Record times for coverage thresholds and maximal spacings

Abstract

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 $\nu_m$ 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 $\nu_n^{1/\sqrt{n}}\to \exp \big(\sqrt{2}\: \big)$ as $n \to \infty$, and likewise for $\tilde{N}_n$ and $\tilde{\nu}_n$, defined analogously in terms of $(\tilde{R}_n)$. But if $B=A$ and $d \geq 3$, then $N_n \sim (1- \frac{1}{d}) (\log n)^2$ and $\nu_n^{1/\sqrt{n}}\to \exp \big( \sqrt{2d/(d-1)} \: \big)$. 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 $A$.