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Distribution of eigenvalues of sample covariance matrices with tensor product samples (1601.07443v2)
Published 27 Jan 2016 in math-ph and math.MP
Abstract: We consider $n2\times n2$ real symmetric and hermitian matrices $M_n$, which are equal to sum of $m_n$ tensor products of vectors $X\mu=B(Y\mu\otimes Y\mu)$, $\mu=1,\dots,m_n$, where $Y\mu$ are i.i.d. random vectors from $\mathbb Rn (\mathbb Cn)$ with zero mean and unit variance of components, and $B$ is an $n2\times n2$ positive definite non-random matrix. We prove that if $m_n/n2\to c\in [0,+\infty)$ and the Normalized Counting Measure of eigenvalues of $BJB$, where $J$ is defined below, 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.