An exceptional max-stable process fully parameterized by its extremal coefficients

TL;DR

This paper introduces Tawn-Molchanov processes, a class of max-stable processes fully characterized by their extremal coefficients, providing new bounds and constructions for max-stable processes.

Contribution

It establishes a complete characterization of extremal coefficient functions via negative definiteness and introduces Tawn-Molchanov processes with maximal dependency sets.

Findings

Tawn-Molchanov processes are uniquely determined by their extremal coefficients.

Sharp lower bounds for finite-dimensional distributions are derived.

New methods to construct valid extremal coefficient functions using Bernstein functions.

Abstract

The extremal coefficient function (ECF) of a max-stable process $X$ on some index set $T$ assigns to each finite subset $A\subset T$ the effective number of independent random variables among the collection $\{X_t\}_{t\in A}$. We introduce the class of Tawn-Molchanov processes that is in a 1:1 correspondence with the class of ECFs, thus also proving a complete characterization of the ECF in terms of negative definiteness. The corresponding Tawn-Molchanov process turns out to be exceptional among all max-stable processes sharing the same ECF in that its dependency set is maximal w.r.t. inclusion. This entails sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. A spectral representation of the Tawn-Molchanov process and stochastic continuity are discussed. We also show how to build new valid ECFs from given ECFs by means of Bernstein functions.