Half-space depth of log-concave probability measures

2022/01/28 by Brazitikos, Silouanos, Giannopoulos, Apostolos, Pafis, Minas
#46B06 #52A23 #52A40 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 60D05 #Probability (math.PR) #Secondary 62H05

paper · doi:10.48550/arxiv.2201.11992

Abstract

Given a probability measure μ on \mathbb Rⁿ, Tukey's half-space depth is defined for any x∈ \mathbb Rⁿ by φ_μ(x)=inf\μ(H):H∈ \cal H(x)\, where \cal H(x) is the set of all half-spaces H of \mathbb Rⁿ containing x. We show that if μ is log-concave then e^-c₁n≤ ∫_ℝⁿφ_μ(x) dμ(x) ≤ e^-c₂n/L_μ2 where L_μ is the isotropic constant of μ and c₁,c₂>0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of L_q-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

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