• Bernoulli
  • Volume 23, Number 4B (2017), 3067-3113.

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

Friedrich Götze, Alexey Naumov, and Alexander Tikhomirov

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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 $\kappa_{4}:=\mathbb{E}(X_{11}^{(1)})^{4}-3$.

Article information

Bernoulli, Volume 23, Number 4B (2017), 3067-3113.

Received: July 2015
Revised: February 2016
First available in Project Euclid: 23 May 2017

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Zentralblatt MATH identifier

central limit theorem characteristic functions Fuss–Catalan distributions products of random matrices


Götze, Friedrich; Naumov, Alexey; Tikhomirov, Alexander. Distribution of linear statistics of singular values of the product of random matrices. Bernoulli 23 (2017), no. 4B, 3067--3113. doi:10.3150/16-BEJ837.

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