  The goal of this paper is an exhaustive investigation of the link between the
tail measure of a regularly varying time series and its spectral tail process,
independently introduced in Owada and Samorodnitsky (2012) and Basrak and
Segers (2009). Our main result is to prove in an abstract framework that there
is a one to one correspondance between these two objets, and given one of them
to show that it is always possible to build a time series of which it will be
the tail measure or the spectral tail process. For non negative time series, we
recover results explicitly or implicitly known in the theory of max-stable
processes.
