Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

For , we consider random matrices of the form where are real numbers and are i.i.d. copies of a normalized isotropic random vector . For every fixed , if the Normalized Counting Measures of converge weakly as , and is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457-483, 1967). For , we define a subclass of good vectors for which the centered linear eigenvalue statistics converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.