Limit theorems for convex hulls of random vectors with regularly varying distribution

Let $Z_1, Z_2, \ldots$ be iid random variables. Suppose that they have multivariate heavy tails. It is known that this is equivalent to the weak convergence of empirical measures of properly normalized random variables to some Poisson random process. We will show that, under certain conditions, this is also equivalent to the weak convergence of the convex hulls of these empirical measures and derive some applications about the properties of the random convex polytopes.