
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 allows the representation of a max-stable distribution on with standard negative margins via a norm, called \emph{-norm}, whose generator is . The set of -norms can be equipped with a commutative multiplication type operation, making it a semigroup with an identity element. This multiplication leads to idempotent -norms. We characterize the set of idempotent -norms. Iterating the multiplication provides a track of -norms, whose limit exists and is again a -norm. If this iteration is repeatedly done on the same -norm, then the limit of the track is idempotent.
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Taxonomy
TopicsStochastic processes and financial applications · Risk and Portfolio Optimization · Mathematical Analysis and Transform Methods
