Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices
Abstract
For fixed l,m 1, let Xn(0),Xn(1),…,Xn(l) be independent random n × n matrices with independent entries, let Fn(0) := Xn(0) (Xn(1))-1 ·s (Xn(l))-1, and let Fn(1),…,Fn(m) be independent random matrices of the same form as Fn(0). We investigate the limiting spectral distributions of the matrices Fn(0) and Fn(1) + … + Fn(m) as n ∞. Our main result shows that the sum Fn(1) + … + Fn(m) has the same limiting eigenvalue distribution as Fn(0) after appropriate rescaling. This extends recent findings by Tikhomirov and Timushev (2014). To obtain our results, we apply the general framework recently introduced in G\"otze, K\"osters and Tikhomirov (2014) to sums of products of independent random matrices and their inverses. We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.