Limit theorems for convex hull peels.
Preprint

Consider the convex hull peels of a finite point set in R^d consecutively obtained by repeating (i) construct the convex hull of the point set and (ii) eliminate the points from the point set that span the hull. This tool naturally ranks points in R^d and is of use in multivariate statistics. We examine the number N_n,i of vertices of the i-th convex peel. The Stein method is employed to derive a rate of convergence to the normal of the conditional law of N_n,i, given the first (i-1) convex peels, for i > 0 below a certain threshold in the case of rotationally invariant samples of size n in R^d as well as the corresponding Poisson point process models. Examples of rotationally invariant random point sets in which a central limit theorem is established include Poisson points from the normal and certain exponentially decaying distributions in R^d for d > 1.