The convex hull of a random sample in

Abstract

Let X1.......Xn be an iid sequence of random vectors in with common distribution function which satisfies a multivariate regular o variation condition. In the metric space of compact convex sets of with metric given by the Hausdorff distance we show that the sample convex hull converges in distribution to the convex hull of the points of a two-dimensional Poisson process. We also give necessary and sufficient conditions for the number of vertices of the limiting convex hull to be finite with probability one. Finally, we discuss weak convergence of the number of vertices of the sample convex hull as well as asymptotic behavior of the expected number of vertices.