  The goal of this paper is two-fold: 1. We review classical and recent
measures of serial extremal dependence in a strictly stationary time series as
well as their estimation. 2. We discuss recent concepts of heavy-tailed time
series, including regular variation and max-stable processes. Serial extremal
dependence is typically characterized by clusters of exceedances of high
thresholds in the series. We start by discussing the notion of extremal index
of a univariate sequence, i.e. the reciprocal of the expected cluster size,
which has attracted major attention in the extremal value literature. Then we
continue by introducing the extremogram which is an asymptotic auto-correlation
function for sequences of extremal events in a time series. In this context, we
discuss regular variation of a time series. This notion has been useful for
describing serial extremal dependence and heavy tails in a strictly stationary
sequence. We briefly discuss the tail process coined by Basrak and Segers to
describe the dependence structure of regularly varying sequences in a
probabilistic way. Max-stable processes with Frechet marginals are an important
class of reg- ularly varying sequences. Recently, this class has attracted
attention for modeling and statistical purposes. We apply the extremogram to
max-stable processes. Finally, we discuss estimation of the extremogram both in
the time and frequency domains.
