(assuming $M\geq N$) matrix with i.i.d. complex Gaussian entries whose real and imaginary parts have variance $\frac{1}{2}$ and zero mean. Let $\Sigma_N$ be an $N\times
N$ positive definite Hermitian matrix with eigenvalues $a_1,\ldots,a_N$ (not necessarily distinct). In this paper, we will consider the case where $\Sigma_N$ has only 2 distinct eigenvalues, 1 and $a$ such that $N_1$ of its eigenvalues are $a$ and $N-N_1$ of them are $1$. We will assume that $\frac{N}{M}\rightarrow c$ and $\frac{N_1}{N}\rightarrow\beta$ as $N,M\rightarrow\infty$ and that $0<c,\beta<1$. To be precise, we will assume the following $$\label{eq:lim}
cM-N=\tau_1=O(1),\quad N\beta-N_1=\tau_2=O(1),\quad
M,N,N_1\rightarrow\infty.$$ Let $\Sigma_N^{\frac{1}{2}}$ be any Hermitian square root of $\Sigma_N$. Then the columns of the matrix $X\Sigma_N^{\frac{1}{2}}$ are random vectors with variances $\frac{1}{2}\sqrt{a_j}$. Let the matrix $B_N$ be the following $$\label{eq:BN}
B_N=\frac{1}{M}\Sigma_N^{\frac{1}{2}}X^{\dag}X\Sigma_N^{\frac{1}{2}},$$ Then $B_N$ is the sample covariance matrix of the columns of $X\Sigma_N^{\frac{1}{2}}$, while $\Sigma_N$ is the covariance matrix. In particular, $B_N$ is a complex Wishart matrix in the class $W_{\mathbb{C}}\left(\Sigma_N,M\right)$.

The sample covariance matrix is a fundamental tool in the studies of multivariate statistics and its distribution is already known to Wishart at around 1928 (See e.g. [@Muir]) $$\label{eq:wishart}
\mathcal{P}(B_N)=\frac{1}{C}e^{-M{\mathrm{tr}}(\Sigma^{-1}B_N)}\left(\det
S\right)^{M-N},\quad M\geq N,$$ for some normalization constant $C$.

Let $y_1,\ldots,y_N>0$ be the eigenvalues of the sample 