Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices

Abstract

For fixed $l,m \ge 1$, let $\mathbf{X}_n^{(0)},\mathbf{X}_n^{(1)},\dots,\mathbf{X}_n^{(l)}$ be independent random $n \times n$ matrices with independent entries, let $\mathbf{F}_n^{(0)} := \mathbf{X}_n^{(0)} (\mathbf{X}_n^{(1)})^{-1} \cdots (\mathbf{X}_n^{(l)})^{-1}$, and let $\mathbf{F}_n^{(1)},\dots,\mathbf{F}_n^{(m)}$ be independent random matrices of the same form as $\mathbf{F}_n^{(0)}$. We investigate the limiting spectral distributions of the matrices $\mathbf{F}_n^{(0)}$ and $\mathbf{F}_n^{(1)} + \dots + \mathbf{F}_n^{(m)}$ as $n \to \infty$. Our main result shows that the sum $\mathbf{F}_n^{(1)} + \dots + \mathbf{F}_n^{(m)}$ has the same limiting eigenvalue distribution as $\mathbf{F}_n^{(0)}$ 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.