Convex Hulls of Regularly Varying Processes

Abstract

We consider the asymptotic behavior of compact convex subsets $ {{\widetilde{W}}_n} $ of ℝ d defined as the closed convex hulls of the ranges of independent and identically distributed (i.i.d.) random processes (X i )1≤i≤n . Under a condition of regular variation on the law of the X i ’s, we prove the weak convergence of the rescaled convex hulls $ {{\widetilde{W}}_n} $ as n → ∞ and analyze the structure and properties of the limit shape. We illustrate our results by several examples of regularly varying processes and show that, in contrast with the Gaussian setting, in many cases, the limit shape is a random polytope of ℝ d .