# Spectral tail processes and max-stable approximations of multivariate regularly varying time series

## TL;DR

This paper explores the relationship between spectral tail processes and max-stable processes in multivariate regularly varying time series, showing that processes satisfying certain properties are spectral tail processes of stationary max-stable processes, offering new insights into extremal behavior.

## Contribution

It demonstrates that any process satisfying the time change formula is a spectral tail process of a stationary max-stable process, linking two key extremal process frameworks.

## Key findings

- Spectral tail processes characterize extremal behavior of multivariate time series.
- Processes satisfying the time change formula are spectral tail processes of max-stable processes.
- Provides a dual perspective on extremal analysis of stationary time series.

## Abstract

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.