu \le 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $\alpha <2(1+{\mu }^{-1})$ and converge to a constant in the case $\alpha >2(1+{\mu }^{-1})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.

23 pages