Estimating the probability that a given vector is in the convex hull of a random sample

Abstract

For a d-dimensional random vector X, let pn,X(θ) be the probability that the convex hull of n independent copies of X contains a given point θ. We provide several sharp inequalities regarding pn,X(θ) and NX(θ) denoting the smallest n for which pn,X(θ) ≥ 1 / 2. As a main result, we derive the totally general inequality 1 / 2 ≤ αX(θ) NX(θ) ≤ 3 d+ 1 , where αX(θ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point θ. We also show several applications of our general results: one is a moment-based bound on NX(E[X]), which is an important quantity in randomized approaches to cubature construction or measure reduction problem. Another application is the determination of the canonical convex body included in a random convex polytope given by independent copies of X, where our combinatorial approach allows us to generalize existing results in random matrix community significantly.