Journal article
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 ) ≤ 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. 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.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, pdf, 578.7KB, Terms of use)
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- Publisher copy:
- 10.1007/s00440-022-01186-1
Authors
- Publisher:
- Springer
- Journal:
- Probability Theory and Related Fields More from this journal
- Volume:
- 185
- Issue:
- 3-4
- Pages:
- 705-746
- Publication date:
- 2023-01-07
- Acceptance date:
- 2022-12-21
- DOI:
- EISSN:
-
1432-2064
- ISSN:
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0178-8051
- Language:
-
English
- Keywords:
- Pubs id:
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1318665
- Local pid:
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pubs:1318665
- Deposit date:
-
2023-01-09
Terms of use
- Copyright holder:
- Hayakawa et al.
- Copyright date:
- 2023
- Rights statement:
- © The Author(s) 2023. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
- Licence:
- CC Attribution (CC BY)
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