Limiting spectral distributions of sums of products of non-Hermitian random matrices

For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of the same form as Fn0 . We show that as n → ∞, the matrices Fn0 and m−l+1/2Fn1 +...+ Fnm have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov 2015 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.