Gaussian fluctuations for linear eigenvalue statistics of products of independent iid random matrices

TL;DR

This paper proves that the linear eigenvalue statistics of products of independent iid random matrices follow a universal Gaussian distribution as the matrix size grows, extending previous results for single matrices.

Contribution

It establishes Gaussian fluctuations for products of iid matrices with a universal variance, generalizing prior results for single matrices.

Findings

Gaussian limits for linear spectral statistics are proven.

Variance is universal, independent of the number of matrices or entry distribution.

Results extend previous work on single iid matrices to matrix products.

Abstract

Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$ iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of $X$ for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on $m$ (the number of factor matrices) or on the distribution of the entries of the matrices. The main result generalizes and improves upon previous limit statements for the linear spectral statistics of a single iid matrix by Rider and Silverstein as well as Renfrew and the second author.