Tail measure and spectral tail process of regularly varying time series.

*(English)*Zbl 1404.60074Summary: 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 [T. Owada and G. Samorodnitsky, “Tail measures of stochastic processes or random fields with regularly varying tails”, Preprint] and [B. Basrak and J. Segers, Stochastic Processes Appl. 119, No. 4, 1055–1080 (2009; Zbl 1161.60319)]. Our main result is to prove in an abstract framework that there is a one-to-one correspondence between these two objects, 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 nonnegative time series, we recover results explicitly or implicitly known in the theory of max-stable processes.

##### MSC:

60G70 | Extreme value theory; extremal stochastic processes |

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