  Consider a sample of a centered random vector with unit covariance matrix. We
show that under certain regularity assumptions, and up to a natural scaling,
the smallest and the largest eigenvalues of the empirical covariance matrix
converge, when the dimension and the sample size both tend to infinity, to the
left and right edges of the Marchenko--Pastur distribution. The assumptions are
related to tails of norms of orthogonal projections. They cover isotropic
log-concave random vectors as well as random vectors with i.i.d. coordinates
with almost optimal moment conditions. The method is a refinement of the rank
one update approach used by Srivastava and Vershynin to produce non-asymptotic
quantitative estimates. In other words we provide a new proof of the Bai and
Yin theorem using basic tools from probability theory and linear algebra,
together with a new extension of this theorem to random matrices with dependent
entries.
