Polar decomposition of regularly varying time series in star-shaped metric spaces

TL;DR

This paper establishes the equivalence of two definitions of regular variation for time series in star-shaped metric spaces and introduces a polar decomposition framework that captures extremal dependence through spectral tail processes.

Contribution

It proves the equivalence of distribution-based and sequence-based regular variation definitions and develops a polar decomposition with a spectral tail process in star-shaped metric spaces.

Findings

Two definitions of regular variation are shown to be equivalent.

A polar decomposition with a modulus and spectral tail process is introduced.

Stationarity induces a transformation formula for the spectral tail process.

Abstract

There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions are shown to be equivalent. The introduction of a norm-like function, called modulus, yields a polar decomposition similar to the one in Euclidean spaces. The angular component of the time series, called angular or spectral tail process, captures all aspects of extremal dependence. The stationarity of the underlying series induces a transformation formula of the spectral tail process under time shifts.