Abstract:Let $M_n$ denote the largest interpoint distance among independent random points $X_1,\dots,X_n$ uniformly distributed in a compact set in $\mathbb{R}^d$. Weak limit laws for $M_n$ are known in several geometric settings, in particular for ellipsoids with a unique major axis. In this paper we treat the simplest nontrivial case in which the largest semi-axis is not unique, namely the rotational ellipsoid $\{(x_1,x_2,x_3)\in\mathbb{R}^3: (x_1^2+x_2^2)/h^2 + x_3^2/a^2 \le 1\}$, where $0<a<h$. The diameter of this ellipsoid is attained by all antipodal pairs on the equatorial circle, so the extremal points are not isolated. We prove that $n^{4/7}(2h-M_n)$ converges in distribution to a Weibull-type limit law with explicit parameter. The proof combines geometric localization arguments with a Chen--Stein Poisson approximation for rare nearly diametral pairs.
| Comments: | 11 pages |
| Subjects: | Probability (math.PR) |
| MSC classes: | 60D05, 60F05, 60G70 |
| Cite as: | arXiv:2605.24627 [math.PR] |
| (or arXiv:2605.24627v1 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24627 arXiv-issued DOI via DataCite (pending registration) |
From: Norbert Henze [view email]
[v1]
Sat, 23 May 2026 15:37:56 UTC (10 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。