Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples

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 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.