  We provide some asymptotic theory for the largest eigenvalues of a sample
covariance matrix of a p-dimensional time series where the dimension p = p_n
converges to infinity when the sample size n increases. We give a short
overview of the literature on the topic both in the light- and heavy-tailed
cases when the data have finite (infinite) fourth moment, respectively. Our
main focus is on the heavytailed case. In this case, one has a theory for the
point process of the normalized eigenvalues of the sample covariance matrix in
the iid case but also when rows and columns of the data are linearly dependent.
We provide limit results for the weak convergence of these point processes to
Poisson or cluster Poisson processes. Based on this convergence we can also
derive the limit laws of various function als of the ordered eigenvalues such
as the joint convergence of a finite number of the largest order statistics,
the joint limit law of the largest eigenvalue and the trace, limit laws for
successive ratios of ordered eigenvalues, etc. We also develop some limit
theory for the singular values of the sample autocovariance matrices and their
sums of squares. The theory is illustrated for simulated data and for the
components of the S&P 500 stock index.
