On Idempotent D-Norms

TL;DR

This paper studies the mathematical structure of D-norms used in max-stable distributions, characterizing idempotent D-norms and analyzing their iterative behavior within a semigroup framework.

Contribution

It introduces a semigroup operation on D-norms, characterizes idempotent D-norms, and explores the limits of iterative multiplication of D-norms.

Findings

Characterization of idempotent D-norms

Existence of limits for iterative D-norm multiplication

Identification of idempotent D-norms as fixed points

Abstract

Replacing the spectral measure by a random vector $\bfZ$ allows the representation of a max-stable distribution on $\R^d$ with standard negative margins via a norm, called \emph{$D$-norm}, whose generator is $\bfZ$. The set of $D$-norms can be equipped with a commutative multiplication type operation, making it a semigroup with an identity element. This multiplication leads to idempotent $D$-norms. We characterize the set of idempotent $D$-norms. Iterating the multiplication provides a track of $D$-norms, whose limit exists and is again a $D$-norm. If this iteration is repeatedly done on the same $D$-norm, then the limit of the track is idempotent.