Canonical spectral representation for exchangeable max-stable sequences

Abstract

The set $\mathfrak {L}$ of infinite-dimensional, symmetric stable tail dependence functions associated with exchangeable max-stable sequences of random variables with unit Fréchet margins is shown to be a simplex. Except for a single element, the extremal boundary of $\mathfrak {L}$ is in one-to-one correspondence with the set $\mathfrak {F}_{1}$ of distribution functions of non-negative random variables with unit mean. Consequently, each $\ell \in \mathfrak {L}$ is uniquely represented by a pair (b, µ) of a constant b and a probability measure µ on $\mathfrak {F}_{1}$. A canonical stochastic construction for arbitrary exchangeable max-stable sequences and a stochastic representation for the Pickands dependence measure of finite-dimensional margins of l 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.