Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure μ̂ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N ×N symmetric matrix YσN whose (i, j) entry is σ( i N , j N )xij where (xij , 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, α ∈ (0, 2), and σ is a deterministic function. For random diagonal DN independent of Y σ N and with appropriate rescaling aN , we prove that μ̂ a −1 N Yσ N +DN 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.