Regularly Varying Measures on Metric Spaces: Hidden Regular Variation and Hidden Jumps

TL;DR

This paper introduces a flexible framework for analyzing hidden regular variation in metric spaces, enabling more precise risk probability estimation across different scales and applying to various stochastic models.

Contribution

It extends existing theories to include hidden regular variation on metric spaces with removed cones, allowing simultaneous multi-scale regular variation analysis.

Findings

Framework applies to iid variables with heavy tails

Framework extends to Lévy processes with regularly varying Lévy measures

Multiple regular variation properties coexist at different scales

Abstract

We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in Hult & Lindskog (2006), Das, Mitra & Resnick (2013). Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in $\mathbb{R}_+^\infty$ with marginal distributions having regularly varying tails and to c\`adl\`ag L\'evy processes whose L\'evy measures have regularly varying tails. In both cases, an infinite number of regular variation properties coexist distinguished by different scaling functions and state spaces.