Skip to content
DraconiticResearch
Search Sources API
197,761 studies31 sourcesUpdated 56 min ago
PreprintCC BY-NC-ND 4.0arXiv

Joint Distributions of Minimum and Maximum Angles on High-Dimensional Spheres

  • Yongcheng Qi
  • Lijian Yang
Open at sourcePDFDOICite

Abstract

Consider n independent random vectors sampled from uniform distribution on (p−1)-dimensional unit sphere. This paper investigates the limiting joint distribution for the minimum and the maximum values of their pairwise angles. It proves that the minimum and the maximum angles are asymptotically independent when both n and p tend to infinity, which solves an open problem raised in Cai, Fan and Jiang (2013) [\emph{Journal of Machine Learning Research} 14, 1837-1864]. Cai, Fan and Jiang (2013) obtained the limiting marginal distributions for both the minimum and the maximum angles under assumption limn→∞ln n/p=β according to whether β=0, β∈ (0,∞), or β=∞. This paper presents unified limits for both joint distributions and marginal distributions regardless of the relative divergence rate of n and p. The paper also derives the limiting distributions for some statistics based on the minimum and the maximum angles,

Related

  1. Maximum Entropy Probability Distributions on Spheres with Fixed Mean Busemann Function and Holomorphic-Information-Geometric Model of Cognition
    Vladimir Jacimovic
    arXivPreprintCC BY 4.0Full textProbabilityarXiv
  2. Minimum Distance Spectral Radius of Tree with Given Maximum Degree
    Yifei Li, Zia Ullah Khan
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs
    Elizaveta Popova
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. Tensor network representations of discrete maximum entropy distributions via mean polytopes
    Alex Goessmann, Martin Eigel
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsStatisticsarXiv
  5. Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees
    Xuanyu Cao, Chunxiang Wang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{qi2026joint,
  title = {{Joint Distributions of Minimum and Maximum Angles on High-Dimensional Spheres}},
  author = {Yongcheng Qi and Lijian Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05330},
  eprint = {2609.05330},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05330},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Joint Distributions of Minimum and Maximum Angles on High-Dimensional Spheres
AU  - Yongcheng Qi
AU  - Lijian Yang
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05330
UR  - https://arxiv.org/abs/2609.05330
AB  - Consider n independent random vectors sampled from uniform distribution on (p−1)-dimensional unit sphere. This paper investigates the limiting joint distribution for the minimum and the maximum values of their pairwise angles. It proves that the minimum and the maximum angles are asymptotically independent when both n and p tend to infinity, which solves an open problem raised in Cai, Fan and Jiang (2013) [\emph{Journal of Machine Learning Research} 14, 1837-1864]. Cai, Fan and Jiang (2013) obtained the limiting marginal distributions for both the minimum and the maximum angles under assumption lim_n→∞ln n/p=β according to whether β=0, β∈ (0,∞), or β=∞. This paper presents unified limits for both joint distributions and marginal distributions regardless of the relative divergence rate of n and p. The paper also derives the limiting distributions for some statistics based on the minimum and the maximum angles,
ER  -