For fixed $l,m ge 1$, let$mathbf{X}_n^{(0)},mathbf{X}_n^{(1)},dots,mathbf{X}_n^{(l)}$ be independentrandom $n imes n$ 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 ofthe same form as $mathbf{F}_n^{(0)}$. We investigate the limiting spectraldistributions of the matrices $mathbf{F}_n^{(0)}$ and $mathbf{F}_n^{(1)} +dots + mathbf{F}_n^{(m)}$ as $n o infty$. Our main result shows that thesum $mathbf{F}_n^{(1)} + dots + mathbf{F}_n^{(m)}$ has the same limitingeigenvalue 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 inG"otze, K"osters and Tikhomirov (2014) to sums of products of independentrandom matrices and their inverses. We establish the universality of thelimiting singular value and eigenvalue distributions, and we provide a closerdescription of the limiting distributions in terms of free probability theory.
展开▼
