On Rapid Variation of Multivariate Probability Densities

TL;DR

This paper investigates how rapid variation in multivariate densities relates to tail decay rates, establishing conditions under which joint tail behaviors are uniformly controlled, with applications to skew-elliptical distributions.

Contribution

It introduces a local uniformity condition linking multivariate density variation to joint tail decay, extending tail dependence analysis via copulas.

Findings

Rapid variation of multivariate densities implies joint tail rapid variation.

Skew-elliptical distributions exhibit rapid tail variation under certain conditions.

Higher-order tail dependence can be characterized using copulas.

Abstract

Multivariate rapid variation describes decay rates of joint light tails of a multivariate distribution. We impose a local uniformity condition to control decay variation of distribution tails along different directions, and using higher-order tail dependence of copulas, we prove that a rapidly varying multivariate density implies rapid variation of the joint distribution tails. As a corollary, rapid variation of skew-elliptical distributions is established under the assumption that the underlying density generators belong to the max-domain of attraction of the Gumbel distribution.