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

Abstrakt

For fixed l≥0 and m≥1, let X_n⁰, X_n¹,..., X_n^l be independent random n × n matrices with independent entries, let F_n⁰ := X_n⁰, X_n^1-1,..., X_n^l-1, and let F_n¹,..., F_n^m be independent random matrices of the same form as F_n⁰ . We show that as n → ∞, the matrices F_n⁰ and m^−l+1/2F_n¹ +...+ F_n^m 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.