The maximum domain of attraction of multivariate extreme value distributions is small

TL;DR

This paper investigates the properties of probability measures in the context of multivariate extreme value theory, showing that measures within the domain of attraction are both dense and meager, with analogous results in free probability.

Contribution

It establishes that the set of measures in the domain of attraction of multivariate extreme value distributions is dense and of the first Baire category, extending to free probability.

Findings

The domain of attraction measures are dense in the space of probability measures.

The domain of attraction measures form a first Baire category set.

Analogous results hold in free probability theory.

Abstract

Consider the set of Borel probability measures on $\mathbf{R}^k$ and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.