Half-space depth of log-concave probability measures

• 成果类型: Article
• 署名作者: Brazitikos, Silouanos; Giannopoulos, Apostolos; Pafis, Minas
• 署名单位: National & Kapodistrian University of Athens
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01236-2
• 发表日期: 2024
• 页码: 309-336
• 关键词: random points

摘要

Given a probability measure mu on R-n, Tukey's half-space depth is defined for any x is an element of R-n by phi(mu)(x) = inf{mu (H) : H is an element of H(x)}, where H(x) is the set of all half-spaces H of R-n containing x. We show that if mu is a non-degenerate log-concave probability measure on R-n thene(-c1n) <= integral (n)(R)phi mu (x) d mu (x) <= e(mu)(-c2n/L2)where L-mu is the isotropic constant of mu and c(1), c(2) > 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.