Spectral measure of heavy tailed band and covariance random matrices

Abstract

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $μ$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^σ$ whose (i,j) entry is $σ(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $α$-stable law, $0<α<2$, and $σ$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^σ$ and with appropriate rescaling $a_N$, we prove that the distribution $μ$ of $a_N^{-1}Y_N^σ+ D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.

31 pages, minor modifications, mainly in the regularity argument for Theorem 1.3. To appear in Communications in Mathematical Physics