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Distribution of Linear Statistics of Singular Values of the Product of Random Matrices (1412.3314v2)
Published 10 Dec 2014 in math.PR
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 $s_12,..., s_n2$ showing that the limiting variance depends on $\kappa_4: = \mathbb E (X_{11}{1})4 - 3$.