Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples

D. Tieplova

V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61022, Ukraine
E-mail: dtieplova@yahoo.com

Received December 23, 2015, revised April 30, 2016

Abstract

We consider the n^{2} × n^{2} real symmetric and hermitian matrices M_{n},
which are equal to the sum m_{n} of tensor products of the vectors X^{μ} = B(Y^{μ} ⊗ Y^{μ}), μ = 1, . . . ,m_{n}, where Y^{μ} are i.i.d. random vectors from ℝ^{n}(ℂ^{n}) with zero mean and unit variance of components, and B is an n^{2} × n^{2} positive
definite non-random matrix. We prove that if m_{n} / n^{2} → c ∈ [0,+∞) and the Normalized Counting Measure of eigenvalues of BJB, where J is defined below in (2.6), converges weakly, then the Normalized Counting Measure of
eigenvalues of M_{n} converges weakly in probability to a non-random limit, and its Stieltjes transform can be found from a certain functional equation.

Mathematics Subject Classification 2000: 15B52. Key words: random matrix, sample covariance matrix, tensor product, distribution of eigenvalues.