The shape of asymptotic dependence

Multivariate risk analysis is concerned with extreme observations. If the underlying distribution has a unimodal density then both the decay rate of the tails and the asymptotic shape of the level sets of the density are of importance for the dependence structure of extreme observations. For heavy-tailed densities, the sample clouds converge in distribution to a Poisson point process with a homogeneous intensity. The asymptotic shape of the level sets of the density is the common shape of the level sets of the intensity. For light-tailed densities, the asymptotic shape of the level sets of the density is the limit shape of the sample clouds. This paper investigates how the shape changes as the rate of decrease of the tails is varied while the copula of the distribution is preserved. Four cases are treated: a change from light tails to light tails, from heavy to heavy, heavy to light and light to heavy tails.