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 ${{\hat{\mu}}}$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix ${{\bf Y}_N^\sigma}$ whose (i, j) entry is ${\sigma\left(\frac{i}{N}, \frac{j}{N}\right) x_{ij}}$ , where (x _ij , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, ${\alpha\in (0,2)}$ , and σ is a deterministic function. For random diagonal D _N independent of ${{\bf Y}_N^\sigma}$ and with appropriate rescaling a _N , we prove that ${{\hat{\mu}}_{a_N^{-1} {\bf Y}_N^\sigma + {\varvec {D}}_N}}$ 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.