The volume of random simplices from elliptical distributions in high dimension

Abstract

Random simplices and more general random convex bodies of dimension p in Rn with p≤n are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if p→∞ and n→∞ in such a way that p/n→γ∈(0,1), a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of p×n random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.