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We are concerned with the behavior of the eigenvalues of renormalized samplecovariance matrices of the formC_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 withi.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-rootof the nonnegative definite Hermitian matrix $A_{p}$, and $B_{n}$ is an$n\times n$ nonnegative definite Hermitian matrix. We show that the empiricalspectral distribution (ESD) of $C_n$ converges a.s. to a nonrandom limitingdistribution under some assumptions. The probability density function of theLSD 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 computationalalgorithm for evaluating this limiting density when the LSD of $A_{p}$ is amixture of point masses. In addition, when the entries of $X_{n}$ aresub-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 samplecovariance matrix and $\lambda_j$ denotes the $j$-th largest eigenvalue, when$F^A$ is a finite mixture of point masses. These results are utilized topropose a test for the covariance structure of the data where the nullhypothesis is that the joint covariance matrix is of the form $A_{p} \otimesB_n$ for $\otimes$ denoting the Kronecker product, as well as $A_{p}$ and thefirst two spectral moments of $B_n$ are specified. The performance of this testis illustrated through a simulation study.

Limiting spectral distribution of renormalized separable sample covariance matrices when $p/n\to 0$