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

  1. Holger Kösters
  2. Alexander Tikhomirov

Abstract

For fixed l≥0 and 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,..., (Xn(l))-1, and let Fn(1),..., Fn(m) be independent random matrices of the same form as Fn(0) . We show that as n → ∞, the matrices Fn(0) and m−(l+1)/2(Fn(1) +...+ Fn(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.

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Probability and Mathematical Statistics

38, z. 2, 2018

Pages from 359 to 384

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