  We are concerned with the behavior of the eigenvalues of renormalized sample
covariance matrices of the form
C_n=\sqrt{\frac{n}{p}}\left(\frac{1}{n}A_{p}^{1/2}X_{n}B_{n}X_{n}^{*}A_{p}^{1/2}-\frac{1}{n}\tr(B_{n})A_{p}\right)
as $p,n\to \infty$ and $p/n\to 0$, where $X_{n}$ is a $p\times n$ matrix with
i.i.d. real or complex valued entries $X_{ij}$ satisfying $E(X_{ij})=0$,
$E|X_{ij}|^2=1$ and having finite fourth moment. $A_{p}^{1/2}$ is a square-root
of the nonnegative definite Hermitian matrix $A_{p}$, and $B_{n}$ is an
$n\times n$ nonnegative definite Hermitian matrix. We show that the empirical
spectral distribution (ESD) of $C_n$ converges a.s. to a nonrandom limiting
distribution under some assumptions. The probability density function of the
LSD of $C_{n}$ is derived and it is shown that it depends on the LSD of $A_{p}$
and the limiting value of $n^{-1}\tr(B_{n}^2)$. We propose a computational
algorithm for evaluating this limiting density when the LSD of $A_{p}$ is a
mixture of point masses. In addition, when the entries of $X_{n}$ are
sub-Gaussian, we derive the limiting empirical distribution of
$\{\sqrt{n/p}(\lambda_j(S_n) - n^{-1}\tr(B_n) \lambda_j(A_{p}))\}_{j=1}^p$
where $S_n := n^{-1} A_{p}^{1/2}X_{n}B_{n}X_{n}^{*}A_{p}^{1/2}$ is the sample
covariance matrix and $\lambda_j$ denotes the $j$-th largest eigenvalue, when
$F^A$ is a finite mixture of point masses. These results are utilized to
propose a test for the covariance structure of the data where the null
hypothesis is that the joint covariance matrix is of the form $A_{p} \otimes
B_n$ for $\otimes$ denoting the Kronecker product, as well as $A_{p}$ and the
first two spectral moments of $B_n$ are specified. The performance of this test
is illustrated through a simulation study.
