On the Regular Variation of Ratios of Jointly Frechet Random Variables

TL;DR

This paper establishes a precise criterion for when the ratio of two jointly alpha-Frechet random variables exhibits regular variation, introducing the concept of a ratio tail index to capture dependence features beyond traditional tail dependence measures.

Contribution

It provides a necessary and sufficient condition for regular variation of ratios of jointly alpha-Frechet variables and introduces the ratio tail index as a new dependence measure.

Findings

Derived the asymptotic behavior of the quotient correlation coefficient in dependent cases.

Established a new type of regular variation for products, distinct from previous models.

Provided a practical spectral representation-based condition for regular variation.

Abstract

We provide a necessary and sufficient condition for the ratio of two jointly alpha-Frechet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al. (2002).