  In this paper we study the joint distributional convergence of the largest
eigenvalues of the sample covariance matrix of a $p$-dimensional time series
with iid entries when $p$ converges to infinity together with the sample size
$n$. We consider only heavy-tailed time series in the sense that the entries
satisfy some regular variation condition which ensures that their fourth moment
is infinite. In this case, Soshnikov [31, 32] and Auffinger et al. [2] proved
the weak convergence of the point processes of the normalized eigenvalues of
the sample covariance matrix towards an inhomogeneous Poisson process which
implies in turn that the largest eigenvalue converges in distribution to a
Fr\'echet distributed random variable. They proved these results under the
assumption that $p$ and $n$ are proportional to each other. In this paper we
show that the aforementioned results remain valid if $p$ grows at any
polynomial rate. The proofs are different from those in [2, 31, 32]; we employ
large deviation techniques to achieve them. The proofs reveal that only the
diagonal of the sample covariance matrix is relevant for the asymptotic
behavior of the largest eigenvalues and the corresponding eigenvectors which
are close to the canonical basis vectors. We also discuss extensions of the
results to sample autocovariance matrices.
