Journal of Mathematical Physics, Analysis, Geometry
2017, vol. 13, No 1, pp. 82-98   https://doi.org/10.15407/mag13.01.082     ( to contents , go back )
https://doi.org/10.15407/mag13.01.082

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 n2 × n2 real symmetric and hermitian matrices Mn, which are equal to the sum mn of tensor products of the vectors X μ = B(Y μY μ), μ = 1, . . . ,mn, where Y μ are i.i.d. random vectors from ℝn(ℂn) with zero mean and unit variance of components, and B is an n2 × n2 positive definite non-random matrix. We prove that if mn / n2c ∈ [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 Mn 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.

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