Distribution of Linear Statistics of Singular Values of the Product of Random Matrices

TL;DR

This paper establishes a central limit theorem for linear statistics of squared singular values of the product of two independent random matrices with i.i.d. entries, revealing the dependence of variance on the fourth moment.

Contribution

It provides the first CLT for linear statistics of singular values of matrix products, explicitly linking variance to the fourth cumulant of entries.

Findings

Proves CLT for squared singular values of matrix products

Shows variance depends on the fourth moment of entries

Extends understanding of spectral statistics in random matrix products

Abstract

In this paper we consider the product of two independent random matrices $\mathbb X^{(1)}$ and $\mathbb X^{(2)}$. Assume that $X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, 2,$ are i.i.d. random variables with $\mathbb E X_{jk}^{(q)} = 0, \mathbb E (X_{jk}^{(q)})^2 = 1$. Denote by $s_1, ..., s_n$ the singular values of $\mathbb W: = \frac{1}{n} \mathbb X^{(1)} \mathbb X^{(2)}$. We prove the central limit theorem for linear statistics of the squared singular values $s_1^2, ..., s_n^2$ showing that the limiting variance depends on $\kappa_4: = \mathbb E (X_{11}^{1})^4 - 3$.