Max-stable random sup-measures with comonotonic tail dependence

TL;DR

This paper explores max-stable random sup-measures with comonotonic tail dependence, connecting Extremes theory to random sets and risk measures, and extends related concepts with streamlined proofs.

Contribution

It introduces a unified framework for max-stable random sup-measures, clarifies the role of Choquet measures, and extends existing notions with simplified proofs and new properties.

Findings

Clarifies the role of Choquet random sup-measures

Provides a LePage representation for max-stable sup-measures

Analyzes properties like continuity and invariance

Abstract

Several objects in the Extremes literature are special instances of max-stable random sup-measures. This perspective opens connections to the theory of random sets and the theory of risk measures and makes it possible to extend corresponding notions and results from the literature with streamlined proofs. In particular, it clarifies the role of Choquet random sup-measures and their stochastic dominance property. Key tools are the LePage representation of a max-stable random sup-measure and the dual representation of its tail dependence functional. Properties such as complete randomness, continuity, separability, coupling, continuous choice, invariance and transformations are also analysed.