Products of Independent Non-Hermitian Random Matrices

Abstract

For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, ..., X_m$ with i.i.d. centered entries with a finite $(2+η)$-th moment, $ η>0.$ As $n$ tends to infinity, we show that the empirical spectral distribution of $n^{-m/2} \*X_1 X_2 ... X_m$ converges, with probability 1, to a non-random, rotationally invariant distribution with compact support in the complex plane. The limiting distribution is the $m$-th power of the circular law.

The paper is accepted for publication in Electronic Journal of Probability. In the final version we fixed a small technical gap in the proofs of Theorem 15 and Lemma 19 and added the reference to the 2009 paper by Girko and Vladimirova