One-Component Regular Variation and Graphical Modeling of Extremes

TL;DR

This paper introduces one-component regular variation and extends graphical modeling techniques to high-dimensional extremes, providing a new framework for understanding the distribution of vectors conditioned on large norms.

Contribution

It develops the concept of one-component regular variation, extends classical theorems to this setting, and generalizes the Hammersley-Clifford theorem for modeling multivariate tail dependencies.

Findings

Extended Karamata's theorem to one-component regular variation.

Established a graphical model framework for multivariate extremes.

Linked asymptotic conditional independence to tail density factorizations.

Abstract

The problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize Hammersley-Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.