Record times for coverage thresholds and maximal spacings

Abstract: Let $X_1,X_2, \ldots$ be independent uniform random points in a bounded region $A \subset {\bf R}<sup>d$ 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 $\tilde{R}<em>n$ 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 $R_n &lt; R_{n-1}$. 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 $B \subset A<sup>o$, we show that almost surely, $N_n \sim \frac12 (\log n)<sup>2$ 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 $N_n \sim (1- \frac{1}{d}) (\log n)<sup>2$ 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 $A$.

• The paper develops rigorous asymptotics for record times of coverage thresholds and maximal spacings in stochastic geometry, showing growth rates like $(\ln n)^2$ and $(\ln n)^2/d$ based on sample dimension.
• The precise constants depend on whether the largest hole is in the interior or on the boundary, changing the coverage threshold equation.
• The methodology uses a head-intuitive generalization inspired by ‘maximum entropy’ heuristics and proofs of the uniqueness of the largest hole.

Overview

This paper, by Mathew D. Penrose (2608.19104), studies the record times of two fundamental random quantities in stochastic geometry: the coverage threshold $R_{n,k}$ (the smallest radius such that balls of that radius centred on $X_1,\dots,X_n$ cover a target set $B$ at least $k$ times) and the maximal $k$-spacing $\theta \tilde R_{n,k}^d$ (the volume of the largest ball in $A$ containing fewer than $k$ sample points). Both are monotone non-increasing in $n$, so all records are lower records. The central question is: how often does adding a new point strictly decrease these quantities?

The main results show that the record count grows like $(\ln n)^2$ rather than the classical Rényi asymptotic of order $X_1,\dots,X_n$0 for i.i.d. records. Specifically, for $X_1,\dots,X_n$1, almost surely

$X_1,\dots,X_n$2

where $X_1,\dots,X_n$3 counts records up to time $X_1,\dots,X_n$4 and $X_1,\dots,X_n$5 is the time of the $X_1,\dots,X_n$6th record. For maximal spacings the same constants hold. But when $X_1,\dots,X_n$7, $X_1,\dots,X_n$8 is $X_1,\dots,X_n$9 and boundary holes dominate (which happens in particular in the uniform case with $B$0), the constants change:

$B$1

This dimension-dependent constant is the most striking quantitative feature of the paper: it reflects the fact that when $B$2 with $B$3, the largest hole is centred near $B$4, where a ball's intersection with $B$5 has volume asymptotically half the full ball, but its probability content scales as $B$6 times the interior value.

Heuristic mechanism

A new point creates a record precisely when it lands inside the current largest hole. Since the probability content of the largest hole is known to be asymptotic to $B$7 [Deheuvels et al.; Penrose 2023], the expected number of records up to time $B$8 behaves like $B$9. For $k$0 with boundary-dominated holes, the relevant content is instead about $k$1, giving the factor $k$2. This contrasts sharply with Rényi's theorem, where the record probability at time $k$3 is exactly $k$4.

A general record lemma

All three theorems follow from a single abstract result. Suppose $k$5 is adapted to a filtration, with $k$6 an upper bound on $k$7, equality holding on $k$8, where $k$9 a.s., and $k$0 a.s. with uniform integrability of $k$1. Then

$k$2

In the applications, $k$3 indicates a record, $k$4 indicates uniqueness of the largest hole, and $k$5 is the probability content of that hole. The proof uses a martingale convergence argument on $k$6 plus a deterministic summation lemma for sequences comparable to $k$7.

Uniqueness of the largest hole

A technically substantial part of the paper establishes that, for large $k$8, the set $k$9 of locations at maximum $\theta \tilde R_{n,k}^d$0-nearest-neighbour distance from the sample is almost surely a singleton — otherwise a point landing in "the" largest hole might not actually reduce the threshold.

For $\theta \tilde R_{n,k}^d$1, uniqueness holds a.s. via a general-position argument: two distinct maximizers would each be equidistant from $\theta \tilde R_{n,k}^d$2 sample points, which occurs with probability zero. For $\theta \tilde R_{n,k}^d$3 with $\theta \tilde R_{n,k}^d$4 boundary, the proof is harder because maximizers can lie on $\theta \tilde R_{n,k}^d$5; the paper shows there exists $\theta \tilde R_{n,k}^d$6 such that ties have probability zero whenever $\theta \tilde R_{n,k}^d$7, using geometric lemmas (sphere condition, tangent-ball arguments) and the fact that no boundary point is equidistant from $\theta \tilde R_{n,k}^d$8 sample points. Notably, uniqueness is proven only for all sufficiently large $\theta \tilde R_{n,k}^d$9, not for every $A$0; this suffices since only finitely many non-unique times occur.

Location of the largest hole

The constant $A$1 in the lemma is determined by where $A$2 concentrates. Using strong laws from Penrose [2023], namely $A$3 for $A$4, the paper shows:

• If $A$5, then $A$6 and $A$7.
• If $A$8 and $A$9 (boundary-dominated, e.g. uniform $k$0), then $k$1 in scaled distance, $k$2, and the ball-volume factor gives $k$3.
• If $k$4 and $k$5 (interior-dominated, always true when $k$6), then $k$7 stays away from the boundary, $k$8, and $k$9.

Uniform integrability of the squared normalized contents follows from an exponential moment bound on $n$0 obtained by a cube-partition argument.

Extensions and related quantities

The methods also apply to convex polytopes $n$1: if the maximum over face dimensions $n$2 of $n$3 is attained at a unique $n$4, then conjecturally $n$5. Similarly, for the largest $n$6-nearest-neighbour link $n$7, the paper identifies a quantity $n$8 of pseudo-records (times when $n$9 decreases) satisfying the same asymptotics as Theorem 3; however, since $(\ln n)^2$0 is not monotone, this is not the conventional record count, and proving equivalence with true records is left open.

Limitations and open questions

The paper explicitly leaves several cases unresolved:

• The uniform case with $(\ln n)^2$1 and $(\ln n)^2$2 is untreated, because the largest hole may lie either in the interior or on the boundary with comparable probability. Based on earlier results, the author conjectures that for $(\ln n)^2$3 the boundary-dominated asymptotics hold, while for $(\ln n)^2$4 the limit of $(\ln n)^2$5 should be a weighted average of $(\ln n)^2$6 and $(\ln n)^2$7 depending on the perimeter of $(\ln n)^2$8.
• The critical regime $(\ln n)^2$9 for general densities is not handled.
• The intermediate case $X_1,\dots,X_n$00 with $X_1,\dots,X_n$01 not contained in $X_1,\dots,X_n$02 is excluded.
• The polytope and pseudo-record extensions are described without proofs.
• Uniqueness of the largest hole is established only eventually (for large $X_1,\dots,X_n$03), not for all $X_1,\dots,X_n$04.

Conclusion

The paper converts the heuristic "records occur when a new point lands in the largest hole" into rigorous almost sure asymptotics, yielding $X_1,\dots,X_n$05 record growth with explicit constants determined by whether the extremal hole lies in the interior or near the boundary. The key technical contributions are the general record lemma and the eventual-uniqueness results for the furthest location from the sample. The remaining open cases — notably $X_1,\dots,X_n$06 uniform coverage of $X_1,\dots,X_n$07, and true records for nearest-neighbour links — define the natural next targets for this line of research.