Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

Abstract

Abstract. We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measureμ of large random real symmetric matrices with heavy tailed entries. Specifically, consider theis 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 D N independent of Y σ N and with appropriate rescaling a N , we prove thatμ aconverges 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.

Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices

“…Set t N := N µ ∈ (0, 1 2(2−α) ) and define b := a½ |a|≤t N and c := a½ |a|>t N . Then (2) is obvious by Hypothesis (8) and the fact that a N = N 1/α up to a slowly varying factor and (3) follows directly from Lemma 5.8 of [11] (in fact, this lemma gives us the right upper bound for the second moment of b, which of course implies that it is true for its variance). Let us now treat the second part of the hypothesis.…”

Circulant type matrices with heavy tailed entries

“…Later Belinschi et al (2009) studied some symmetric band matrices and the sample variance covariance matrices with heavy tailed inputs. In both these articles the LSD was shown to be nonrandom and their methods were based on computation of Stieltjes transform of the empirical distribution of eigenvalues.…”

Delocalization at Small Energy for Heavy-Tailed Random Matrices

“…Assume that the tail of A ij is such that, for some 0 < α < 2, nP(|A ij | ≥ t) ≃ t→∞ t −α , (in a sense which will be made precise later on). Then, for z ∈ C\R, consider the following fractional moment of the resolvent: For β = 1, as n goes to infinity and then z goes to E ∈ R on the real line, y n z (1)/π converges towards the spectral density which turns out to be positive [7,6,11]. However, we proved in [12] that for β = α/2, and for sufficiently heavy tails (0 < α < 2/3), as n goes to infinity and then z goes to E ∈ R large enough, y n z (α/2) goes to zero.…”