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

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.