Conditional independence among max-stable laws

TL;DR

This paper proves that for max-stable random vectors with positive continuous density, conditional independence among disjoint sub-vectors implies their joint independence, limiting the Markov structures in such models.

Contribution

It establishes a fundamental link between conditional and joint independence in max-stable vectors, showing certain Markov structures are impossible.

Findings

Conditional independence implies joint independence in max-stable vectors.

Most tractable max-stable models cannot have interesting Markov structures.

The result applies to vectors with positive continuous density.

Abstract

Let $X$ be a max-stable random vector with positive continuous density. It is proved that the conditional independence of any collection of disjoint sub-vectors of $X$ given the remaining components implies their joint independence. We conclude that a broad class of tractable max-stable models cannot exhibit an interesting Markov structure.