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

2014/12/10 by Götze, Friedrich, Naumov, Alexey, Tikhomirov, Alexander
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1412.3314

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 ≤ j,k ≤ 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 s₁, ..., s_n the singular values of \mathbb W: = (1)/(n) \mathbb X⁽¹⁾ \mathbb X⁽²⁾. We prove the central limit theorem for linear statistics of the squared singular values s₁², ..., s_n² showing that the limiting variance depends on κ₄: = \mathbb E (X₁₁¹)⁴ - 3.

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