Distribution of linear statistics of singular values of the product of random matrices

Abstract

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