  In this paper, we show that the diagonal of a high-dimensional sample
covariance matrix stemming from $n$ independent observations of a
$p$-dimensional time series with finite fourth moments can be approximated in
spectral norm by the diagonal of the population covariance matrix. We assume
that $n,p\to \infty$ with $p/n$ tending to a constant which might be positive
or zero. As applications, we provide an approximation of the sample correlation
matrix ${\mathbf R}$ and derive a variety of results for its eigenvalues. We
identify the limiting spectral distribution of ${\mathbf R}$ and construct an
estimator for the population correlation matrix and its eigenvalues. Finally,
the almost sure limits of the extreme eigenvalues of ${\mathbf R}$ in a
generalized spiked correlation model are analyzed.
