  Given a large sample covariance matrix $S_N=\frac 1n\Gamma_N^{1/2}Z_N
Z_N^*\Gamma_N^{1/2}\, ,$ where $Z_N$ is a $N\times n$ matrix with i.i.d.
centered entries, and $\Gamma_N$ is a $N\times N$ deterministic Hermitian
positive semidefinite matrix, we study the location and fluctuations of
$\lambda_{\max}(S_N)$, the largest eigenvalue of $S_N$ as $N,n\to\infty$ and
$Nn^{-1} \to r\in(0,\infty)$ in the case where the empirical distribution
$\mu^{\Gamma_N}$ of eigenvalues of $\Gamma_N$ is tight (in $N$) and
$\lambda_{\max}(\Gamma_N)$ goes to $+\infty$. These conditions are in
particular met when $\mu^{\Gamma_N}$ weakly converges to a probability measure
with unbounded support on $\mathbb{R}^+$. We prove that asymptotically
$\lambda_{\max}(S_N)\sim \lambda_{\max}(\Gamma_N)$. Moreover when the
$\Gamma_N$'s are block-diagonal, and the following {\em spectral gap condition}
is assumed:$$\limsup_{N\to\infty}
\frac{\lambda_2(\Gamma_N)}{\lambda_{\max}(\Gamma_N)}<1,$$where
$\lambda_2(\Gamma_N)$ is the second largest eigenvalue of $\Gamma_N$, we prove
Gaussian fluctuations for $\lambda_{\max}(S_N)/\lambda_{\max}(\Gamma_N)$ at the
scale $\sqrt{n}$.In the particular case where $Z_N$ has i.i.d. Gaussian entries
and $\Gamma_N$ is the $N\times N$ autocovariance matrix of a long memory
Gaussian stationary process $({\mathcal X}_t)_{t\in\mathbb{Z}}$, the columns of
$\Gamma_N^{1/2} Z_N$ can be considered as $n$ i.i.d. samples of the random
vector $({\mathcal X}_1,\dots,{\mathcal X}_N)^T$. We then prove that $\Gamma_N$
is similar to a diagonal matrix which satisfies all the required assumptions of
our theorems, hence our results apply to this case.
