Tail measure and tail spectral process of regularly varying time series

TL;DR

This paper establishes a one-to-one correspondence between the tail measure and spectral tail process of regularly varying time series, providing a comprehensive theoretical framework and construction methods for these objects.

Contribution

It proves the equivalence between tail measure and spectral tail process and offers a way to construct time series from either object, extending known results for non-negative series.

Findings

Proves a one-to-one correspondence between tail measure and spectral tail process.

Provides a method to construct time series from either object.

Recovers known results in max-stable process theory for non-negative series.

Abstract

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.