Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples
V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv 61022, Ukraine
Received December 23, 2015, revised April 30, 2016
We consider the n2 × n2 real symmetric and hermitian matrices Mn,
which are equal to the sum mn of tensor products of the vectors Xμ = B(Yμ ⊗ Yμ), μ = 1, . . . ,mn, where Yμ 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 / n2 → 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 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.