The volume of random simplices from elliptical distributions in high dimension

Abstract

Random simplices and more general random convex bodies of dimension p in R n 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.

Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls

“…More recent developments also treat concentration inequalities, small ball probabilities, large deviations or weak limit theorems. In particular central limit theorems for the volume or log-volume in the respective models were established in [9,81,69,2,30,29]. The three former references treat convex polytopes in a fixed dimension, whereas the other ones random simplices for the dimension tending to infinity.…”