Canonical spectral representation for exchangeable max-stable sequences

TL;DR

This paper characterizes the structure of exchangeable max-stable sequences, providing a canonical spectral representation and stochastic construction, which advances understanding of their dependence structure and infinite divisibility.

Contribution

It introduces a canonical spectral representation for exchangeable max-stable sequences and provides a unique representation via a pair of a constant and a probability measure.

Findings

The set of stable tail dependence functions forms a simplex.

Each element is represented by a constant and a probability measure.

A canonical stochastic construction for these sequences is established.

Abstract

The set of infinite-dimensional, symmetric stable tail dependence functions associated with exchangeable max-stable sequences of random variables with unit Fr\'echet margins is shown to be a simplex. Except for a single element, the extremal boundary is in one-to-one correspondence with the set of distribution functions of non-negative random variables with unit mean. Consequently, each element is uniquely represented by a pair of a constant and a probability measure on the space of distribution functions of non-negative random variables with unit mean. A canonical stochastic construction for arbitrary exchangeable max-stable sequences and a stochastic representation for the Pickands dependence measure of finite-dimensional margins are immediate corollaries. As by-products, a canonical analytical description and an associated canonical Le Page series representation for non-decreasing stochastic processes that are strongly infinitely divisible with respect to time are obtained.