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Browsing by Author Satoshi, Hayakawa

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    • Estimating the probability that a given vector is in the convex hull of a random sample
    Authors: Satoshi, Hayakawa; Terry, Lyons; Harald, Oberhauser;  Advisor: -;  Co-Author: - (2023)

    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)≤3d+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.