Multivariate Regular Variation on Cones: Application to Extreme Values, Hidden Regular Variation and Conditioned Limit Laws

TL;DR

This paper unifies three key areas of heavy tail and extreme value analysis—limit laws, hidden regular variation, and conditioned limit laws—using the framework of multivariate regular variation on cones.

Contribution

It introduces a unified approach to analyze heavy tails and extreme values through multivariate regular variation on specific cones, connecting different subareas.

Findings

Unified framework for heavy tail analysis

Clarified relationships between limit laws and regular variation

Provided insights into asymptotic independence and conditioned laws

Abstract

We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory: limit laws for componentwise maxima of iid random variables;hidden regular variation and asymptotic independence;conditioned limit laws when one component of a random vector is extreme. The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones $[0,\infty]^d\setminus \{\boldsymbol 0\};(0,\infty]^d;$ and $[0,\infty]\times (0,\infty]$.