Decomposable Tail Graphical Models

TL;DR

This paper develops an asymptotic theory for extreme value analysis in decomposable graphical models, showing that the tail behavior inherits the graphical structure and applies to various dependence types.

Contribution

It introduces a comprehensive asymptotic framework for extremes in decomposable graphical models, including new results on tail limits and dependence structures.

Findings

Tail graphical models inherit the original graph structure.

Weak limits of normalized extremes are characterized for various copula classes.

Convergence to tail noise is established in block graphs.

Abstract

We develop an asymptotic theory for extremes in decomposable graphical models by presenting results applicable to a range of extremal dependence types. Specifically, we investigate the weak limit of the distribution of suitably normalised random vectors, conditioning on an extreme component, where the conditional independence relationships of the random vector are described by a chordal graph. Under mild assumptions, the random vector corresponding to the distribution in the weak limit, termed the tail graphical model, inherits the graphical structure of the original chordal graph. Our theory is applicable to a wide range of decomposable graphical models including asymptotically dependent and asymptotically independent graphical models. Additionally, we analyze combinations of copula classes with differing extremal dependence in cases where a normalization in terms of the conditioning variable is not guaranteed by our assumptions. We show that, in a block graph, the distribution of the random vector normalized in terms of the random variables associated with the separators converges weakly to a distribution we term tail noise. In particular, we investigate the limit of the normalized random vectors where the clique distributions belong to two widely used copula classes, the Gaussian copula and the max-stable copula.