发表机构
University of Minnesota Duluth; Tsinghua University（明尼苏达大学德卢斯分校; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究高维球面上随机向量两两夹角的最小角与最大角的极限联合分布，证明二者渐近独立，解决了Cai等人2013年提出的开放问题，给出了不依赖n和p相对发散速率的统一极限。
AI 中文摘要
考虑从(p-1)维单位球面上的均匀分布中抽取的n个独立随机向量，本文研究它们两两夹角的最小值与最大值的极限联合分布。证明了当n和p都趋于无穷时，最小角与最大角渐近独立，解决了Cai、Fan和Jiang（2013）在《Journal of Machine Learning Research》第14卷1837-1864页提出的开放问题。Cai、Fan和Jiang（2013）在假设limₙ→∞(ln n)/p=β的情况下，根据β=0、β∈(0,∞)或β=∞分别得到了最小角与最大角的极限边际分布；本文则无论n和p的相对发散速率如何，都给出了联合分布和边际分布的统一极限，还推导了基于最小角与最大角的某些统计量的极限分布。
英文摘要
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 $\lim_{n\to\infty}{\ln n}/p=β$ according to whether $β=0$, $β\in (0,\infty)$, or $β=\infty$. 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,
