  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.
