均匀随机样本最大填充距离的Gumbel收敛
Gumbel convergence for maximal packing distances from uniform random samples
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中文总结 AI 辅助
本文针对均匀随机样本，在光滑紧子流形上建立了最大填充最小距离的Gumbel型极值定律，显式推导仅依赖几何的尺度序列，提供可解释的置信界，并给出圆、球面、环面示例与模拟验证。
中文摘要 AI 辅助
设$(X_n)_n$为一列独立同分布的$R^d$值随机变量，均匀分布在$R^d$的紧子集$M$上。本文研究该紧子集$M$的最大填充的渐近行为。在关于$M$的某些假设下（特别是$M$为$R^d$中无边界的光滑子流形），我们建立了从样本$(X_1,\cdots,X_n)$到最大填充中心的最小距离的精确极值定律（Gumbel型）。该结果导出了该最大填充以及支撑集$M$的显式渐近置信界。我们工作的一个显著特征在于显式推导了Gumbel收敛中的尺度序列，该序列仅依赖于支撑集$M$的几何性质。这些公式为$M$提供了可解释的置信界，是对Fasy等人（2014）等先前方法的新补充。我们的方法桥接了几何与组合概率论证，并依赖于诸如Lambert $W$函数等分析工具。此外，我们提供了方向统计和圆形统计中主要样本空间（圆、球面、环面）的示例，以及说明和支持理论发现的模拟。
英文摘要
Let $(X _n )_n$ be a sequence of i.i.d. $R^d$-valued random variables uniformly distributed on a compact subset $M$ of $R^d$ . In this work, we study the asymptotic behavior of maximal packings of this compact subset $M$. Under some assumptions on $M$ (in particular, that it is a smooth submanifold of $R^d$ without boundary), we establish a precise extreme value law (of Gumbel type) for the minimal distances from the sample $(X_1 , \cdots, X_ n)$ to the centers of maximal packings. This result leads to explicit asymptotic confidence bounds for this maximal packing and thus to the support $M$. A distinctive feature of our contribution is the explicit derivation of the scaling sequences in the Gumbel convergence, depending only on the geometry of the support M. These formulas provide interpretable confidence bounds for $M$, which represent a novel complement to previous approaches such as those by Fasy et al. (2014). Our approach bridges geometric and combinatorial probability arguments and relies on analytic tools such as the Lambert $W$ function. In addition, we provide examples of main sample spaces in directional and circular statistics (circle, sphere, torus), along with simulations that illustrate and support the theoretical findings.
发表机构
- Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、拉格朗日数学研究所）
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