覆盖阈值与最大间距的记录时间

中文摘要 AI 辅助

设$X_1,X_2,\ldots$是$\mathbb{R}^d$中具有光滑边界的有界区域$A$内的独立均匀随机点，$d\geq1$；设$B\subset A$为紧集。$B$的覆盖阈值$R_n$是使$B$被以$X_1,\ldots,X_n$为中心、半径为$r$的球覆盖的最小$r$；最大间距$\tilde{R}_n$是$A\setminus\{X_1,\ldots,X_n\}$中最大球的体积。我们研究序列$(R_n)$中记录时间的渐近频率，即满足$R_n<R_{n-1}$的时刻$n$。设$N_m$为序列$(R_n)$到时刻$m$的记录数，$\nu_m$为序列$(R_n)$第$m$个记录值出现的时刻。对于$B\subset A^o$，我们证明几乎必然有$N_n\sim\frac{1}{2}(\log n)^2$，且$\nu_n^{1/\sqrt{n}}\to\exp(\sqrt{2})$（当$n\to\infty$时），对由$(\tilde{R}_n)$类似定义的$\tilde{N}_n$和$\tilde{\nu}_n$也成立。但当$B=A$且$d\geq3$时，有$N_n\sim(1-\frac{1}{d})(\log n)^2$，且$\nu_n^{1/\sqrt{n}}\to\exp(\sqrt{2d/(d-1)})$。我们还讨论了固定$k\in\mathbb{N}$时向$k$覆盖阈值、最大$k$间距以及$A$中非均匀分布点$X_i$的推广情况。

英文摘要

Let $X_1,X_2, \ldots $ be independent uniform random points in a bounded region $A \subset {\bf R}^d$ having a smooth boundary, $d \geq 1$. Let $B \subset A$ be compact. The _coverage threshold_ of $B$, $R_n$, is the smallest $r$ such that $B$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. The _maximal spacing_ $\tilde{R}_n$ is the volume of the largest ball contained in $A \setminus \{X_1,\ldots,X_n\}$. We investigate the asymptotic frequency of _record times_ in the sequence $(R_n)$, that is times $n$ for which $R_n < R_{n-1}$. Let $N_m$ denote the number of records in the sequence $(R_n)$ up to time $m$, and let $ν_m$ be the time at which the $m$th record value of the sequence $(R_n)$ occurs. For $B \subset A^o$, we show that almost surely, $N_n \sim \frac12 (\log n)^2$ and $ν_n^{1/\sqrt{n}}\to \exp \big(\sqrt{2}\: \big)$ as $n \to \infty$, and likewise for $\tilde{N}_n$ and $\tildeν_n$, defined analogously in terms of $(\tilde{R}_n)$. But if $B=A$ and $d \geq 3$, then $N_n \sim \frac12 (1- \frac{1}{d}) (\log n)^2$ and $ν_n^{1/\sqrt{n}}\to \exp \big( \sqrt{2d/(d-1)} \: \big)$. We also discuss the generalization (for fixed $k \in {\bf N}$) to $k$-coverage thresholds, maximal $k$-spacings and non-uniformly distributed points $X_i$ in $A$.

发表机构

• University of Bath（巴斯大学）

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