  A regularly varying time series as introduced in Basrak and Segers (2009) is
a (multivariate) time series such that all finite dimensional distributions are
multivariate regularly varying. The extremal behavior of such a process can
then be described by the index of regular variation and the so-called spectral
tail process, which is the limiting distribution of the rescaled process, given
an extreme event at time 0. As shown in Basrak and Segers (2009), the
stationarity of the underlying time series implies a certain structure of the
spectral tail process, informally known as the "time change formula". In this
article, we show that on the other hand, every process which satisfies this
property is in fact the spectral tail process of an underlying stationary
max-stable process. The spectral tail process and the corresponding max-stable
process then provide two complementary views on the extremal behavior of a
multivariate regularly varying stationary time series.
