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 X(1) and X(2). Assume that X(q)jk, 1 ≤ j, k ≤ n, q = 1, 2, are i.i.d. random variables with EX(q)jk= 0, VarX(q)jk = 1. Denote by s1(W), . , sn(W) the singular values of W:= 1/n X(1) and X(2). We prove the central limit theorem for linear statistics of the squared singular values s2/1 (W), . , s2/n(W) showing that the limiting variance depends on κ4 := E(X(1)11)4 -3.