Estimating the probability that a given vector is in the convex hull of a random sample
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For a d-dimensional random vector X, let p_n, X be the probability that the convex hull of n i.i.d. copies of X contains a given point x. We provide several sharp inequalities regarding p_n, X and N_X, which denotes the smallest n with p_n, X≥ 1/2. As a main result, we derive a totally general inequality which states 1/2 ≤α_X N_X ≤ 16d, where α_X (a.k.a. the Tukey depth) is the infimum of the probability that X is contained in a fixed closed halfspace including the point x. We also provide some applications of our results, one of which gives a moment-based bound of N_X via the Berry-Esseen type estimate.
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