#  Random coverage from within with variable radii, and Johnson-Mehl cover
times††thanks: Supported by EPSRC grant EP/T028653/1

Mathew D. Penrose  Department of Mathematical Sciences, University of Bath,
Bath BA2 7AY, United Kingdom<EMAIL_ADDRESS>Frankie Higgs Department
of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.
<EMAIL_ADDRESS>
###### Abstract

Given a compact planar region $A$, let $\tau_{A}$ be the (random) time it
takes for $A$ to be fully covered by a spatial birth-growth process in $A$
with seeds arriving as a unit-intensity Poisson point process in
$A\times[0,\infty)$, where upon arrival each seed grows at unit rate in all
directions. We show that if $\partial A$ is smooth or polygonal then
$\mathbb{P}[\pi\tau_{sA}^{3}-6\log s-4\log\log s\leq x]$ tends to
$\exp(-(\frac{81}{4\pi})^{1/3}|A|e^{-x/3}\linebreak-(\frac{9}{2\pi^{2}})^{1/3}|\partial
A|e^{-x/6})$ in the large-$s$ limit; the second term in the exponent is due to
boundary effects, the importance of which was not recognized in earlier work
on this model. We present similar results in higher dimensions (where boundary
effects dominate). These results are derived using new results on the
asymptotic probability of covering $A$ with a high-intensity spherical Poisson
Boolean model _restricted to $A$_ with grains having iid small random radii,
which generalize recent work of the author that dealt only with grains of
deterministic radius.

##  1 Introduction

The Johnson-Mehl (J-M) tessellation is a classic model of a random
tessellation in $\mathbb{R}^{d}$, where $d\in\mathbb{N}$. Seeds are generated
as a homogeneous Poisson point process ${\cal
H}_{\rho}=\\{(x_{i},t_{i})\\}_{i\geq 1}$ of intensity $\rho$ in space-time
$\mathbb{R}^{d}\times[0,\infty)$. If location $x_{i}$ is not already claimed
by time $t_{i}$, the seed $i$ becomes a cell at that instant, which
immediately starts to grow from $x_{i}$ at unit rate in all directions,
claiming previously unclaimed territory as part of that cell. Whenever the
growing cell hits another cell, it stops growing in that direction. Ultimately
the whole of $\mathbb{R}^{d}$ is tessellated by cells.

This model dates back at least to work of Kolmogorov and of Johnson and Mehl
in the 1930s on modelling crytallization processes; other applications include
growth of surface film on metals, and more recent applications in neurobiology
are described in [2]. For further discussion and references, see also [7, 8].

A natural variant is the restricted Johnson-Mehl tessellation, which we define
as follows. Given a specified compact region $A\subset\mathbb{R}^{d}$, we
allow only for seeds that arrive inside the region $A$, that is, the seeds are
generated by a Poisson process in $A\times[0,\infty)$. Cells grow by the same
rules as described before, but stop growing whenever they hit the boundary of
$A$. Ultimately the cells tessellate $A$.

We are interested here in the time at which $A$ becomes completely covered,
i.e. the first time at which every point of $A$ has been claimed by a cell,
either for the J-M model or for the restricted J-M model. In particular we are
concerned with the limiting distribution of the cover time, denoted
$\tilde{T}_{\rho}$ for the J-M model and $T_{\rho}$ for the restricted J-M
model, as $\rho\to\infty$. Equivalently, one can keep $\rho$ fixed (say,
$\rho=1$), and consider the distribution of the cover times for an expanding
sequence of windows $(A_{L})_{L>0}$ given by dilations of a fixed set $A$ by a
factor of $L$, in the large-$L$ limit. In the later case, the restricted
Johnson-Mehl tessellation uses only seeds that arrive inside $A_{L}$, and we
refer to the cover times in this limiting regime as $\tau_{L}$ (for the
restricted J-M model) and $\tilde{\tau}_{L}$ (for the J-M model).

These cover times have previously been considered by S. N. Chiu in [1], for
$A=[0,1]^{d}$. In fact, Chiu considers a more general class of J-M models,
where the rate at which seeds arrive is allowed to be non-homogeneous in time,
and where moreover the seeds do not necessarily all grow at the same rate. We
do not consider such generalizations here.

In [1, page 893], Chiu rightly distinguishes between the J-M and restricted
J-M models. However, he goes on to assert that ‘all theorems in this paper are
valid for both models’ because ‘almost surely edge effects do not play a role
in the limiting behaviour of $T_{L}$’ (Chiu’s $T_{L}$ is equivalent to our
$\tau_{L}$). In other words, Chiu seems to be asserting that $\tau_{L}$ and
$\tilde{\tau}_{L}$ have the same limiting distribution.

It is our contention that this assertion is incorrect. In dimensions $d\geq
2$, the limiting behaviour of $T_{\rho}$ is different from that of
$\tilde{T}_{\rho}$, and the limiting behaviour of $\tau_{L}$ is different from
that of $\tilde{\tau}_{L}$; in other words, edge effects _do_ play a role. We
justify this assertion with results identifying the limiting distribution of
$T_{\rho}$ (suitably scaled and centred) and showing it is different from that
of $\tilde{T}_{\rho}$; our limiting result for $\tilde{T}_{\rho}$ is
consistent with the result in [1]. Likewise we show that the limiting
distribution of $\tau_{L}$ (suitably scaled and centred) is different from
that of $\tilde{\tau}_{L}$. Our results for $T_{\rho}$ and $\tau_{L}$ apply
when $A$ is polygonal in $d=2$, or when it has a smooth (more precisely,
$C^{2}$) boundary for general $d\geq 2$.

When $A=[0,1]^{d}$, we do not provide detailed limiting distributions for
$T_{\rho}$ and $\tau_{L}$ except in the case $d=2$. To give a detailed limit
distribution when $d\geq 3$ would require careful consideration of all faces
of all dimensions and is beyond the scope of this paper. However, the time to
cover all the $(d-1)$-dimensional faces will be a lower bound for the actual
cover time so boundary effects can certainly not be neglected in this case
either.

We shall derive our results on the J-M model from new results, of independent
interest, on the spherical Poisson Boolean model (SPBM), which is defined to
be a collection of Euclidean balls (referred to as grains) of i.i.d. random
radii, centred on the points of a homogeneous Poisson process in the whole of
$\mathbb{R}^{d}$. For the the restricted SPBM, we take a Poisson process on
$A$ rather than on all of $\mathbb{R}^{d}$.

We shall determine the probability that the restricted SPBM covers the whole
of $A$, in the limit when the Poisson intensity becomes large and the radii of
balls become small in a linked manner. In [10] we derived results of this
nature for balls with deterministic radius; here we generalize them to allow
for balls of random radius, which is needed to derive our results on the J-M
cover time. Our results on coverage of $A$ by the restricted SPBM complement
the classic results of [3, 5] on the limiting probability of covering $A$ with
an unrestricted SPBM.

##  2 Statement of results

Throughout this paper, we assume that we are given $d\in\mathbb{N}$, and a
compact, Riemann measurable set $A\subset\mathbb{R}^{d}$ (Riemann
measurability of a bounded set in $\mathbb{R}^{d}$ amounts to its boundary
having zero Lebesgue measure).

For $x\in\mathbb{R}^{d}$ and $r>0$ set
$B(x,r):=\\{y\in\mathbb{R}^{d}:\|y-x\|\leq r\\}$ where $\|\cdot\|$ denotes the
Euclidean norm. (We write $B_{(d)}(x,r)$ for this if we wish to emphasise the
dimension.) For $D\subset\mathbb{R}^{d}$, let $\overline{D}$ and $D^{o}$
denote the closure of $D$ and interior of $D$, respectively, and set $\partial
D:=\overline{D}\cap\overline{\mathbb{R}^{d}\setminus D}$, the topological
boundary of $D$. For $r>0$, let $D^{(r)}:=\\{x\in A:B(x,r)\subset D^{o}\\}$,
the ‘$r$-interior’ of $D$. Let $|D|$ denote the Lebesgue measure (volume) of
$D$, and $|\partial D|$ the perimeter of $D$, i.e. the $(d-1)$-dimensional
Hausdorff measure of $\partial D$, when these are defined. Write $\log\log t$
for $\log(\log t)$, $t>1$.

Define the set $D^{[r]}$ to be the interior of the union of all hypercubes of
the form $\prod_{i=1}^{d}[n_{i}r,(n_{i}+1)r]$, with
$n_{1},\ldots,n_{d}\in\mathbb{Z}$, that are contained in $D^{o}$ (the set
$D^{[r]}$ resembles $D^{(r)}$ but is guaranteed to be Riemann measurable).

We say that $D$ has $C^{2}$ boundary (for short: $\partial D\in C^{2}$) if for
each $x\in\partial D$ there exists a neighbourhood $U$ of $x$ and a real-
valued function $f$ that is defined on an open set in $\mathbb{R}^{d-1}$ and
twice continuously differentiable, such that $\partial D\cap U$, after a
rotation, is the graph of the function $f$. We say that $\partial D\in
C^{1,1}$ (a weaker condition) if for each $x$ the function $f$ satisfies only
that $f$ is contiuously differentiable with Lipschitz first order partial
derivatives.

###  2.1 Results for the spherical Poisson Boolean model

As discsussed in the introduction, we shall derive our results for the J-M
model from results on coverage by the restricted SPBM, which are of
independent interest and which we now state. Given a nonnegative random
variable $Y$ and given $n>0$, suppose we have a collection of balls of
independent random radii with the distribution of $Y$, centred on the points
of a homogeneous Poisson point process in $A$ of intensity $n$.

Given also $k\in\mathbb{N}$, let $Z_{A,k}(n,Y)$ be the set of points
$x\in\mathbb{R}^{d}$ such that $x$ is covered by at least $k$ of the balls, so
in particular $Z_{A,1}(n,Y)$ is the union of the balls (we shall provide a
more detailed description of $Z_{A,k}(n,Y)$ in Section 3.2.) Throughout this
paper, $n$ is not necessarily an integer.

######  Theorem 2.1.

Suppose $Y$ is a nonnegative random variable with
$0<\mathbb{E}\,[Y^{2+\varepsilon}]<\infty$ for some $\varepsilon>0$. Suppose
that $d=2$ and $A$ is polygonal. Let $k\in\mathbb{N},\beta\in\mathbb{R}$, and
suppose $(r_{n})_{n>0}$ are nonnegative numbers satisfying $n\pi
r_{n}^{2}\mathbb{E}\,[Y^{2}]-\log n-(2k-1)\log\log n\to\beta$. as
$n\to\infty$. Then as $n\to\infty$,

$\displaystyle\mathbb{P}[A\subset
Z_{A,k}(n,r_{n}Y)]\to\exp\Big{(}-\Big{(}\frac{(\mathbb{E}\,[Y])^{2}}{\mathbb{E}\,[Y^{2}]}\Big{)}{\bf
1}_{\\{k=1\\}}|A|e^{-\beta}-\Big{(}\frac{c_{2,k}\mathbb{E}\,[Y]|\partial
A|}{(\mathbb{E}\,[Y^{2}])^{1/2}}\Big{)}e^{-\beta/2}\Big{)},$ (2.1)

where we set $c_{2,k}:=\pi^{-1/2}(1/2)^{k-1}/(k-1)!$.

Given $d\in\mathbb{N}$, let $\omega_{d}:=\pi^{d/2}/\Gamma(1+d/2)$, the volume
of the unit ball in $\mathbb{R}^{d}$, and set

$\displaystyle
c_{d}:=\frac{1}{d!}\left(\frac{\sqrt{\pi}\;\Gamma(1+d/2)}{\Gamma((d+1)/2)}\right)^{d-1}.$
(2.2)

Note that $c_{1}=c_{2}=1$, and $c_{3}=3\pi^{2}/32$. Moreover, using Stirling’s
formula one can show that $c_{d}^{1/d}\sim e(\pi/(2d))^{1/2}$ as $d\to\infty$.
The constant $c_{d}$ is denoted $\psi_{d}$ in [1, page 894], while
$\omega_{d}$ is denoted $\theta_{d}$ in [10]. Later it will be useful to have
$\omega_{0}$ defined: we set $\omega_{0}:=1$.

Our result for $A$ with $\partial A\in C^{2}$ involves a further constant
$c_{d,1}$ defined (as in [10]) for each $k\in\mathbb{N}$ by

$\displaystyle c_{d,k}$
$\displaystyle:=c_{d-1}\omega_{d}^{2-d-1/d}\omega_{d-1}^{2d-3}\omega_{d-2}^{1-d}(1-1/d)^{d+k-3+1/d}2^{-1+1/d}/(k-1)!.$
(2.3)

It can be checked that

$\displaystyle
c_{d,1}=((d-1)!)^{-1}2^{1-d}(d-1)^{d-2+1/d}\pi^{(d/2)-1}\Gamma\left(\frac{d+1}{2}\right)^{1-d}\Gamma\left(\frac{d}{2}\right)^{d-1+1/d},$
(2.4)

and $c_{d,k}=c_{d,1}(1-1/d)^{k-1}/(k-1)!$. By (2.3) we have
$c_{2,1}=\pi^{-1/2}$ and $c_{3,1}=2^{-4}\pi^{5/3}$. In the right hand side of
(2.4) the first factor of $((d-1)!)^{-1}$ was given, incorrectly, as
$(d!)^{-1}$ in [10].

######  Theorem 2.2.

Suppose that $d\geq 2$ and $\partial A\in C^{2}$ with $\overline{A^{o}}=A$.
Suppose $Y$ is a nonnegative random variable with
$0<\mathbb{E}\,[Y^{d+\varepsilon}]<\infty$ for some $\varepsilon>0$. Let
$k\in\mathbb{N},\beta\in\mathbb{R}$, and suppose $(r_{n})_{n>0}$ are
nonnegative numbers satisfying

$\displaystyle n\omega_{d}r_{n}^{d}\mathbb{E}\,[Y^{d}]-(2-2/d)\log
n-2(d+k-3+1/d)\log\log n\to\beta$ (2.5)

as $n\to\infty$. If $d=2$ then (2.1) holds, while if $d\geq 3$ then

$\displaystyle\lim_{n\to\infty}\mathbb{P}[A\subset
Z_{A,k}(n,r_{n}Y)]=\exp\Big{(}-c_{d,k}\Big{(}\frac{(\mathbb{E}\,[Y^{d-1}])^{d-1}}{(\mathbb{E}\,[Y^{d}])^{d-2+1/d}}\Big{)}|\partial
A|e^{-\beta/2}\Big{)}.$ (2.6)

######  Remark 2.3.

Taking $Y\equiv 1$ in these results gives [10, Theorem 3.2] and [10, Theorem
3.1]. [10] also includes parallel results (in the case $Y\equiv 1$) for a
situation where the number of grains in the restricted SPBM is a large
deterministic constant rather than a Poisson variable. We expect that a result
along these lines could be given for general $Y$ using the methods of this
paper.

######  Remark 2.4.

The condition $\overline{A^{o}}=A$ should have been included in the statement
of [10, Theorem 3.1]. For example, if $d=2$ and $A$ is the union of a disk
(filled in) and a circle (not filled in, and disjoint from the disk) then that
theorem does not apply.

######  Remark 2.5.

In Theorem 2.2, we conjecture that the condition $\partial A\in C^{2}$ can be
relaxed to $\partial A\in C^{1,1}$.

######  Remark 2.6.

If $\mathbb{E}\,[Y^{d}]=\infty$ then $\mathbb{P}[A\subset
Z_{\mathbb{R}^{d},k}(n,rY)]=1$ for any $k\in\mathbb{N}$ and any $r>0$; see
e.g. [6, Theorem 16.4] for the case $k=1$. However, one could still look for a
sequence $(r_{n})$ such that $\mathbb{P}[A\subset Z_{A}(n,r_{n}Y)]$ converges
to a nontrivial limit. It seems plausible in this case that $A$ is most likely
to be covered (if at all) by a single ball. This looks like an interesting
problem that lies beyond the scope of this paper.

######  Remark 2.7.

A series of videos demonstrating Theorem 2.1 can be viewed at this
link222https://www.youtube.com/playlist?list=PLiaV5rk6Gk7qEXpLOU7FSvN4b8dy1_GJn.
In each of these videos, each frame is a picture of $[0,1]^{2}$ with each
$x\in[0,1]^{2}$ assigned a colour based on $\mathrm{argmin}_{i}\inf\\{t\geq
0:x\in B(p_{i},tY_{i})\\}$, where $p_{i}$ is the centre of the $i$th ball in
the collection and $Y_{i}$ is the associated independent copy of $Y$.

The $Y_{i}$s are Pareto distributed, so have a heavy tail. As the video
progresses, the tail becomes heavier, with the growth rates in subsequent
frames coupled so that the $Y_{i}$s change continuously. It starts when
$\mathbb{E}\,[Y_{i}^{3}]=\infty$ but
$\mathbb{E}\,[Y_{i}^{3-\varepsilon}]<\infty$ for all $\varepsilon>0$ (this
meets the conditions of Theorem 2.1), and gradually reduces the number of
finite moments, finishing when $\mathbb{E}\,[Y_{i}^{0.2}]=\infty$.

The video after the point where $\mathbb{E}\,[Y_{i}^{2}]=\infty$ illustrates
Remark 2.6, and shows that even when second moments do not exist, the
restricted coverage model can have an interesting structure, with multiple
discs covering large areas, but no single enormous disc covering most of
$[0,1]^{2}$, apart from when $Y_{i}$ is extremely heavy-tailed.

###  2.2 Results for the Johnson-Mehl model

Assume that on a common probability space $(\mathbb{S},{\cal F},\mathbb{P})$,
we have a family of Poisson point processes ${\cal H}_{\rho},\rho>0$. Here
${\cal H}_{\rho}$ is homogeneous of intensity $\rho$ on
$\mathbb{R}^{d}\times\mathbb{R}_{+}$, where $\mathbb{R}_{+}:=[0,\infty)$. We
write ${\cal H}_{\rho,A}$ for ${\cal H}_{\rho}\cap(A\times\mathbb{R}_{+})$.

Given $\rho\in(0,\infty)$, define the coverage times $T_{\rho}$ and
$\tilde{T}_{\rho}$ by

$\displaystyle T_{\rho}:=\inf\left\\{t>0:A\subset\cup_{(x,s)\in{\cal
H}_{\rho}\cap(A\times[0,t])}B(x,t-s)\right\\};$ (2.7)
$\displaystyle\tilde{T}_{\rho}:=\inf\left\\{t>0:A\subset\cup_{(x,s)\in{\cal
H}_{\rho}\cap(\mathbb{R}^{d}\times[0,t])}B(x,t-s)\right\\}.$ (2.8)

Also, for $L>0$ set $A_{L}:=\\{Lx:x\in A\\}$ and let

$\displaystyle\tau_{L}:=\inf\left\\{t>0:A_{L}\subset\cup_{(x,s)\in{\cal
H}_{1}\cap(A_{L}\times[0,t])}B(x,t-s)\right\\};$ (2.9)
$\displaystyle\tilde{\tau}_{L}:=\inf\left\\{t>0:A_{L}\subset\cup_{(x,s)\in{\cal
H}_{1}\cap(\mathbb{R}^{d}\times[0,t])}B(x,t-s)\right\\}.$ (2.10)

Then $T_{\rho}$, $\tilde{T}_{\rho}$ are the cover times of $A$ for the
restricted J-M model and for the J-M model, respectively, with intensity
$\rho$. Likewise $\tau_{L}$, $\tilde{\tau}_{L}$, are the cover time of $A_{L}$
for the restricted J-M model and for the J-M model, respectively, with
intensity $1$.

Our results in this subsection are concerned with the asymptotic distribution
of the cover times $T_{\rho}$ and $\tilde{T}_{\rho}$ as $\rho\to\infty$, and
the asymptotic distribution of $\tau_{L}$ and $\tilde{\tau}_{L}$ as
$L\to\infty$. We start with a result in general $d$ for the unrestricted cover
time $\tilde{T}_{\rho}$ (and for $\tilde{\tau}_{L})$. This is simpler to deal
with than $T_{\rho}$ because boundary effects are avoided.

######  Proposition 2.8.

Suppose $A\subset\mathbb{R}^{d}$ is compact and Riemann measurable with
$|A|>0$. Let $\beta\in\mathbb{R}$. Then as $\rho\to\infty$,

$\displaystyle\mathbb{P}\left[\omega_{d}\rho\tilde{T}_{\rho}^{d+1}-d\log\rho-d^{2}\log\log\rho\leq\beta\right]\to\exp(-c_{d}(d^{d}\omega_{d})^{-1/(d+1)}|A|e^{-\beta/(d+1)}).$
(2.11)

Also as $L\to\infty$,

$\displaystyle\mathbb{P}\left[\omega_{d}\tilde{\tau}_{L}^{d+1}-d(d+1)\log
L-d^{2}\log\log L\leq\beta\right]$
$\displaystyle\to\exp\Big{(}-c_{d}\Big{(}\frac{(d+1)^{d^{2}}}{d^{d}\omega_{d}}\Big{)}^{1/(d+1)}|A|e^{-\beta/(d+1)}\Big{)}.$
(2.12)

In Appendix A, we shall verify that this result is consistent with the case
$\gamma=v=1$ of [1, Theorem 4]. The proof in [1] refers to an unpublished
preprint; for completeness we shall present the (fairly short) proof of
Proposition 2.8 in Section 4.

Our main new results in this section are concerned with $T_{\rho}$ and
$\tau_{L}$, the cover times for the restricted J-M model. We first give a
result for the case where $A$ is a polygon in $d=2$.

######  Theorem 2.9.

Suppose that $d=2$ and $A$ is polygonal. Let $|\partial A|$ denote the length
of the boundary of $A$. Let $\beta\in\mathbb{R}$. Then as $\rho\to\infty$,

$\displaystyle\mathbb{P}[\pi\rho
T_{\rho}^{3}-2\log\rho-4\log\log\rho\leq\beta]\to\exp(-(4\pi)^{-1/3}|A|e^{-\beta/3}-(2\pi^{2})^{-1/3}|\partial
A|e^{-\beta/6}).$ (2.13)

Also as $L\to\infty$,

$\displaystyle\mathbb{P}[\pi\tau_{L}^{3}-6\log L-4\log\log
L\leq\beta]\to\exp\Big{(}-\Big{(}\frac{81}{4\pi}\Big{)}^{1/3}|A|e^{-\beta/3}-\Big{(}\frac{9}{2\pi^{2}}\Big{)}^{1/3}|\partial
A|e^{-\beta/6}\Big{)}.$ (2.14)

Our result for general $d$ and $A$ with $C^{2}$ boundary involves the constant
$c^{\prime}_{d}$ defined by

$\displaystyle c^{\prime}_{d}$
$\displaystyle:=\frac{c_{d-1}\omega_{d-1}^{2d-3}}{\omega_{d-2}^{d-1}d^{d-1}}\Big{(}\frac{(d-1)^{d}}{2\omega_{d}^{d}}\Big{)}^{(d-1)/(d+1)}.$
(2.15)

######  Theorem 2.10.

Suppose that $d\geq 2$ and $\partial A\in C^{2}$ with $\overline{A^{o}}=A$.
Let $|\partial A|$ denote the $(d-1)$-dimensional Hausdorff measure of the
boundary of $A$. Let $\beta\in\mathbb{R}$. Then as $\rho\to\infty$, if $d=2$
then (2.13) holds while if $d\geq 3$ then

$\displaystyle\mathbb{P}[\omega_{d}\rho
T_{\rho}^{d+1}-2(d-1)\log\rho-2d(d-1)\log\log\rho\leq\beta]\to\exp(-c^{\prime}_{d}|\partial
A|e^{-\beta/(2d+2)}).$ (2.16)

Also as $L\to\infty$, if $d=2$ then (2.14) holds while if $d\geq 3$ then

$\displaystyle\mathbb{P}[\omega_{d}\tau_{L}^{d+1}-2(d^{2}-1)\log
L-2d(d-1)\log\log L\leq\beta]$
$\displaystyle\to\exp(-(d+1)^{d(d-1)/(d+1)}c^{\prime}_{d}|\partial
A|e^{-\beta/(2d+2)}).$ (2.17)

In short, when $d=2$, for $\rho$ large the last point of $A$ to get covered in
the restricted J-M model could lie either on the boundary or in the interior
of $A$, while for $d\geq 3$, when $\rho$ is large the last point to get
covered is very likely to lie on the boundary of $A$.

######  Remark 2.11.

A series of videos of very high intensity Johnson-Mehl processes inside
$A=[0,1]^{2}$ can be viewed
here333https://youtu.be/hCj9td6x2WY?list=PLiaV5rk6Gk7rh6nvAts1w4WZrHE0TKeW2&index=2
An interactive tool for generating Johnson-Mehl tessellations is available at
https://frankiehiggs.pyscriptapps.com/johnson-mehl-plot/latest. To illustrate
the above remark about which point is covered last when $d=2$, we have
included one video in the playlist where the last point to be covered is at
the boundary, and one where it is in the interior.

######  Remark 2.12.

The results stated above are actually proved for a slight variant of the
restricted J-M model, in which a growing cell, when it hits the boundary of
$A$, does not stop growing in that direction but extends beyond $A$ (but cells
are still restricted to be born only inside $A$); then $T_{\rho}$ denotes the
time at which the union of cells first covers $A$.

When $A$ is convex, the value of $T_{\rho}$ is the same for this variant of
the restricted J-M model as it is for the restricted J-M model first described
at the start of this paper. If $A$ is not convex, we would expect our results
to still be true for the restricted J-M model as originally described, but we
prove them only for the variant described just now.

######  Remark 2.13.

Suppose we take $d=2$ and $A=[0,1]^{2}$. Comparing (2.11) with (2.13) we see
that the limiting distribution of $T_{\rho}^{3}$ (appropriately scaled and
centred) is different from that of $\tilde{T}_{\rho}^{3}$ (scaled and centred
the same way). Similarly, comparing (2.12) with (2.14) we see that the
limiting distribution of $\tau_{L}^{3}$ (appropriately scaled and centred) is
different from that of $\tilde{\tau}_{L}^{3}$ (scaled and centred the same
way).

When $d\geq 3$ and $\partial A\in C^{2}$, we see from (2.11) and (2.16) that
even the centring constants required to get a nondegenerate limit distribution
for $\omega_{d}\rho T_{\rho}^{d+1}$ are different (larger) compared to the
corresponding constants for $\omega_{d}\rho\tilde{T}_{\rho}^{d+1}$. Again
similar remarks apply for $\tau_{L}$ and $\tilde{\tau}_{L}$.

######  Remark 2.14.

As mentioned earlier, [1] considers cover times for more general J-M models
where the growth rate of cells, or the arrival rates of seeds, are not
necessarily constants. If growth rates are non-constant, then it is rather
problematic to model the covered region at a given time as simply a SPBM; one
has to modify our initial description of the cell growth model to allow a
faster-growing cell to pass through a slower-growing cell and come out the
other side. If one is willing to do this (which seems to be the approach in
[1]), the methods of this paper should be applicable for dealing with boundary
effects for these more general J-M models too.

######  Remark 2.15.

The distribution of the maximum of two independent Gumbel variables with
different scale parameters is known as a two-component extreme value (TCEV)
distribution in the hydrology literature [12].

Theorems 2.9 and 2.10 tell us that when $d=2$ and $A$ is polygonal or
$\partial A\in C^{2}$, the random variables $\omega_{d}\rho T_{\rho}^{d+1}$
and of $\omega_{d}\tau_{L}^{d+1}$, suitably centred, are asymptotically TCEV
distributed, while if $d\geq 3$ and $\partial A\in C^{2}$, they are
asymptotically Gumbel distributed.

###  2.3 Strategy of proof

Next, we briefly describe the strategy for the proof, in Sections 3 and 4, of
the weak convergence results that were stated above.

We shall use a known result (Lemma 3.2) giving the limiting probability of
covering a bounded region of $\mathbb{R}^{d}$ by an SPBM on the whole of
$\mathbb{R}^{d}$, in the limit of high intensity $n$ and small balls of random
radius, i.e. distributed as $r_{n}Y$ for a specified nonnegative random
variable $Y$ and for constants $r_{n}$ that become small as $n$ becomes large.

We shall consider for large $n$ and suitable choice of $r_{n}$, the SPBM with
balls of radii distributed as $r_{n}Y$, restricted to a $d$-dimensional half-
space $\mathbb{H}$. In Lemma 3.5 we determine the limiting probability that a
given bounded set within the hyperplane $\partial\mathbb{H}$ is covered, by
applying Lemma 3.2 in $d-1$ dimensions. Moreover we will show that the
probability that a region in the half-space within distance $n^{\zeta}r_{n}$
of that set is covered with the same limiting probability, where ${\zeta}$ is
a small positive constant.

We shall then prove Theorem 2.1 by applying Lemma 3.5 to determine the
limiting probability that the region near the edges of a polygonal set $A$ is
covered, and Lemma 3.2 directly to determine the limiting probability that the
interior region is covered, along with a separate argument to show the regions
near the corners of $A$ are covered with high probability.

To prove Theorem 2.2 we approximate to $A$ by a polytopal set $A_{n}$ with
faces of width $O(n^{9{\zeta}}r_{n})$ and follow similar steps to those just
mentioned for polygonal $A$.

In the case where $Y$ is unbounded, to obtain the required independence, for
example between coverage events for different faces of our polyhedron or
polygon, we consider only those balls of radius at most $n^{\zeta}r_{n}$, with
a separate argument (Lemmma 3.3) to show this does not affect the limiting
coverage probabilities.

Applying Lemma 3.2 for an SPBM with $Y$ uniformly distributed on $[0,1]$
yields a proof of Proposition 2.8. Similarly, applying Theorems 2.1 and 2.2
for an SPBM with $Y$ uniformly distributed on $[0,1]$ yields proofs of
Theorems 2.9 and 2.10 respectively.

##  3 Proof of Theorems 2.1 and 2.2

Throughout this section, we assume $A\subset\mathbb{R}^{d}$ is compact with
$|A|>0$, and that we are given a nonnegative random variable $Y$ satisfying
$0<\mathbb{E}\,[Y^{\gamma}]<\infty$ for some $\gamma>d$.

###  3.1 Preliminaries

We now give some further notation used throughout. Let $o$ denote the origin
in $\mathbb{R}^{d}$. Set $\mathbb{H}:=\mathbb{R}^{d-1}\times[0,\infty)$ and
$\partial\mathbb{H}:=\mathbb{R}^{d-1}\times\\{0\\}$.

Given two sets $\mathcal{X},{\cal Y}\subset\mathbb{R}^{d}$, we set
$\mathcal{X}\triangle{\cal Y}:=(\mathcal{X}\setminus{\cal Y})\cup({\cal
Y}\setminus\mathcal{X})$, the symmetric difference between $\mathcal{X}$ and
${\cal Y}$. Also, we write $\mathcal{X}\oplus{\cal Y}$ for the set
$\\{x+y:x\in\mathcal{X},y\in{\cal Y}\\}$, and $\\#(\mathcal{X})$ for the
number of elements of $\mathcal{X}$ (possibly $+\infty$). Given also
$x\in\mathbb{R}^{d}$ we write $x+{\cal Y}$ for $\\{x\\}\oplus{\cal Y}$.

Given a Borel measure $\mu$ on $\mathbb{R}^{d}$, and Borel
$D\subset\mathbb{R}^{d}$, let $\mu|_{D}$ denote the restriction of the measure
$\mu$ to $D$ (a Borel measure on $D$).

Given $x,y\in\mathbb{R}^{d}$, we denote by $[x,y]$ the line segment from $x$
to $y$, that is, the convex hull of the set $\\{x,y\\}$. We write $a\wedge b$
(respectively $a\vee b$) for the minimum (resp. maximum) of any two numbers
$a,b\in\mathbb{R}$.

Given $m\in\mathbb{N}$ and functions $f:\mathbb{N}\cap[m,\infty)\to\mathbb{R}$
and $g:\mathbb{N}\cap[m,\infty)\to(0,\infty)$, we write $f(n)=O(g(n))$ as
$n\to\infty$ if $\limsup_{n\to\infty}|f(n)|/g(n)<\infty$. We write
$f(n)=o(g(n))$ as $n\to\infty$ if $\lim_{n\to\infty}f(n)/g(n)=0$. We write
$f(n)=\Theta(g(n))$ as $n\to\infty$ if $f$ takes only positive values, and
both $f(n)=O(g(n))$ and $g(n)=O(f(n))$. Given $s>0$ and functions
$f:(0,s)\to\mathbb{R}$ and $g:(0,s)\to(0,\infty)$, we write $f(r)=O(g(r))$ as
$r\downarrow 0$ $\limsup_{r\downarrow 0}|f(r)|/g(r)<\infty$, and write
$g(r)=\Omega(g(r))$ as $r\downarrow 0$, if $\liminf_{r\downarrow
0}f(r)/g(r)>0$. We write $f(r)=o(g(r))$ as $r\downarrow 0$ if
$\lim_{r\downarrow 0}f(r)/g(r)=0$, and $f(r)\sim g(r)$ as $r\downarrow 0$ if
this limit is 1.

From time to time we shall use the following geometrical lemma.

######  Lemma 3.1 (Geometrical lemma).

Suppose $\partial A\in C^{2}$, and $A=\overline{A^{o}}$. Given
$\varepsilon>0$, there exists $r_{0}=r_{0}(d,A,\varepsilon)>0$ such that

$\displaystyle|B(x,r)\cap
A|\geq((\omega_{d}/2)-\varepsilon)r^{d},~{}~{}~{}~{}\forall x\in
A,r\in(0,r_{0}).$ (3.1)

###### Proof.

See [4, Lemma 3.2(a)] ∎

We shall repeatedly use the following result from [10], which is based on
results in Janson [5] or Hall [3]. Recall that $c_{d}$ and $Z_{A}(n,Y)$ were
defined at (2.2) and just before Theorem 2.1 respectively.

######  Lemma 3.2 (Limiting probability of coverage by an unrestricted SPBM).

Let $d\in\mathbb{N}$. Let $\alpha:=\omega_{d}\mathbb{E}\,[Y^{d}]$. Let
$\beta\in\mathbb{R}$. Suppose $\delta(\lambda)\in(0,\infty)$ is defined for
all $\lambda>0$, and satisfies

$\displaystyle\lim_{\lambda\to\infty}\left(\alpha\delta(\lambda)^{d}\lambda-\log\lambda-(d+k-2)\log\log\lambda\right)=\beta.$
(3.2)

Let $B\subset\mathbb{R}^{d}$ be compact and Riemann measurable, and for each
$\lambda>0$ let $B_{\lambda}\subset B$ be Riemann measurable with the
properties that $B_{\lambda}\subset B_{\lambda^{\prime}}$ whenever
$\lambda\leq\lambda^{\prime}$, and that $\cup_{\lambda>0}B_{\lambda}\supset
B^{o}$. Then

$\displaystyle\lim_{\lambda\to\infty}\mathbb{P}[B_{\lambda}\subset
Z_{\mathbb{R}^{d}}(\lambda,\delta(\lambda)Y)=\exp\left(-\left(\frac{c_{d}(\mathbb{E}\,[Y^{d-1}])^{d}}{(k-1)!(\mathbb{E}\,[Y^{d}])^{d-1}}\right)|B|e^{-\beta}\right).$
(3.3)

###### Proof.

See [10, Lemma 7.2]. There, it was assumed that $Y$ is bounded, but the proof
carries over if this condition is relaxed to the $(d+\varepsilon)$-th moment
condition used here. ∎

###  3.2 Coverage in the boundary by a SPBM in a half-space

In this subsection, we fix $d,k\in\mathbb{N}$ with $d\geq 2$. Then we assume
$(r_{n})_{n>0}$ satisfies (2.5) for some $\beta\in\mathbb{R}$. Then as
$n\to\infty$,

$\displaystyle r_{n}=\left(\frac{(2-2/d)\log n+2(d+k-3+1/d)\log\log
n+\beta+o(1)}{n\omega_{d}\mathbb{E}\,[Y^{d}]}\right)^{1/d},$ (3.4)

and

$\displaystyle\exp(-\omega_{d}nr_{n}^{d}\mathbb{E}\,[Y^{d}])\sim
n^{-(2-2/d)}(\log n)^{-2(d+k-3+1/d)}e^{-\beta}.$ (3.5)

We shall use the following notation throughout the sequel. Fix
$d\in\mathbb{N}$. Given $n>0$ let ${\cal U}_{n}$ be a Poisson point process in
$\mathbb{R}^{d}\times\mathbb{R}_{+}$ (viewed as a random set of points in
$\mathbb{R}^{d+1}$) with intensity measure $n{\rm Leb}_{d}\otimes\mu_{Y}$,
where ${\rm Leb}_{d}$ denotes $d$-dimensional Lebesgue measure and $\mu_{Y}$
denotes the distribution of $Y$. Given Borel $D\subset\mathbb{R}^{d}$, and
given any point process $\mathcal{X}$ in $\mathbb{R}^{d}\times\mathbb{R}_{+}$,
we define

$\displaystyle{\cal U}_{n,D}$ $\displaystyle:={\cal
U}_{n}\cap(D\times\mathbb{R}_{+});$ (3.6) $\displaystyle Z_{n}(\mathcal{X})$
$\displaystyle:=\\{y\in\mathbb{R}^{d}:\\#\\{(x,s)\in\mathcal{X}:y\in
B(x,r_{n}s)\\}\geq k\\};$ (3.7) $\displaystyle Z_{n}^{o}(\mathcal{X})$
$\displaystyle:=(Z_{n}(\mathcal{X}))^{o}.$

Thus, ${\cal U}_{n,D}$ is a Poisson process in $D\times\mathbb{R}_{+}$ with
intensity measure $n{\rm Leb}_{d}\otimes\mu_{Y}$ (strictly speaking, with
intensity measure $n{\rm Leb}_{d}|_{D}\otimes\mu_{Y}$). Equivalently, it can
be viewed as a homogeneous marked Poisson process in $D$ of intensity $n$ with
marks having the distribution of $Y$ (see e.g. [6, Theorem 5.6].)

The set $Z_{n}({\cal U}_{n,D})$ is the region that is covered at least $k$
times (for short: the $k$-covered region) for the restricted SPBM in $D$ with
intensity $n$ and with radii having the distribution of $r_{n}Y$. It is the
same as $Z_{D,k}(n,r_{n}Y)$ in earlier notation; we now consider $k$ to be
fixed and suppress it from the notation. The set $Z_{n}^{o}({\cal U}_{n,D})$
is the region covered at least $k$ times by a union of open balls.

Recall that $\mathbb{H}:=\mathbb{R}^{d-1}\times[0,\infty)$. The main result of
this section is Lemma 3.5 below, which determines the limiting probability of
covering a $(d-1)$-dimensional region of the form $\Omega\times\\{0\\}$ in the
hyperplane $\partial\mathbb{H}:=\mathbb{R}^{d-1}\times\\{0\\}$ by $Z_{n}({\cal
U}_{n,\mathbb{H}})$, or of covering the $a_{n}r_{n}$-neighbourhood of
$\Omega\times\\{0\\}$ in $\mathbb{H}$, for $a_{n}$ not growing too fast with
$n$. It is crucial for dealing with boundary regions in the proof of Theorem
2.10.

Before getting to that, we present another idea which will be important later.
Let ${\zeta}>0$ to be chosen later (think of ${\zeta}$ as a small constant).
Given Borel $D\subset\mathbb{R}^{d}$, we consider two separate, independent
Poisson processes ${\cal U}^{\prime}_{n,D}$ and ${\cal U}^{\prime}_{n,D}$,
defined by

$\displaystyle{\cal U}^{\prime}_{n,D}={\cal
U}_{n}\cap(D\times[0,n^{\zeta}]);~{}~{}~{}~{}~{}~{}~{}~{}~{}{\cal
U}^{\prime\prime}_{n,D}={\cal U}_{n}\cap(D\times(n^{\zeta},\infty)).$ (3.8)

For various sets $D$ and ultimately for $D=A$, we shall be interested in
coverage of certain regions by the set $Z_{n}({\cal U}_{n,D})$; the next lemma
says that the limiting probability of coverage is unaffected if we take
$Z_{n}({\cal U}^{\prime}_{n,D})$ instead of $Z_{n}({\cal U}_{n,D})$; this is
useful because the random sets $Z_{n}({\cal U}^{\prime}_{n,D})$ have better
spatial independence properties, considered as a function of $D$.

######  Lemma 3.3 (Asymptotic equivalence of coverage events using ${\cal
U}_{n,D}$ or ${\cal U}^{\prime}_{n,D}$).

Let $n_{0}\in(0,\infty)$. Suppose that Borel $D_{n}\subset\mathbb{R}^{d}$ and
$E_{n}\subset\mathbb{R}^{d}$ are defined for each $n\geq n_{0}$. Then as
$n\to\infty$,

$\displaystyle\mathbb{P}[\\{E_{n}\subset Z_{n}({\cal
U}_{n,D_{n}})\\}\setminus\\{E_{n}\subset Z_{n}({\cal
U}^{\prime}_{n,D_{n}})\\}]\to 0.$ (3.9)

###### Proof.

Recall that we are assuming $\mathbb{E}\,[Y^{\gamma}]<\infty$ for some
$\gamma>d$. Observe that for any fixed point $x\in\mathbb{R}^{d}$, if $N_{n}$
is the number of points $(y,t)\in{\cal U}^{\prime\prime}_{n,\mathbb{R}^{d}}$
such that $\|y-x\|\leq r_{n}t$, then by the Hölder and Markov inequalities,

$\displaystyle\mathbb{P}[N_{n}\geq 1]\leq\mathbb{E}\,[N_{n}]$
$\displaystyle=n\int_{(n^{\zeta},\infty)}|B(x,r_{n}t)|\mu_{Y}(dt)$
$\displaystyle=n\omega_{d}r_{n}^{d}\mathbb{E}\,[Y^{d}{\bf
1}_{\\{Y>n^{\zeta}\\}}]$ $\displaystyle\leq
n\omega_{d}r_{n}^{d}(\mathbb{E}\,[Y^{\gamma}])^{d/\gamma}(\mathbb{P}[Y>n^{\zeta}])^{(\gamma-d)/\gamma}$
$\displaystyle\leq\omega_{d}(\mathbb{E}\,[Y^{\gamma}])^{d/\gamma}nr_{n}^{d}(\mathbb{E}\,[Y^{\gamma}]/n^{{\zeta}\gamma})^{(\gamma-d)/\gamma},$

which tends to zero since $nr_{n}^{d}=O(\log n)$ by (3.4). Therefore

$\displaystyle\sup_{x\in\mathbb{R}^{d}}\mathbb{P}[x\in Z_{n}({\cal
U}^{\prime\prime}_{n,\mathbb{R}^{d}})]\to 0~{}~{}{\rm as}~{}n\to\infty.$
(3.10)

Let $n\geq n_{0}$. Since $E_{n}\subset\mathbb{R}^{d}$, $E_{n}$ is separable;
see e.g. [11, page 20]. Let $\\{x_{n,i}\\}_{i\in\mathbb{N}}$ be an enumeration
of a countable dense set in $E_{n}$. Define the
($\mathbb{N}\cup+\infty$)-valued random variable $J_{n}$ to be the first $i$
such that $x_{n,i}\notin Z_{n}({\cal U}^{\prime}_{n,D_{n}})$, or
$J_{n}:=+\infty$ if there is no such $i$. Since $E_{n}\setminus Z_{n}({\cal
U}^{\prime}_{n,D_{n}})$ is open in $E_{n}$, if $E_{n}\setminus Z_{n}({\cal
U}^{\prime}_{n,D_{n}})\neq\varnothing$ then $J_{n}<\infty$. Therefore

$\displaystyle\mathbb{P}[E_{n}\subset Z_{n}({\cal
U}_{n,D_{n}})|\\{E_{n}\subset Z_{n}({\cal
U}^{\prime}_{n,D_{n}})\\}^{c}]\leq\mathbb{P}[x_{n,J_{n}}\in Z_{n}({\cal
U}^{\prime\prime}_{n,D_{n}})],$

which tends to zero by (3.10) and the independence of ${\cal
U}^{\prime}_{n,D_{n}}$ and ${\cal U}^{\prime\prime}_{n,D_{n}}$, and (3.9)
follows. ∎

The following terminology and notation will be used in the proof of Lemma 3.5,
and again later on. Given $x\in\mathbb{R}^{d}$, we let
$\pi_{1}(x),\ldots,\pi_{d}$ denote the co-ordinates of $x$, and refer to
$\pi_{d}(x)$ as the height of $x$. Given ${\bf
x}_{1}=(x_{1},s_{1}),\ldots,{\bf
x}_{d}=(x_{d},s_{d})\in\mathbb{R}^{d}\times\mathbb{R}_{+}$, if
$\cap_{i=1}^{d}\partial B(x_{i},s_{i})$ consists of exactly two points, we
refer to these as $p({\bf x}_{1},\ldots,{\bf x}_{d})$ and $q({\bf
x}_{1},\ldots,{\bf x}_{d})$ with $p({\bf x}_{1},\ldots,{\bf x}_{d})$ at a
smaller height than $q({\bf x}_{1},\ldots,{\bf x}_{d})$ (or if they are at the
same height, take $p({\bf x}_{1},\ldots,{\bf x}_{d})<q({\bf x}_{1},\ldots,{\bf
x}_{d})$ in the lexicographic ordering). Define the indicator function

$\displaystyle h({\bf x}_{1},\ldots,{\bf x}_{d}):=$ $\displaystyle{\bf
1}\\{\pi_{d}(x_{1})\leq\min(\pi_{d}(x_{2}),\ldots,\pi_{d}(x_{d}))\\}$
$\displaystyle\times{\bf 1}\\{\\#(\cap_{i=1}^{d}\partial
B(x_{i},r_{i}))=2\\}{\bf 1}\\{\pi_{d}(x_{1})<\pi_{d}(q({\bf x}_{1},\ldots,{\bf
x}_{d}))\\}.$ (3.11)

Given also $n>0$ we also define

$\displaystyle h_{n}({\bf x}_{1},\ldots,{\bf
x}_{d}):=h((x_{1},r_{n}s_{1}),\ldots,(x_{d},r_{n}s_{d}));$ (3.12)
$\displaystyle p_{n}({\bf x}_{1},\ldots,{\bf
x}_{d}):=p((x_{1},r_{n}s_{1}),\ldots,(x_{d},r_{n}s_{d}));$ (3.13)
$\displaystyle q_{n}({\bf x}_{1},\ldots,{\bf
x}_{d}):=q((x_{1},r_{n}s_{1}),\ldots,(x_{d},r_{n}s_{d})).$ (3.14)

######  Lemma 3.4 (Integrability of $h$).

There exists a constant $c$ depending only on $d$ such that for all
$x_{1}\in\mathbb{R}^{d}$ and all $s_{1},\ldots,s_{d}\in\mathbb{R}_{+}$,

$\displaystyle\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}h((x_{1},s_{1}),\ldots,(x_{d},s_{d}))dx_{2}\cdots
dx_{d}\leq c\prod_{i=1}^{d}(1\vee s_{i}^{d-1}).$ (3.15)

###### Proof.

Denote the left hand side of (3.15) by $I(x_{1},s_{1},\ldots,s_{d})$. Divide
$\mathbb{R}^{d}$ into rectilinear unit hypercubes $Q_{z},z\in\mathbb{Z}^{d}$,
with $Q_{z}$ centred at $z$ for each $z$. Then

$\displaystyle I(x_{1},s_{1},\ldots,s_{d})$
$\displaystyle\leq\sum_{z\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}{\bf
1}\\{\cap_{i=1}^{d}\partial B(x_{i},s_{i})\cap
Q_{z}\neq\varnothing\\}dx_{2}\cdots dx_{d}$
$\displaystyle\leq\sum_{z\in\mathbb{Z}^{d}}\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}\prod_{i=1}^{d}{\bf
1}\\{\partial B(x_{i},s_{i})\cap Q_{z}\neq\varnothing\\}dx_{2}\cdots dx_{d}$
$\displaystyle=\sum_{z\in\mathbb{Z}^{d}}{\bf 1}\\{\partial B(x_{1},s_{1})\cap
Q_{z}\neq\varnothing\\}\times\prod_{i=2}^{d}\int_{\mathbb{R}^{d}}{\bf
1}\\{\partial B(x_{i},s_{i})\cap Q_{z}\neq\varnothing\\}dx_{i}.$

Given $s\geq 0$, the value of the integral $\int_{\mathbb{R}^{d}}{\bf
1}\\{\partial B(x,s)\cap Q_{z}\neq\varnothing\\}dx$ does not depend on $z$,
and is bounded by $c^{\prime}(1\vee s^{d-1})$ for some constant $c^{\prime}$,
independent of $s$. Moreover the sum $\sum_{z\in\mathbb{Z}^{d}}{\bf
1}\\{\partial B(x,s)\cap Q_{z}\neq\varnothing\\}$ is bounded by
$c^{\prime\prime}(1\vee s^{d-1})$ for some further constant
$c^{\prime\prime}$, independent of $s$ and $x$. Applying these two
observations gives us (3.15). ∎

We are now ready to present the main result of this section. In it, $|\Omega|$
denotes the $(d-1)$-dimensional Lebesgue measure of $\Omega$. Recall the
definition of $c_{d,1}$ at (2.3).

######  Lemma 3.5 (Coverage near the boundary of a half-space).

Let $\Omega\subset\mathbb{R}^{d-1}$ and (for each $n>0$)
$\Omega_{n}\subset\Omega$ be closed and Riemann measurable. Assume that (2.5)
holds for some $\beta\in\mathbb{R}$ or $\beta=+\infty$, and also that
$\limsup_{n\to\infty}(nr_{n}^{d}/(\log n))<\infty$. Then

$\displaystyle\lim_{n\to\infty}(\mathbb{P}[\Omega\times\\{0\\}\subset
Z_{n}({\cal
U}_{n,\mathbb{H}})])=\exp\left(-c_{d,k}\Big{(}\frac{(\mathbb{E}\,[Y^{d-1}])^{d-1}}{(\mathbb{E}\,[Y^{d}])^{d-2+1/d}}\Big{)}|\Omega|e^{-\beta/2}\right).$
(3.16)

Also, given $a_{n}\in(0,\infty)$ for each $n>0$, such that
$a_{n}=O(n^{(1/d)-\varepsilon})$ as $n\to\infty$ for some $\varepsilon>0$, we
have

$\displaystyle\lim_{n\to\infty}(\mathbb{P}[\\{(\Omega_{n}\times\\{0\\})\cup($
$\displaystyle(\partial\Omega_{n})\times[0,a_{n}r_{n}])\subset Z^{o}_{n}({\cal
U}^{\prime}_{n,\mathbb{H}})\\}$
$\displaystyle\setminus\\{(\Omega_{n}\times[0,a_{n}r_{n}])\subset Z_{n}({\cal
U}^{\prime}_{n,\mathbb{H}})\\}])=0.$ (3.17)

######  Remark 3.6.

Usually we shall use Lemma 3.5 in the case where $\beta<\infty$. In this case
the extra condition $\limsup_{n\to\infty}(nr_{n}^{d}/(\log n))<\infty$ is
automatic. When $\beta=\infty$, in (3.16) we use the convention
$e^{-\infty}:=0$.

###### Proof of Lemma 3.5.

Assume for now that $\beta<\infty$. Considering the slices of balls induced by
the points of ${\cal U}_{n,\mathbb{H}}$ that intersect the hyperplane
$\mathbb{R}^{d-1}\times\\{0\\}$, we have a $(d-1)$-dimensional SPBM, where we
claim the parameters (in the notation of Lemma 3.2) are

$\displaystyle\delta=r_{n},~{}~{}~{}\lambda=nr_{n}\mathbb{E}\,[Y],~{}~{}~{}~{}\alpha=\omega_{d}\mathbb{E}\,[Y^{d}]/(2\mathbb{E}\,[Y]).$
(3.18)

We justify these claims as follows. The intensity of balls intersecting the
hyperplane $\partial\mathbb{H}_{d}$ per unit unit ‘area’ (i.e. per unit
$(d-1)$-dimensional Lebesgue measure), is equal to
$n\int_{0}^{\infty}\mathbb{P}[r_{n}Y\geq t]dt=nr_{n}\mathbb{E}\,[Y]$. Also if
$W$ denotes the radius of a slice, divided by $r_{n}$, the distribution of $W$
is that of $\tilde{Y}(1-U^{2})^{1/2}$, where $\tilde{Y}$ has the size-biased
distribution of $Y$ (i.e., $\mathbb{P}[\tilde{Y}\in dt]=t\mathbb{P}[Y\in
dt]/\mathbb{E}\,[Y]$), and $U$ is uniformly distributed on $[0,1]$,
independent of $\tilde{Y}$. Therefore

$\displaystyle\mathbb{E}\,[W^{d-1}]$
$\displaystyle=\mathbb{E}\,[\tilde{Y}^{d-1}]\int_{0}^{1}(1-u^{2})^{(d-1)/2}du$
$\displaystyle=(\mathbb{E}\,[Y^{d}]/\mathbb{E}\,[Y])\omega_{d}/(2\omega_{d-1}),$
(3.19)

and the asserted value of $\alpha$ follows.

By (3.18), followed by (2.5),

$\displaystyle\alpha\delta^{d-1}\lambda$
$\displaystyle=\Big{(}\frac{\omega_{d}\mathbb{E}\,[Y^{d}]}{2\mathbb{E}\,[Y]}\Big{)}r_{n}^{d-1}nr_{n}\mathbb{E}\,[Y]=(\omega_{d}/2)nr_{n}^{d}\mathbb{E}\,[Y^{d}]$
$\displaystyle=(1-1/d)\log n+(d+k-3+1/d)\log\log n+(\beta/2)+o(1).$

Also by (3.18) and (3.4),

$\displaystyle\log\lambda$ $\displaystyle=(1-1/d)\log
n+\Big{(}\frac{1}{d}\Big{)}\log\log
n+\log\Big{(}\Big{(}\frac{2-2/d}{\omega_{d}\mathbb{E}\,[Y^{d}]}\Big{)}^{1/d}\mathbb{E}\,[Y]\Big{)}+o(1).$

Also $\log\log\lambda=\log\log n+\log(1-1/d)+o(1).$ Hence

$\displaystyle\alpha\delta^{d-1}\lambda-\log\lambda-(d+k-3)\log\log\lambda=(\beta/2)-\log(c^{\prime\prime}_{d,k,Y})+o(1),$

where we take

$\displaystyle c^{\prime\prime}_{d,k,Y}$
$\displaystyle:=\left(\frac{2-2/d}{\omega_{d}\mathbb{E}\,[Y^{d}]}\right)^{1/d}\mathbb{E}\,[Y](1-1/d)^{d-2}=(1-1/d)^{d+k-3+1/d}\left(\frac{2(\mathbb{E}\,[Y])^{d}}{\omega_{d}\mathbb{E}\,[Y^{d}]}\right)^{1/d}.$

By (3.19), and a similar calculation for $\mathbb{E}\,[W^{d-2}]$,

$\displaystyle\frac{(\mathbb{E}\,[W^{d-2}])^{d-1}}{(\mathbb{E}\,[W^{d-1}])^{d-2}}=\left(\frac{\omega_{d-1}\mathbb{E}\,[Y^{d-1}]}{2\omega_{d-2}\mathbb{E}\,[Y]}\right)^{d-1}\left(\frac{\omega_{d}\mathbb{E}\,[Y^{d}]}{2\omega_{d-1}\mathbb{E}\,[Y]}\right)^{2-d}=\frac{\omega_{d-1}^{2d-3}(\mathbb{E}\,[Y^{d-1}])^{d-1}}{2\omega_{d-2}^{d-1}\omega_{d}^{d-2}(\mathbb{E}\,[Y^{d}])^{d-2}\mathbb{E}\,[Y]}.$

Thus by Lemma 3.2 we obtain that

$\displaystyle\lim_{n\to\infty}(\mathbb{P}[\Omega\times\\{0\\}\subset
Z_{n}({\cal
U}_{n,\mathbb{H}})])=\exp\Big{(}-\Big{(}\frac{c_{d-1}\omega_{d-1}^{2d-3}(\mathbb{E}\,[Y^{d-1}])^{d-1}}{(k-1)!2\omega_{d-2}^{d-1}\omega_{d}^{d-2}\mathbb{E}\,[Y](\mathbb{E}\,[Y^{d}])^{d-2}}\Big{)}$
$\displaystyle\times c^{\prime\prime}_{d,k,Y}|\Omega|e^{-\beta/2}\Big{)},$

and hence (3.16).

Having now verified (3.16) in the case where $\beta<\infty$, we can then
easily deduce (3.16) in the other case too.

It remains to prove (3.17); we now consider general $\beta\in(0,\infty]$. For
the duration of this proof only, let $F_{n}$ be the event displayed in (3.17)
but with ${\cal U}^{\prime}_{n,\mathbb{H}}$ changed to ${\cal
U}_{n,\mathbb{H}}$ both times, i.e.

$\displaystyle
F_{n}:=\\{(\Omega_{n}\times\\{0\\})\cup((\partial\Omega_{n})\times[0,a_{n}r_{n}])\subset
Z^{o}_{n}({\cal
U}_{n,\mathbb{H}})\\}\setminus\\{(\Omega_{n}\times[0,a_{n}r_{n}])\subset
Z_{n}({\cal U}_{n,\mathbb{H}})\\}.$

By Lemma 3.3, to prove (3.17) it suffices to prove that $\mathbb{P}[F_{n}]\to
0$ as $n\to\infty$.

Let $E_{n}$ be the (exceptional) event that there exist $d$ distinct marked
points $(x_{1},s_{1}),\ldots,$ $(x_{d},s_{d})$ of ${\cal U}_{n,\mathbb{H}}$
such that $\cap_{i=1}^{d}\partial B(x_{i},r_{n}s_{i})$ has non-empty
intersection with the hyperplane $\partial\mathbb{H}$. Then
$\mathbb{P}[E_{n}]=0$.

Suppose that $F_{n}\setminus E_{n}$ occurs. Let $w$ be a location of minimal
height (i.e., $d$-coordinate) in the closure of
$(\Omega_{n}\times[0,a_{n}r_{n}])\setminus Z_{n}({\cal U}_{n,\mathbb{H}})$.
Since we assume $(\partial\Omega_{n})\times[0,a_{n}r_{n}]\subset
Z^{o}_{n}({\cal U}_{n,\mathbb{H}})$ occurs, $w$ must lie in
$\Omega_{n}^{o}\times(0,a_{n}r_{n}]$. Also we claim that $w$ must be a
‘corner’ given by the meeting point of the boundaries of exactly $d$ balls
$B(x_{1},r_{n}s_{1}),\ldots,B(x_{d},r_{n}s_{d})$ with
$\\{(x_{i},s_{i})\\}_{i=1}^{d}\subset{\cal U}_{n,\mathbb{H}}$, and with
$x_{1}$ the lowest of the $d$ points $x_{1},\ldots,x_{d}$, and with
$\\#(\cap_{i=1}^{d}\partial B(x_{i},r_{n}s_{i}))=2$, and $w\notin Z_{n}({\cal
U}_{n,\mathbb{H}}\setminus\\{(x_{1},s_{1}),\ldots,(x_{d},s_{d})\\})$.

Indeed, if $w$ is not at the boundary of any such ball, then for some
$\delta>0$ we have $B(w,\delta)\cap Z_{n}({\cal
U}_{n,\mathbb{H}})=\varnothing$, and then we could find a location in
$(\Omega_{n}\times[0,ar_{n}])\setminus Z_{n}({\cal U}_{n,\mathbb{H}})$ lower
than $w$, a contradiction. Next, suppose instead that $w$ lies at the boundary
of fewer than $d$ such balls. Then denoting by $L$ the intersection of the
supporting hyperplanes at $w$ of each of these balls, we have that $L$ is an
affine subspace of $\mathbb{R}^{d}$, of dimension at least 1. Take $\delta>0$
small enough so that $B(w,\delta)$ does not intersect any of the boundaries of
balls centred at points of ${\cal U}_{n,\mathbb{H}}$ (viewed here as a marked
Poisson point process in $\mathbb{R}^{d}$) and with radius given by their
marks times $r_{n}$, other than those which meet at $w$. Taking $w^{\prime}\in
L\cap B(w,\delta)\setminus\\{w\\}$ such that $w^{\prime}$ is at least as low
as $w$, we have that $w^{\prime}$ lies in the interior of $Z_{n}({\cal
U}_{n,\mathbb{H}})^{c}$. Hence for some $\delta^{\prime}>0$,
$B(w^{\prime},\delta^{\prime})\cap Z_{n}({\cal U}_{n,\mathbb{H}})=\varnothing$
and we can find a location in $B(w^{\prime},\delta^{\prime})$ that is lower
than $w$, yielding a contradiction for this case too. Finally, with
probability 1 there is no set of more than $d$ points of ${\cal
U}_{n,\mathbb{H}}$ such that the boundaries of the associated balls have non-
empty intersection, so $w$ is not at the boundary of more than $d$ such balls.
Thus we have justified the claim.

Moreover $w$ must be the point $q_{n}({\bf x}_{1},\ldots,{\bf x}_{d})$ rather
than $p_{n}({\bf x}_{1},\ldots,{\bf x}_{d})$, where for $1\leq i\leq d$ we
write ${\bf x}_{i}$ for $(x_{i},r_{i})$, because otherwise by extending the
line segment from $q_{n}({\bf x}_{1},\ldots,{\bf x}_{d})$ to $p_{n}({\bf
x}_{1},\ldots,{\bf x}_{d})$ slightly beyond $p_{n}({\bf x}_{1},\ldots,{\bf
x}_{d})$ we could find a point in $(\Omega_{n}\times[0,ar_{n}])\setminus
Z_{n}({\cal U}_{n,\mathbb{H}})$ lower than $w$, contradicting the statement
that $w$ is a location of minimal height in the closure of
$(\Omega_{n}\times[0,ar_{n}])\setminus Z_{n}({\cal U}_{n,\mathbb{H}})$.
Moreover, $w$ must be strictly higher than $x_{1}$, since if
$\pi_{d}(w)\leq\min(\pi_{d}(x_{1}),\ldots,\pi_{d}(x_{d}))$, then locations
just below $w$ would lie in $(\Omega_{n}\times[0,ar_{n}])\setminus Z_{n}({\cal
U}_{n,\mathbb{H}})$, contradicting the statement that $w$ is a point of
minimal height in the closure of $(\Omega_{n}\times[0,ar_{n}])\setminus
Z_{n}({\cal U}_{n,\mathbb{H}})$. Hence, $h_{n}({\bf x}_{1},\ldots,{\bf
x}_{d})=1$, where $h_{n}(\cdot)$ was defined at (3.12).

Thus if $F_{n}\setminus E_{n}$ occurs, then there exists at least one
$d$-tuple of marked points ${\bf x}_{1}=(x_{1},s_{1}),\ldots,{\bf
x}_{d}=(x_{d},s_{d})\in{\cal U}_{n,\mathbb{H}}$, such that $h_{n}({\bf
x}_{1},\ldots,{\bf x}_{d})=1$, and moreover $q_{n}({\bf x}_{1},\ldots,{\bf
x}_{d})\in(\Omega_{n}\times(0,a_{n}r_{n}])\setminus Z_{n}({\cal
U}_{\rho,\mathbb{H}}\setminus\\{{\bf x}_{1},\ldots,{\bf x}_{d}\\})$. Let
$M_{n}(a_{n})$ denote the number of such $d$-tuples.

Assume for now that for some constant $a\in(0,\infty)$ we have $a_{n}=a$ for
all $n$. We shall prove (3.17) first in this case.

Choose $\delta>0$ such that $\mathbb{P}[Y>\delta]>\delta$. Then choose
$c^{\prime}>0$ so that

$\omega_{d-1}\delta^{d}2^{1-d}((\delta/2)\wedge u)\geq
c^{\prime}u,~{}~{}~{}\forall u\in[0,a].$

For any $y\in\mathbb{H}$, let $N_{n}(y)$ denote the number of balls making up
$Z_{n}({\cal U}_{n,\mathbb{H}})$ which cover $y$, i.e. the number of
$(x,s)\in{\cal U}_{n,\mathbb{H}}$ such that $\|x-y\|\leq r_{n}s$. Also, let
$\mathbb{H}_{y}:=\\{z\in\mathbb{H}:\pi_{d}(z)>\pi_{d}(y)\\}$. Then for all $n$
and all $y\in\mathbb{H}$ with $0\leq\pi_{d}(y)\leq ar_{n}$, since for all
$s>0$, the half-ball $B(y,r_{n}s)\cap\mathbb{H}_{y}$, and also the cylinder
with radius $sr_{n}/2$, upper face centred on $y$ and height
$(sr_{n}/2)\wedge\pi_{d}(y)$, are disjoint and contained in
$B(y,r_{n}s)\cap\mathbb{H}$,

$\displaystyle\mathbb{E}\,[N_{n}(y)]$
$\displaystyle=n\int_{\mathbb{R}_{+}}|B(y,r_{n}s)\cap\mathbb{H}|\mu_{Y}(ds)$
$\displaystyle\geq
n\int_{\mathbb{R}_{+}}(\omega_{d}r_{n}^{d}s^{d}/2)\mu_{Y}(ds)+n\int_{(\delta,\infty)}\omega_{d-1}(sr_{n}/2)^{d-1}((sr_{n}/2)\wedge\pi_{d}(y))\mu_{Y}(ds)$
$\displaystyle\geq(n\omega_{d}/2)r_{n}^{d}\mathbb{E}\,[Y^{d}]+n\delta\omega_{d-1}(\delta
r_{n}/2)^{d-1}r_{n}((\delta/2)\wedge(\pi_{d}(y)/r_{n}))$
$\displaystyle\geq(n\omega_{d}/2)r_{n}^{d}\mathbb{E}\,[Y^{d}]+c^{\prime}nr_{n}^{d-1}\pi_{d}(y).$
(3.20)

By the Mecke formula (see e.g. [6]), taking $Y_{1},\ldots,Y_{d}$ to be
independent random variables with the distribution of $Y$, there is a constant
$c$ such that

$\displaystyle\mathbb{E}\,[M_{n}(a)]\leq$ $\displaystyle
cn^{d}\int_{\mathbb{H}}\cdots\int_{\mathbb{H}}\mathbb{E}\,[h_{n}((x_{1},Y_{1}),\ldots,(x_{d},Y_{d}))$
$\displaystyle\times{\bf
1}\\{q_{n}((x_{1},Y_{1}),\ldots,(x_{d},Y_{d}))\in\Omega_{n}\times[0,ar_{n}]\\}(k(n\omega_{d}r_{n}^{d})^{k-1})$
$\displaystyle\times
e^{-n(\omega_{d}/2)\mathbb{E}\,[Y^{d}]r_{n}^{d}-c^{\prime}nr_{n}^{d-1}\pi_{d}(q_{n}((x_{1},Y_{1}),\ldots,(x_{d},Y_{d})))}]dx_{d}\cdots
dx_{1}.~{}~{}~{}$ (3.21)

Now we change variables to $y_{i}=r_{n}^{-1}(x_{i}-x_{1})$ for $2\leq i\leq
d$, noting that

$\displaystyle\pi_{d}(q_{n}((x_{1},Y_{1}),\ldots,(x_{d},Y_{d})))=\pi_{d}(x_{1})+\pi_{d}(q_{n}((o,Y_{1}),(x_{2}-x_{1},Y_{2}),\ldots,(x_{d}-x_{1},Y_{d}))).$

With these changes of variable, we get a factor of $r_{n}^{d(d-1)}$ on
changing from $(x_{2},\ldots,x_{d})$ to $(y_{2},\ldots,y_{d})$. Hence by
(3.5), there is a constant $c^{\prime\prime}$ such that the expression on the
right hand side of (3.21) is at most

$\displaystyle c^{\prime\prime}n^{d+k-1}r_{n}^{d(d+k-2)}n^{-(1-1/d)}(\log
n)^{-(d+k-3+1/d)}\int_{0}^{ar_{n}}e^{-c^{\prime}nr_{n}^{d-1}u}du$
$\displaystyle\times\int_{\mathbb{H}}\cdots\int_{\mathbb{H}}\mathbb{E}\,[h_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d},Y_{d}))e^{-c^{\prime}nr_{n}^{d-1}\pi_{d}(q_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d},Y_{d})))}]$
$\displaystyle dy_{d}\ldots
dy_{2}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (3.22)

In the last expression the first line is bounded by a constant times the
expression

$\displaystyle n^{d+k-3+1/d}(r_{n}^{d})^{d+k-2-(d-1)/d}(\log
n)^{-(d+k-3+1/d)}=(nr_{n}^{d}/(\log n))^{d+k-3+1/d},$

which is bounded because we assume $nr_{n}^{d}=O(\log n)$. Using Fubini’s
theorem and the definition of $h_{n}(\cdot)$, the expression in the second and
third lines of (3.22) can be rewritten as

$\displaystyle\mathbb{E}\,\Big{[}\int_{\mathbb{H}}\cdots\int_{\mathbb{H}}h((o,Y_{1}),(y_{2},Y_{2}),\ldots,(y_{d},Y_{d}))e^{-c^{\prime}nr_{n}^{d}\pi_{d}(q((o,Y_{1}),(y_{2},Y_{2}),\ldots,(y_{d},Y_{d})))}dy_{d}\ldots
dy_{2}\Big{]}$
$\displaystyle=\int_{\mathbb{R}_{+}^{d}}\int_{\mathbb{H}^{d-1}}h((o,u_{1}),(y_{2},u_{2}),\ldots,(y_{d},u_{d}))e^{-c^{\prime}nr_{n}^{d}\pi_{d}(q((o,u_{1}),(y_{2},u_{2}),\ldots,(y_{d},u_{d})))}$
$\displaystyle d(y_{2},\ldots,y_{d})\mu_{Y}^{d}(d(u_{1},\ldots,u_{d})).$

This tends to zero by dominated convergence because $nr_{n}^{d}\to\infty$ by
(2.5) and because the indicator function $({\bf u},{\bf y})\mapsto
h((o,u_{1}),(y_{2},u_{2}),\ldots,(y_{d},u_{d}))$ is integrable and is zero
when $\pi_{d}(q((o,u_{1}),(y_{2},u_{2}),\ldots,(y_{d},u_{d})))\leq 0$; the
integrability follows from Lemma 3.4 and the fact that
$\mathbb{E}\,[Y^{d-1}]<\infty$ since we assume
$\mathbb{E}\,[Y^{\gamma}]<\infty$ for some $\gamma>d$.

Therefore for any fixed constant $a$ we have

$\displaystyle\lim_{n\to\infty}(\mathbb{E}\,[M_{n}(a)])=0,$ (3.23)

so by Markov’s inequality we obtain that
$\mathbb{P}[F_{n}]\leq\mathbb{P}[M_{n}(a)\geq 1]\to 0$, and hence (3.17), in
the case where $a_{n}\equiv a$.

Now assuming only that $a_{n}=O(n^{(1/d)-\varepsilon})$ for some
$\varepsilon>0$, choose $a>0$ such that

$\int_{(a,\infty)}x^{d}\mu_{Y}(dx)<\varepsilon\mathbb{E}\,[Y^{d}]/4.$

Assume without loss of generality that $a_{n}>a$. For $y\in\mathbb{H}$ with
$ar_{n}\leq\pi_{d}(y)\leq a_{n}r_{n}$, instead of (3.20) we use the estimate

$\displaystyle\mathbb{E}\,[N_{n}(y)]$
$\displaystyle=n\int_{\mathbb{R}_{+}}|B(y,r_{n}s)\cap\mathbb{H}|\mu_{Y}(ds)$
$\displaystyle\geq n\int_{(0,a)}(\omega_{d}r_{n}^{d}s^{d})\mu_{Y}(ds)$
$\displaystyle\geq n\omega_{d}r_{n}^{d}\mathbb{E}\,[Y^{d}](1-\varepsilon/4).$

By this, and the Mecke formula, there is a constant $c$ such that

$\displaystyle\mathbb{E}\,[M_{n}(a_{n})-M_{n}(a)]\leq
cn^{d}(nr_{n}^{d})^{k-1}\int_{\mathbb{H}}\cdots\int_{\mathbb{H}}\mathbb{E}\,[h_{n}((x_{1},Y_{1}),\ldots,(x_{d},Y_{d}))$
$\displaystyle\times{\bf
1}\\{q_{n}((x_{1},Y_{1}),\ldots,(x_{d},Y_{d}))\in\Omega_{n}\times[ar_{n},a_{n}r_{n}]\\}e^{-n\omega_{d}r_{n}^{d}\mathbb{E}\,[Y^{d}](1-\varepsilon/4)}]dx_{d}\cdots
dx_{1}.~{}~{}~{}$ (3.24)

Changing variables as before to $y_{i}=r_{n}^{-1}(x_{i}-x_{1})$ for $2\leq
i\leq d$, and using (3.5), we find there is a constant
$c^{\prime\prime\prime}$ such that the expression in the right hand side of
(3.24) is at most

$\displaystyle
c^{\prime\prime\prime}n^{d+k-1}r_{n}^{d(d+k-2)}n^{-(2-2/d)+\varepsilon/2}\mathbb{E}\,\Big{[}\int_{\mathbb{H}}\cdots\int_{\mathbb{H}}h_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d},Y_{d}))$
$\displaystyle\times{\bf
1}_{\Omega_{n}\times[ar_{n},a_{n}r_{n}]}(x_{1}+q_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d},Y_{d})))dx_{1}dy_{d}\ldots
dy_{2}\Big{]}$ $\displaystyle\leq
c^{\prime\prime\prime}(nr_{n}^{d})^{d+k-2}n^{-1+(2/d)+\varepsilon/2}|\Omega_{n}|(a_{n}-a)r_{n}$
$\displaystyle\times\mathbb{E}\,\Big{[}\int_{\mathbb{H}^{d-1}}h((o,Y_{1}),(y_{2},Y_{2}),\ldots,(y_{d},Y_{d}))d((y_{2},\ldots,y_{d}))\Big{]}.$

Since $\mathbb{E}\,[Y^{d-1}]<\infty$ the expectation in the last line is
finite by Lemma 3.4, and since also $a_{n}=O(n^{(1/d)-\varepsilon})$ we obtain
that

$\mathbb{E}\,[M_{n}(a_{n})-M_{n}(a)]=O((nr_{n}^{d})^{d+k-2+1/d}n^{-1+(2/d)+(\varepsilon/2)-\varepsilon}),$

which tends to zero. Combined with (3.23) this shows that
$\mathbb{E}\,[M_{n}(a_{n})]\to 0$, and therefore by Markov’s inequality we
obtain that $\mathbb{P}[F_{n}]\leq\mathbb{P}[M_{n}(a_{n})\geq 1]\to 0$, and
hence (3.17). ∎

###  3.3 Proof of Theorem 2.1

In this subsection, we set $d=2$ and take $A$ to be polygonal. Denote the
vertices of $A$ by $q_{1},\ldots,q_{\kappa}$, and the angles subtended at
these vertices by $\alpha_{1},\ldots,\alpha_{\kappa}$ respectively.

For the duration of this subsection (and the next) we fix
$\beta\in\mathbb{R}$. Assume we are given real numbers $(r_{n})_{n>0}$
satisfying the case $d=2$ of (2.5), i.e.

$\displaystyle\lim_{n\to\infty}\left(\pi nr_{n}^{2}\mathbb{E}\,[Y^{2}]-\log
n-\log\log n\right)=\beta.$ (3.25)

Fix ${\zeta}>0$. For $n>0$, in this subsection we define the ‘corner regions’
$Q_{n}$ and $Q_{n}^{-}$ by

$Q_{n}:=\cup_{j=1}^{\kappa}B(q_{j},n^{3{\zeta}}r_{n})\cap
A;~{}~{}~{}~{}~{}Q^{-}_{n}:=\cup_{j=1}^{\kappa}B(q_{j},n^{2{\zeta}}r_{n})\cap
A.$

Also we define ${\cal U}_{n,A}$, $Z_{n}({\cal U}_{n,A})$ and $Z^{o}_{n}({\cal
U}_{n,A})$ by (3.6) and (3.7).

######  Lemma 3.7 (Coverage of regions near the corners of $A$).

Provided ${\zeta}$ is taken to be small enough, $\mathbb{P}[Q_{n}\subset
Z_{n}^{o}({\cal U}_{n,A})]\to 1$ as $n\to\infty$.

###### Proof.

Let $\delta>0$ with $\mathbb{P}[Y>3\delta]>\delta$. Then the restricted SPBM
$Z^{o}_{n}({\cal U}_{n,A})$ stochastically dominates a Poisson Boolean model
$Z^{\prime}_{n}$ with closed balls of deterministic radius $2\delta r_{n}$,
centred on the points of a homogeneous Poisson point process in $A$ of
intensity $\delta n$. Let $Z^{\prime\prime}_{n}$ be a Poisson Boolean model
with closed balls of deterministic radius $\delta r_{n}$, centred on the
points of a homogeneous Poisson point process in $A$ of intensity $\delta n$.

There is a constant $K$, independent of $n$, such that for all $n\geq 1$ there
exist points $y_{n,1},\ldots,y_{n,\lfloor Kn^{6{\zeta}}\rfloor}\in A$ such
that $Q_{n}\subset\cup_{j=1}^{\lfloor Kn^{6{\zeta}}\rfloor}B(y_{n,j},\delta
r_{n})$. Also there is a constant $a>0$ such that for all $n\geq 1$ and all
$y\in A$ we have $|B(y,\delta r_{n})\cap A|\geq ar_{n}^{2}$. Then

$\displaystyle\mathbb{P}[Q_{n}\setminus Z_{n}^{o}({\cal
U}_{n,A})\neq\varnothing]$ $\displaystyle\leq\mathbb{P}[Q_{n}\setminus
Z^{\prime}_{n}\neq\varnothing]$
$\displaystyle\leq\mathbb{P}[\cup_{j=1}^{\lfloor
Kn^{6{\zeta}}\rfloor}\\{y_{n,j}\notin Z^{\prime\prime}_{n}\\}]$
$\displaystyle\leq Kn^{6{\zeta}}k(n\pi r_{n}^{2})^{k-1}\exp(-a\delta
nr_{n}^{2}),$ (3.26)

and since $(nr_{n}^{2})/\log n\to 1/(\pi\mathbb{E}\,[Y^{2}])$ by (3.25),
provided we take ${\zeta}<a\delta/(6\pi\mathbb{E}\,[Y^{2}])$ the expression in
(3.26) tends to zero, and the result follows. ∎

Recall from (3.6) the definition of the point processes ${\cal
U}^{\prime}_{n,D}$ for $n>0,D\subset\mathbb{R}^{d}$.

######  Lemma 3.8 (Coverage of regions near edges of $A$).

Assume ${\zeta}<1/d$. Then

$\displaystyle\lim_{n\to\infty}(\mathbb{P}[(\partial A\setminus
Q^{-}_{n})\subset Z_{n}({\cal
U}^{\prime}_{n,A})])=\exp\Big{(}-\Big{(}\frac{c_{2,k}\mathbb{E}\,[Y]}{(\mathbb{E}\,[Y^{2}])^{1/2}}\Big{)}|\partial
A|e^{-\beta/2}\Big{)}.$ (3.27)

Also,

$\displaystyle\lim_{n\to\infty}(\mathbb{P}[\\{A^{(3n^{\zeta}r_{n})}\subset
Z_{n}({\cal U}^{\prime}_{n,A})\\}\cap\\{(\partial A\cup Q_{n})\subset
Z^{o}_{n}({\cal U}^{\prime}_{n,A})\\}\setminus\\{A\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}])=0.$ (3.28)

###### Proof.

Denote the line segments making up $\partial A$ by $I_{1},\ldots,I_{\kappa}$,
and for $n>0$ and $1\leq i\leq\kappa$ set $I_{n,i}:=I_{i}\setminus Q^{-}_{n}$.

Let $i,j,k\in\\{1,\ldots,\kappa\\}$ be such that $i\neq j$ and the edges
$I_{i}$ and $I_{j}$ are both incident to $q_{k}$. Provided $n$ is large
enough, if $x\in I_{n,i}$ and $y\in I_{n,j}$, then
$\|x-y\|\geq(n^{2\zeta}r_{n})\sin\alpha_{k}\geq 3n^{\zeta}r_{n}$. Hence for
all large enough $n$ the events $\\{I_{n,1}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\},\ldots,\\{I_{n,\kappa}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}$ are mutually independent. Therefore

$\displaystyle\mathbb{P}[(\partial A\setminus Q^{-}_{n})\subset Z_{n}({\cal
U}^{\prime}_{n,A})]=\prod_{i=1}^{\kappa}\mathbb{P}[I_{n,i}\subset Z_{n}({\cal
U}^{\prime}_{n,A})],$

and by Lemmas 3.5 and 3.3, this converges to the right hand side of (3.27).

Now we prove (3.28). For $n>0$, and $i\in\\{1,2,\ldots,\kappa\\}$, let
$S_{n,i}$ denote the rectangular block of dimensions $|I_{n,i}|\times
3n^{\zeta}r_{n}$, consisting of all points in $A$ at perpendicular distance at
most $3n^{\zeta}r_{n}$ from $I_{n,i}$. Let $\partial_{{\rm side}}S_{n,i}$
denote the union of the two ‘short’ edges of $S_{n,i}$, i.e. the two edges
bounding $S_{n,i}$ which are perpendicular to $I_{n,i}$.

Then for $n$ large, $A\setminus(A^{(3n^{\zeta}r_{n})}\cup
Q_{n})\subset\cup_{i=1}^{\kappa}S_{n,i}$, and also $\partial_{\rm
side}S_{n,i}\subset Q_{n}$ for $1\leq i\leq\kappa$, so that

$\displaystyle\\{(A^{(3n^{\zeta}r_{n})}\cup\partial A)\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}\cap\\{Q_{n}\subset Z^{o}_{n}({\cal
U}^{\prime}_{n,A})\\}\setminus\\{A\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\subset\cup_{i=1}^{\kappa}[\\{I_{n,i}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}\cup\\{\partial_{{\rm side}}S_{n,i}\subset
Z_{n}^{o}({\cal U}^{\prime}_{n,A})\\}\setminus\\{S_{n,i}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}].$

For $i\in\\{1,\ldots,\kappa\\}$, let $I^{\prime}_{n,i}$ denote an interval of
length $|I_{n,i}|$ in $\mathbb{R}$. Using the Mapping theorem [6] and the
translation and rotation invariance of Lebesgue measure one sees that

$\displaystyle\mathbb{P}[\\{I_{n,i}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}\cap\\{\partial_{{\rm side}}S_{n,i}\subset
Z^{o}_{n}({\cal U}^{\prime}_{n,A})\\}\setminus\\{S_{n,i}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}]$
$\displaystyle=\mathbb{P}[\\{(I^{\prime}_{n,i}\times\\{0\\})\subset
Z_{n}({\cal U}^{\prime}_{n,\mathbb{H}})\\}\cap\\{(\partial
I^{\prime}_{n,i})\times[0,3n^{\zeta}r_{n}]\subset Z_{n}^{o}({\cal
U}^{\prime}_{n,\mathbb{H}})\\}$
$\displaystyle\setminus\\{(I^{\prime}_{n,i}\times[0,3n^{\zeta}r_{n}])\subset
Z_{n}({\cal U}^{\prime}_{n,\mathbb{H}})\\}].$

By (3.17) from Lemma 3.5, this probability tends to zero, and hence (3.28). ∎

###### Proof of Theorem 2.1.

Suppose $(r_{n})_{n>0}$ satisfies (3.25). We assert that provided
${\zeta}<1/2$,

$\displaystyle\lim_{n\to\infty}\mathbb{P}[A^{(3n^{\zeta}r_{n})}\subset
Z_{n}({\cal U}^{\prime}_{n,A})]=\exp\Big{(}-{\bf
1}_{\\{k=1\\}}\Big{(}\frac{(\mathbb{E}\,[Y])^{2}}{\mathbb{E}\,[Y^{2}]}\Big{)}|A|e^{-\beta}\Big{)}.$
(3.29)

Indeed, setting $\lambda=n$ and $\delta(\lambda)=r_{n}$, we have that if $k=1$
then (3.2) holds, while if $k\geq 2$ then the left hand side of (3.2) tends to
$+\infty$. Also $\mathbb{P}[A^{(3n^{\zeta}r_{n})}\subset Z_{n}({\cal
U}^{\prime}_{n,A})]$ is bounded from below by $\mathbb{P}[A\subset Z_{n}({\cal
U}^{\prime}_{n,\mathbb{R}^{2}})]$, which converges to the right hand side of
(3.29) by Lemmas 3.2 and 3.3. Also, if $k=1$, then given $\varepsilon>0$, for
$n$ large the event $\\{A^{(3n^{\zeta}r_{n})}\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}$ is contained in $\\{A^{[\varepsilon]}\subset
Z_{n}({\cal U}^{\prime}_{n,A})\\}$, so by by Lemmas 3.2 and 3.3 again

$\displaystyle\limsup_{n\to\infty}\mathbb{P}[A^{(n^{\zeta}r_{n})}\subset
Z_{n}({\cal U}^{\prime}_{n,A})]$
$\displaystyle\leq\limsup_{n\to\infty}\mathbb{P}[A^{[\varepsilon]}\subset
Z_{n}({\cal U}^{\prime}_{n,A})]$
$\displaystyle=\exp\Big{(}-\Big{(}\frac{(\mathbb{E}\,[Y])^{2}}{\mathbb{E}\,[Y^{2}]}\Big{)}|A^{[\varepsilon]}|e^{-\beta}\Big{)},$

and since $|A^{[\varepsilon]}|\to|A|$ as $\varepsilon\downarrow 0$, this gives
us the assertion (3.29).

Also, by (3.27) and Lemma 3.7,

$\displaystyle\lim_{n\to\infty}(\mathbb{P}[\partial A\subset Z_{n}({\cal
U}^{\prime}_{n,A})])=\exp\Big{(}-\Big{(}\frac{c_{2,k}\mathbb{E}\,[Y]}{\sqrt{\mathbb{E}\,[Y^{2}]}}\Big{)}|\partial
A|e^{-\beta/2}\Big{)}.$ (3.30)

Note $\mathbb{P}[\\{(Q_{n}\cup\partial A)\subset Z_{n}({\cal
U}^{\prime}_{n,A})\\}\setminus\\{(Q_{n}\cup\partial A)\subset Z^{o}_{n}({\cal
U}^{\prime}_{n,A})\\}]=0$. Therefore using (3.28) followed by Lemma 3.7, and
then Lemma 3.3, we obtain that

$\displaystyle\lim_{n\to\infty}\mathbb{P}[A\subset Z_{n}({\cal
U}^{\prime}_{n,A})]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}[(A^{(3n^{\zeta}r_{n})}\cup\partial
A\cup Q_{n})\subset Z_{n}({\cal U}^{\prime}_{n,A})]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}[(A^{(3n^{\zeta}r_{n})}\cup\partial
A)\subset Z_{n}({\cal U}^{\prime}_{n,A})],$ (3.31)

provided these limits exist.

The events $\\{\partial A\subset Z_{n}({\cal U}^{\prime}_{n,A})\\}$ and
$\\{A^{(3n^{\zeta}r_{n})}\subset Z_{n}({\cal U}^{\prime}_{n,A})\\}$ are
independent since the first of these events is determined by the configuration
of Poisson points with projection onto $\mathbb{R}^{2}$ distant at most
$n^{\zeta}r_{n}$ from $\partial A$, while the second event is determined by
the Poisson points with projection distant at least $2n^{\zeta}r_{n}$ from
$\partial A$. Therefore the limit in (3.31) does indeed exist, and is the
product of the limits arising in (3.29) and (3.30). By Lemma 3.3 we get the
same limit for $\mathbb{P}[A\subset Z_{n}({\cal U}_{n,A})]$ as for
$\mathbb{P}[A\subset Z_{n}({\cal U}^{\prime}_{n,A})]$. This gives us (2.1). ∎

###  3.4 First steps toward proving Theorem 2.2

In this subsection we assume that $d\geq 2$ and $\partial A\in C^{2}$. Let
$k\in\mathbb{N},\beta\in\mathbb{R}$, and assume that $(r_{n})_{n>0}$ satisfies
(2.5). Given any $n>0$, Borel $D\subset\mathbb{R}^{d}$, and any point process
$\mathcal{X}$ in $\mathbb{R}^{d}\times\mathbb{R}_{+}$, define
$F_{n}(D,\mathcal{X})$ to be the event that $D$ is ‘fully $k$-covered’ by the
SPBM based on $\mathcal{X}$ with radii scaled by $r_{n}$, that is,

$\displaystyle F_{n}(D,\mathcal{X}):=\\{D\subset Z_{n}(\mathcal{X})\\},$
(3.32)

where $Z_{n}(\cdot)$ was defined at (3.7). Also, given $r>0$, define the
‘covering number’

$\displaystyle\kappa(D,r):=\min\\{m\in\mathbb{N}:\exists x_{1},\ldots,x_{m}\in
D~{}{\rm with}~{}D\subset\cup_{i=1}^{m}B(x_{i},r)\\}.$ (3.33)

Given $x\in\partial A$ we can express $\partial A$ locally in a neighbourhood
of $x$, after a rotation, as the graph of a $C^{2}$ function from
$\mathbb{R}^{d-1}$ to $\mathbb{R}$ with all of its partial derivatives taking
the value zero at $x$. We shall approximate to that function by the graph of a
piecewise affine function (in $d=2$, a piecewise linear function).

For each $x\in\partial A$, we can find an open neighbourhood ${\cal N}_{x}$ of
$x$, a number $r(x)>0$ such that $B(x,3r(x))\subset{\cal N}_{x}$ and a
rotation $\theta_{x}$ about $x$ such that $\theta_{x}(\partial A\cap{\cal
N}_{x})$ is the graph of a real-valued $C^{2}$ function $f$ defined on an open
ball $D\subset\mathbb{R}^{d-1}$, with $\langle f^{\prime}(x),e\rangle=0$ and
$\langle f^{\prime}(u),e\rangle\leq 1/9$ for all $u\in D$ and all unit vectors
$e$ in $\mathbb{R}^{d-1}$, where $\langle\cdot,\cdot\rangle$ denotes the
Euclidean inner product in $\mathbb{R}^{d-1}$ and
$f^{\prime}(u):=(\partial_{1}f(u),\partial_{2}f(u),\ldots,\partial_{d-1}f(u))$
is the derivative of $f$ at $u$. Moreover, by taking a smaller neighbourhood
if necessary, we can also assume that there exist $\varepsilon>0$ and
$a\in\mathbb{R}$ such that $f(u)\in[a+\varepsilon,a+2\varepsilon]$ for all
$u\in D$ and also $\theta_{x}(A)\cap(D\times[a,a+3\varepsilon])=\\{(u,z):u\in
D,a\leq z\leq f(u)\\}$.

By a compactness argument, we can and do take a finite collection of points
$x_{1},\ldots,x_{J}\in\partial A$ such that

$\displaystyle\partial A\subset\cup_{j=1}^{J}B(x_{j},r(x_{j})).$ (3.34)

Then there are constants $\varepsilon_{j}>0$, and rigid motions
$\theta_{j},1\leq j\leq J$, such that for each $j$ the set
$\theta_{j}(\partial A\cap{\cal N}_{x_{j}})$ is the graph of a $C^{2}$
function $f_{j}$ defined on a ball $I_{j}$ in $\mathbb{R}^{d-1}$, with
$\langle f^{\prime}_{j}(u),e\rangle\leq 1/9$ for all $u\in I_{j}$ and all unit
vectors $e\in\mathbb{R}^{d-1}$, and also with $\varepsilon_{j}\leq
f_{j}(u)\leq 2\varepsilon_{j}$ for all $u\in I_{j}$ and
$\theta_{j}(A)\cap(I_{j}\times[0,3\varepsilon_{j}])=\\{(u,z):u\in I_{j},0\leq
z\leq f(u)\\}$.

Later we shall refer to each tuple $(x_{j},r(x_{j}),\theta_{j},f_{j})$, $1\leq
j\leq J$, as a chart. More loosely we shall also refer to each set
$B(x_{j},r(x_{j}))$ as a chart.

Let $\Gamma\subset\partial A$ be a closed set such that $\Gamma\subset
B(x_{j},r(x_{j}))$ for some $j\in\\{1,\ldots,J\\}$, and such that
$\kappa(\partial\Gamma,r)=O(r^{2-d})$ as $r\downarrow 0$, where we set
$\partial\Gamma:=\Gamma\cap\overline{\partial A\setminus\Gamma}$, the boundary
of $\Gamma$ relative to $\partial A$. To simplify notation we shall assume
that $\Gamma\subset B(x_{1},r(x_{1}))$, and moreover that $\theta_{1}$ is the
identity map. Then $\Gamma=\\{(u,f_{1}(u)):u\in U\\}$ for some bounded set
$U\subset\mathbb{R}^{d-1}$. Also, writing $\phi(\cdot)$ for $f_{1}(\cdot)$
from now on, we assume

$\displaystyle\phi(U)\subset[\varepsilon_{1},2\varepsilon_{1}]$ (3.35)

and

$\displaystyle A\cap(U\times[0,3\varepsilon_{1}])=\\{(u,z):u\in U,0\leq
z\leq\phi(u)\\}.$ (3.36)

Note that for any $u,v\in U$, by the Mean Value theorem we have for some
$w\in[u,v]$ that

$\displaystyle|\phi(v)-\phi(u)|=|\langle
v-u,\phi^{\prime}(w)\rangle|\leq(1/9)\|v-u\|.$ (3.37)

Choose (and keep fixed for the rest of this paper) a constant ${\zeta}$ with

$\displaystyle 0<{\zeta}<1/(198d(18+d)).$ (3.38)

Henceforth we shall use this choice of ${\zeta}$ in the definition of ${\cal
U}^{\prime}_{n,D}$ at (3.8). We shall use $n^{c{\zeta}}r_{n}$, for various
choices of constant $c$, to provide various different length scales.

We shall approximate to $\Gamma$ by a polytopal surface $\Gamma_{n}$ with face
diameters that are $\Theta(n^{9{\zeta}}r_{n})$, taking all the faces of
$\Gamma_{n}$ to be $(d-1)$-dimensional simplices. Later we shall fit together
a finite number of surfaces like $\Gamma$ to make up the whole of $\partial
A$.

To do the polytopal approximation, divide $\mathbb{R}^{d-1}$ into hypercubes
of dimension $d-1$ and side $n^{9{\zeta}}r_{n}$, and divide each of these
hypercubes into $(d-1)!$ simplices (we take these simplices to be closed). Let
$U_{n}$ be the union of all those simplices in the resulting tessellation of
$\mathbb{R}^{d-1}$ into simplices, that are contained within $U$, and let
$U_{n}^{-}$ be the union of those simplices in the tessellation which are
contained within $U^{(n^{10{\zeta}}r_{n})}$, where for $r>0$ we set $U^{(r)}$
to be the set of $x\in U$ at a Euclidean distance more than $r$ from
$\mathbb{R}^{d-1}\setminus U$. If $d=2$, the simplices are just intervals. See
Figure 1.

Figure 1:  Example in $d=3$. The outer crescent-shaped region is $U$, while
the inner crescent is $U^{(10{\zeta}r_{n})}$ The outer polygon is $U_{n}$,
while the inner polygon is $U_{n}^{-}$

Let $\sigma^{-}(n)$ denote the number of simplices making up $U_{n}^{-}$.
Choose $n_{0}>0$ such that $\sigma^{-}(n)>0$ for all $n\geq n_{0}$.

Let $\psi_{n}:U_{n}\to\mathbb{R}$ be the function that is affine on each of
the simplices making up $U_{n}$, and agrees with the function $\phi$ on each
of the vertices of these simplices. Our approximating surface (or polygonal
line if $d=2$) will be defined by
$\Gamma_{n}:=\\{(u,\psi_{n}(u)-Kn^{-9{\zeta}}r_{n}):u\in U^{-}_{n}\\}$, with
the constant $K$ given by the following lemma. This lemma uses Taylor
expansion to show that $\psi_{n}$ a good approximation to $\phi$.

######  Lemma 3.9 (Polytopal approximation).

Set $K:=\sup_{n\geq n_{0},u\in
U_{n}}((n^{9{\zeta}}r_{n})^{-2}|\phi(u)-\psi_{n}(u)|)$. Then $K<\infty$.

###### Proof.

See [10, Lemma 7.18]. The notation $t$ there is equivalent to $n$ here. the
length scale of $n^{9\xi}r_{n}$ here plays the role of $t^{-\gamma}$ there. ∎

We now subtract a constant from $\psi_{n}$ to obtain a piecewise affine
function $\phi_{n}$ that approximates $\phi$ from below. For $n\geq n_{0}$ and
$u\in U_{n}$, define $\phi_{n}(u):=\psi_{n}(u)-Kn^{18{\zeta}}r_{n}^{2}$, with
$K$ given by Lemma 3.9. Then for all $n\geq n_{0},u\in U_{n}$ we have
$|\psi_{n}(u)-\phi(u)|\leq Kn^{18{\zeta}}r_{n}^{2}$ so that

$\displaystyle\phi_{n}(u)\leq\phi(u)\leq\phi_{n}(u)+2Kn^{18{\zeta}}r_{n}^{2}.$
(3.39)

Define the set $\Gamma_{n}:=\\{(u,\phi_{n}(u)):u\in U_{n}^{-}\\}$. We refer to
each $(d-1)$-dimensional face of $\Gamma_{n}$ (given by the graph of
$\phi_{n}$ restricted to one of the simplices in our triangulation of
$\mathbb{R}^{d-1}$) as simply a face of $\Gamma_{n}$. Denote these faces of
$\Gamma_{n}$ by $H_{n,1},\ldots,H_{n,\sigma^{-}(n)}$. The number of faces,
$\sigma^{-}(n)$, is $\Theta((n^{9{\zeta}}r_{n})^{1-d})$ as $n\to\infty$. The
perimeter (i.e., the $(d-2)$-dimensional Hausdorff measure of the boundary) of
each individual face is $\Theta((n^{9{\zeta}}r_{n})^{d-2})$.

For $n\geq n_{0}$, define subsets $A_{n},A_{n}^{-},W_{n},A_{n}^{*},W_{n}^{*}$
of $\mathbb{R}^{d}$ (illustrated in Figure 2) by

$\displaystyle A_{n}:=\\{(u,z):u\in U_{n},0\leq
z\leq\phi(u)\\},~{}~{}~{}W_{n}:=\\{(u,z):u\in U_{n},0\leq
z\leq\phi_{n}(u)\\},$ $\displaystyle A_{n}^{-}:=\\{(u,z):u\in U_{n}^{-},0\leq
z\leq\phi(u)\\},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle A_{n}^{*}:=\\{(u,z):u\in
U_{n}^{-},\phi_{n}(u)-(3/2)n^{\zeta}r_{n}\leq
z\leq\phi(u)\\},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle W_{n}^{*}:=\\{(u,z):u\in
U_{n}^{-},\phi_{n}(u)-(3/2)n^{\zeta}r_{n}\leq
z\leq\phi_{n}(u)\\}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.40)

Thus $A_{n}$ is a ‘thick slice’ of $A$ near the boundary region $\Gamma$,
$W_{n}$ is an approximating region having $\Gamma_{n}$ as its upper boundary,
and $A_{n}^{*}$, $W_{n}^{*}$ are ‘thin slices’ of $A$ also having $\Gamma$,
respectively $\Gamma_{n}$, as upper boundary. By (3.39), (3.35) and (3.36),
$W_{n}^{*}\subset A_{n}^{*}\subset A_{n}^{-}\subset A_{n}\subset A$, and
$W_{n}^{*}\subset W_{n}\subset A_{n}$.

Figure 2:  When $d=2$ the sets $A_{n},W_{n},A_{n}^{-},A_{n}^{*},W_{n}^{*}$ are
approximately rectangular but with a curved upper boundary for
$A_{n},A_{n}^{-}$ and $A_{n}^{*}$, a polygonal upper boundary for $W_{n}^{*}$
and $W_{n}$, and a polygonal lower boundary for $A_{n}^{*}$ and $W_{n}^{*}$.

The rest of this subsection, and the next subsection, are devoted to proving
the following intermediate step towards a proof of Theorem 2.10. Recall the
definition of $c_{d,k}$ at (2.3), and define

$\displaystyle
c_{d,k,Y}:=\frac{c_{d,k}(\mathbb{E}\,[Y^{d-1}])^{d-1}}{(\mathbb{E}\,[Y^{d}])^{d-2+1/d}}.$
(3.41)

######  Proposition 3.10 (Limiting coverage probability for approximating
polytopal shell).

It is the case that $\lim_{n\to\infty}\mathbb{P}[F_{n}(W_{n}^{*},{\cal
U}^{\prime}_{n,W_{n}})]=\exp(-c_{d,Y}|\Gamma|e^{-\beta/2}),$ where $|\Gamma|$
denotes the $(d-1)$-dimensional Hausdorff measure of $\Gamma$.

The following corollary of Lemma 3.9 is a first step towards proving this.

######  Lemma 3.11.

(a) It is the case that $|A_{n}\setminus W_{n}|=O(n^{18{\zeta}}r_{n}^{2})$ as
$n\to\infty$.

(b) Let $K$ be as given in Lemma 3.9, and fix $L>0$. Then for all $n\geq
n_{0}$ and $x\in U_{n}^{(Lr_{n})}\times\mathbb{R}$, $|B(x,Lr_{n})\cap
A_{n}\setminus W_{n}|\leq 2K\omega_{d-1}L^{d-1}n^{18{\zeta}}r_{n}^{d+1}$.

###### Proof.

Since $|A_{n}\setminus W_{n}|=\int_{U_{n}}(\phi(u)-\phi_{n}(u))du$, where this
is a $(d-1)$-dimensional Lebesgue integral, part (a) comes from (3.39).

For (b), let $x\in U_{n}^{(Lr_{n})}\times\mathbb{R}$, and let $u\in
U_{n}^{(Lr_{n})}$ be the projection of $x$ onto the first $d-1$ coordinates.
Then if $y\in B(x,Lr_{n})\cap A_{n}\setminus W_{n}$, we have $y=(v,s)$ with
$\|v-u\|\leq Lr_{n}$ and $\phi_{n}(v)<s\leq\phi(v)$. Therefore using (3.39)
yields

$|B(x,Lr_{n})\cap A_{n}\setminus
W_{n}|\leq\int_{B_{(d-1)}(u,Lr_{n})}(\phi(v)-\phi_{n}(v))dv\leq
2K\omega_{d-1}n^{18{\zeta}}r_{n}^{2}L^{d-1}r_{n}^{d-1},$

where the integral is a $(d-1)$-dimensional Lebesgue integral. This gives part
(b). ∎

The next lemma says that small balls centred in $A_{n}^{*}$ have almost half
of their volume in $W_{n}$.

######  Lemma 3.12.

Let $\varepsilon\in(0,1)$. Then for all large enough $n$, all $x\in
A_{n}^{*}$, and all $s\in[\varepsilon r_{n},r_{n}/\varepsilon]$, we have
$|B(x,s)\cap W_{n}|>(1-\varepsilon)(\omega_{d}/2)s^{d}$.

###### Proof.

For all large enough $n$, all $x\in A_{n}^{*}$ and $s\in[\varepsilon
r_{n},r_{n}/\varepsilon]$, we have $B(x,s)\cap A\subset A_{n}$, so $B(x,s)\cap
A_{n}=B(x,s)\cap A$, and hence by Lemma 3.1 and Lemma 3.11(b),

$\displaystyle|B(x,s)\cap W_{n}|$ $\displaystyle=$ $\displaystyle|B(x,s)\cap
A_{n}|-|B(x,s)\cap A_{n}\setminus W_{n}|$ $\displaystyle\geq$
$\displaystyle(1-\varepsilon/2)(\omega_{d}/2)s^{d}-O(n^{18{\zeta}}r_{n}^{d+1}).$

Since $n^{18{\zeta}}r_{n}\to 0$ by (2.5) and (3.38), this gives us the result.
∎

Recall that $\mathbb{H}$ and ${\cal U}^{\prime}_{n,D}$ were defined in Section
3.1 and at (3.8), respectively. The next lemma provides a bound on the
probability that a region of diameter $O(r_{n})$ within $A$ or $A_{n}^{*}$ is
not fully covered. This will be used for dealing with ‘exceptional’ regions
such as those near the boundaries of faces in the polytopal approximation.

######  Lemma 3.13.

Let $\varepsilon\in(0,1)$, $K_{1}>0$. Then as $n\to\infty$,

$\displaystyle\sup_{z\in\mathbb{R}^{d}}\mathbb{P}[F_{n}(B(z,K_{1}r_{n})\cap
A_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})^{c}]=O(n^{\varepsilon-(d-1)/d}),$
(3.42)
$\displaystyle\sup_{z\in\mathbb{R}^{d}}\mathbb{P}[F_{n}(B(z,K_{1}r_{n})\cap\mathbb{H},{\cal
U}^{\prime}_{n,\mathbb{H}})^{c}]=O(n^{\varepsilon-(d-1)/d}),$ (3.43)

and

$\displaystyle\sup_{z\in\mathbb{R}^{d}}\mathbb{P}[F_{n}(B(z,K_{1}r_{n})\cap
A,{\cal U}^{\prime}_{n,A})^{c}]=O(n^{\varepsilon-(d-1)/d}).$ (3.44)

###### Proof.

Let $\delta\in(0,1/2)$ with $(1-\delta)^{d+1}\mathbb{E}\,[Y^{d}{\bf
1}_{\\{\delta\leq Y\leq 1/\delta\\}}]>(1-\varepsilon/2)\mathbb{E}\,[Y^{d}]$.
For any point set $\mathcal{X}\subset\mathbb{R}^{d}\subset[\delta,\infty)$,
define $\tilde{Z}_{n}(\mathcal{X})$ similarly to $Z_{n}(\mathcal{X})$ but with
slightly smaller balls, namely

$\tilde{Z}_{n}(\mathcal{X}):=\\{y\in\mathbb{R}^{d}:\\#(\\{(x,s)\in\mathcal{X}:y\in
B(x,(1-\delta)r_{n}s)\\})\geq k\\},$

and note that for any $y\in\tilde{Z}_{n}(\mathcal{X})$ we have
$B(y,\delta^{2}r_{n})\subset Z_{n}(\mathcal{X})$.

There is a constant $k$ independent of $n$ such that given
$z\in\mathbb{R}^{d}$ and given $n$, we can (and do) choose
$x_{1},\ldots,x_{k}\in A_{n}^{*}$ with $B(z,K_{1}r_{n})\cap
A^{*}_{n}\subset\cup_{i=1}^{k}B(x_{i},\delta^{2}r_{n})$. Then for all $n$
large enough, and for $1\leq i\leq k$, using Lemma 3.12 in the third line
below we have

$\displaystyle 1-\mathbb{P}[F_{n}(B(x_{i},\delta^{2}r_{n})\cap A^{*}_{n},{\cal
U}^{\prime}_{n,W_{n}})]\leq\mathbb{P}[x_{i}\notin\tilde{Z}_{n}({\cal
U}^{\prime}_{n,W_{n}}\cap(W_{n}\times[\delta,1/\delta]))]$ $\displaystyle\leq
k(n\omega_{d}(r_{n}/\delta)^{d})^{k-1}\exp\Big{(}-n\int_{[\delta,1/\delta]}|W_{n}\cap
B(x_{i},(1-\delta)r_{n}t)|\mu_{Y}(dt)\Big{)}$ $\displaystyle\leq
k(n\omega_{d}(r_{n}/\delta)^{d})^{k-1}\exp\Big{(}-n\int_{[\delta,1/\delta]}(1-\delta)^{d+1}(\omega_{d}/2)r_{n}^{d}t^{d}\mu_{Y}(dt)\Big{)}$
$\displaystyle\leq
k(n\omega_{d}(r_{n}/\delta)^{d})^{k-1}\exp\Big{(}-(\omega_{d}/2)nr_{n}^{d}(1-\varepsilon/2)\mathbb{E}\,[Y^{d}]\Big{)}.$

Therefore by(3.5), for $n$ large

$\displaystyle\mathbb{P}[\\{F_{n}(B(x_{i},\delta^{2}r_{n})\cap A_{n}^{*},{\cal
U}^{\prime}_{n,W_{n}})\\}^{c}]\leq n^{\varepsilon-(d-1)/d}.$

Taking the union of the above events for $1\leq i\leq k$, and applying the
union bound, gives us (3.42). The proofs of (3.43) and (3.44) are similar. ∎

Let
$\partial_{d-2}\Gamma_{n}:=\cup_{i=1}^{\sigma^{-}(t)}\partial_{d-2}H_{n,i}$,
the union of all $(d-2)$-dimensional faces in the boundaries of the faces
making up $\Gamma_{n}$ (the $H_{n,i}$ were defined just after (3.39)). Recall
from (3.38) that ${\zeta}\in(0,1/(198d(18+d)))$. Given $n>0$, define the set
$Q^{+}_{n}\subset\mathbb{R}^{d}$ by

$\displaystyle Q^{+}_{n}:=(\partial_{d-2}\Gamma_{n}\oplus
B(o,8dn^{4{\zeta}}r_{n}))\cap A_{n}^{*}.$ (3.45)

Thus $Q_{n}^{+}$ is a region near the corners of our polygon approximating
$\partial A$ (if $d=2$) or near the boundaries of the faces of our polytopal
surface approximating $\partial A$ (if $d\geq 3$). In the next lemma we show
that $Q_{n}^{+}$ is fully covered with high probability.

######  Lemma 3.14.

It is the case that $\mathbb{P}[F_{n}(Q^{+}_{n},{\cal
U}^{\prime}_{n,W_{n}})]\to 1$ as $n\to\infty$.

###### Proof.

Let $\varepsilon:={\zeta}/2$. For each face $H_{n,i}$ of $\Gamma_{n}$, $1\leq
i\leq\sigma^{-}(n)$, we claim that we can take
$x_{i,1},\ldots,x_{i,k_{n,i}}\in\mathbb{R}^{d}$ with $\max_{1\leq
i\leq\sigma^{-}(n)}k_{n,i}=O(n^{9d{\zeta}-10{\zeta}})$, such that

$\displaystyle(\partial_{d-2}H_{n,i})\oplus
B(o,8dn^{4{\zeta}}r_{n})\subset\cup_{j=1}^{k_{n,i}}B(x_{i,j},r_{n}).$ (3.46)

Indeed, we can cover $\partial_{d-2}H_{n,i}$ by $O(n^{5{\zeta}(d-2)})$ balls
of radius $n^{4{\zeta}}r_{n}$, denoted $B_{i,\ell}^{(0)}$ say. Replace each
ball $B_{i,\ell}^{(0)}$ with a ball $B^{\prime}_{i,\ell}$ with the same centre
as $B_{i,\ell}^{(0)}$ and with radius $9dn^{4{\zeta}}r_{n}$. Then cover
$B^{\prime}_{i,\ell}$ by $O((n^{4{\zeta}})^{d})$ balls of radius $r_{n}$.
Every point in the set on the left side of (3.46) lies in one of these balls
of radius $r_{n}$, and the claim follows.

By the definitions of $Q_{n}^{+}$ and $\partial_{d-1}\Gamma_{n}$, and then
(3.46), we have

$Q_{n}^{+}\subset\cup_{i=1}^{\sigma^{-}(n)}((\partial_{d-2}H_{n,i})\oplus
B(o,8dn^{4{\zeta}}r_{n}))\subset\cup_{i=1}^{\sigma^{-}(n)}\cup_{j=1}^{k_{n,i}}B(x_{i,j},r_{n}),$

so by the union bound,

$\displaystyle\mathbb{P}[F_{n}(Q^{+}_{n},{\cal
U}^{\prime}_{n,W_{n}})^{c}]\leq\sum_{i=1}^{\sigma^{-}(n)}\sum_{j=1}^{k_{n,i}}\mathbb{P}[F_{n}(B(x_{i,j},r_{n})\cap
A_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})^{c}].$

Thus using Lemma 3.13 and the fact that
$\sigma^{-}(n)=O((n^{9{\zeta}}r_{n})^{1-d})$, we obtain that

$\displaystyle\mathbb{P}[F_{n}(Q^{+}_{n},{\cal U}^{\prime}_{n,W_{n}})^{c}]$
$\displaystyle=O((n^{9{\zeta}}r_{n})^{1-d}n^{(9d-10){\zeta}}n^{\varepsilon-1+1/d})$
$\displaystyle=O(n^{-{\zeta}}r_{n}^{1-d}n^{\varepsilon-1+1/d})=O(n^{-{\zeta}/2}).$

Thus $\mathbb{P}[F_{n}(Q^{+}_{n},{\cal U}^{\prime}_{n,W_{n}})^{c}]$ tends to
zero. ∎

###  3.5  Induced coverage process and proof of Proposition 3.10

We shall conclude the proof of Proposition 3.10 by means of a device we refer
to as the induced coverage process. This is obtained by taking the parts of
$W_{n}$ near the flat parts of $\Gamma_{n}$, along with any Poisson points
therein, and rearranging them into a flat region of macroscopic size.

Partition each face $H_{n,i}$, $1\leq i\leq\sigma^{-}(n)$ into a collection of
$(d-1)$-dimensional hypercubes of side $n^{3{\zeta}}r_{n}$ contained in
$H_{n,i}$ and distant more than $n^{4{\zeta}}r_{n}$ from
$\partial_{d-2}H_{n,i}$, together with a ‘border region’ contained within
$\partial_{d-2}H_{n,i}\oplus B(o,2n^{4{\zeta}}r_{n})$. Let $P_{n}$ be the
union (over all faces) of the boundaries of the $(d-1)$-dimensional hypercubes
in this partition (see Figure 3: the $P$ stands for ‘plaid’). Set

$\displaystyle P^{+}_{n}:=[P_{n}\oplus B(o,9n^{{\zeta}}r_{n})]\cap W_{n}^{*}.$
(3.47)

Figure 3:  Part of the ‘plaid’ region $P_{n}$ when $d=3$. The outer triangle
represents one face $H_{n,i}$, and the part of $P_{n}$ within $H_{n,i}$ is
given by the union of the boundaries of the squares. The outer triangle has
sides of length $\Theta(n^{9{\zeta}}r_{n})$, while the squares have sides of
length $n^{3{\zeta}}r_{n}$. The region between the two triangles has thickness
$n^{4{\zeta}}r_{n}$ and is contained in $Q_{n}^{+}$.

Enumerate the $(d-1)$-dimensional hypercubes in the above subdivision of the
faces $H_{n,i},1\leq i\leq\sigma^{-}(n)$, as
$I^{+}_{n,1},\ldots,I^{+}_{n,\lambda(n)}$. For $1\leq i\leq\lambda(n)$ let
$I_{n,i}:=I^{+}_{n,i}\setminus(P_{n}\oplus B(o,n^{\zeta}r_{n}))$, which is a
$(d-1)$-dimensional hypercube of side length $(n^{3{\zeta}}-2n^{\zeta})r_{n}$
with the same centre and orientation as $I_{n,i}^{+}$. We claim that the total
$(d-1)$-dimensional Lebesgue measure of these $(d-1)$-dimensional hypercubes
satisfies

$\displaystyle\lim_{n\to\infty}(|\cup_{i=1}^{\lambda(n)}I_{n,i}|)=|\Gamma|.$
(3.48)

Indeed, for $1\leq i\leq\lambda(n)$ we have
$|I_{n,i}|/|I_{n,i}^{+}|=((n^{3{\zeta}}-n^{\zeta})/n^{3{\zeta}})^{d-1}$, which
tends to one, so the proportionate amount removed near the boundaries of the
$(d-1)$-dimensional hypercubes $I^{+}_{n,i}$ to give $I_{n,i}$ vanishes. Also
the ‘border region’ of a face $H_{n,i}$ that is not contained in any of the of
the $I^{+}_{n,j}$s has $(d-1)$-dimensional Lebesgue measure that is
$O((n^{9{\zeta}}r_{n})^{d-2}n^{4{\zeta}}r_{n})$, so that the total
$(d-1)$-dimensional measure of the removed regions near the boundaries of the
faces is
$O((n^{9{\zeta}}r_{n})^{1-d}\times(n^{9{\zeta}}r_{n})^{d-2}n^{4{\zeta}}r_{n})=O(n^{-5{\zeta}})$,
which tends to zero. Thus the claim (3.48) is justified.

For $1\leq i\leq\lambda(n)$, let $S_{n,i}^{+}$, repectively $S_{n,i}$, be a
$d$-dimensional cube of side $n^{3{\zeta}}r_{n}$, respectively of side
$(n^{3{\zeta}}-2n^{\zeta})r_{n}$, having $I^{+}_{n,i}$, respectively
$I_{n,i}$, as its ‘upper face’, i.e. having $I_{n,i}^{+}$ (resp. $I_{n,i}$) as
a face and lying within $W_{n}$. We shall verify in Lemma 3.16 that
$S_{n,1}^{+},\ldots,S_{n,\lambda(n)}^{+}$ are disjoint.

Define a region $D_{n}\subset\mathbb{R}^{d-1}$ that is approximately a
rectilinear hypercube with lower left corner at the origin, and obtained as
the union of $\lambda(n)$ disjoint $(d-1)$-dimensional hypercubes of side
$n^{3{\zeta}}r_{n}$. We can and do arrange that
$D_{n}\subset[0,|\Gamma_{n}|^{1/(d-1)}+n^{3{\zeta}}r_{n}]^{d-1}$ for each $t$,
and $|D_{n}|\to|\Gamma|$ as $t\to\infty$. Define the flat slabs (or if $d=2$,
flat strips)

$\displaystyle
S_{n}:=D_{n}\times[0,(n^{3{\zeta}}-2n^{\zeta})r_{n}];~{}~{}~{}~{}~{}~{}S^{+}_{n}:=D_{n}\times[0,n^{3{\zeta}}r_{n}],$
(3.49)

and denote the lower boundary of $S_{n}$ (that is, the set
$D_{n}\times\\{0\\}$) by $L_{n}$.

Now choose rigid motions $\theta_{n,i}$ of $\mathbb{R}^{d}$, $1\leq
i\leq\lambda(n)$, such that under applications of these rigid motions the
blocks $S^{+}_{n,i}$ are reassembled to form the slab $S_{n}^{+}$, with the
square face $I^{+}_{n,i}$ of the $i$-th block transported to part of the lower
boundary $L_{n}$ of $S_{n}$. In other words, choose the rigid motions so that
the sets $\theta_{n,i}(S^{+}_{n,i})$, $1\leq i\leq\lambda(n)$, have pairwise
disjoint interiors and their union is $S_{n}^{+}$, and also
$\theta_{n,i}(I^{+}_{n,i})\subset L_{n}$ for $1\leq i\leq\lambda(n)$.

Given $n\geq n_{0}$ and $i\leq\lambda(n)$, define
$\tilde{\theta}_{n,i}:\mathbb{R}^{d+1}\to\mathbb{R}^{d+1}$ by
$\tilde{\theta}_{n,i}(x,r)=(\theta_{n,i}(x),r)$ for all
$x\in\mathbb{R}^{d},r\in\mathbb{R}$. By the Restriction, Mapping and
superposition theorems (see e.g. [6]), the point process $\tilde{{\cal
U}}_{n,S^{+}_{n}}:=\cup_{i=1}^{\lambda(n)}\tilde{\theta}_{n,i}({\cal
U}_{n,W_{n}}\cap(S^{+}_{n,i}\times\mathbb{R}_{+}))$ is a Poisson process in
$S_{n}^{+}\times\mathbb{R}_{+}$ with intensity measure $n{\rm
Leb}_{d}\otimes\mu_{Y}$ (strictly speaking, with intensity measure $n{\rm
Leb}_{d}|_{S_{n}^{+}}\otimes\mu_{Y}$).

We extend $\tilde{\cal U}_{n,S^{+}_{n}}$ to a Poisson process $\tilde{\cal
U}_{n,\mathbb{H}}$ in $\mathbb{H}\times\mathbb{R}_{+}$, where
$\mathbb{H}:=\mathbb{R}^{d-1}\times[0,\infty)$, as follows. Let $\tilde{\cal
U}_{n,\mathbb{H}\setminus S^{+}_{n}}$ be a Poisson process with intensity
measure $n{\rm Leb}_{d}\otimes\mu_{Y}$ in $(\mathbb{H}\setminus
S^{+}_{n})\times\mathbb{R}_{+}$, independent of ${\cal U}_{n,W_{n}}$, and set

$\displaystyle\tilde{\cal U}_{n,\mathbb{H}}:=\tilde{\cal
U}_{n,S_{n}^{+}}\cup\tilde{\cal U}_{n,\mathbb{H}\setminus
S_{n}^{+}};~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tilde{\cal
U}^{\prime}_{n,\mathbb{H}}:=\tilde{\cal
U}_{n,\mathbb{H}}\cap(\mathbb{R}^{d}\times[0,n^{\zeta}r_{n}]).$ (3.50)

By the Superposition theorem (see e.g. [6, Theorem 3.3]), $\tilde{\cal
U}_{n,\mathbb{H}}$ is a homogeneous Poisson process in
$\mathbb{H}\times[0,\infty)$ with intensity measure $n{\rm
Leb}_{d}\times\mu_{Y}$. We call the collection of balls of radius $r_{n}t$
centred on $x$, where $\\{(x,t)\\}$ are the points of this point process, the
induced coverage process.

The next lemma says that the region $P_{n}^{+}$ is covered with high
probability. It is needed because locations in $P_{n}^{+}$ lie near the
boundary of blocks $S_{n,i}$, so that coverage of these locations by
$Z_{n}({\cal U}_{n,A})$ does not necessarily correspond to coverage of their
images in the induced coverage process.

######  Lemma 3.15.

It is the case that $\lim_{t\to\infty}\mathbb{P}[F_{n}(P^{+}_{n},{\cal
U}^{\prime}_{n,W_{n}})]=1.$

###### Proof.

We have $\lambda(n)=O((n^{3{\zeta}}r_{n})^{1-d})$, and for $1\leq
i\leq\lambda(n)$, the number of balls of radius $n^{\zeta}r_{n}$ required to
cover the boundary of the $(d-1)$-dimensional hypercube $I_{n,i}^{+}$ is
$O((n^{3{\zeta}}/n^{\zeta})^{d-2})$. Thus we can take
$x_{n,1},\ldots,x_{n,k_{n}}\in\mathbb{R}^{d}$, with
$k_{n}=O(r_{n}^{1-d}n^{-{\zeta}-d{\zeta}})$, such that
$P_{n}\subset\cup_{i=1}^{k_{n}}B(x_{n,i},n^{\zeta}r_{n})$.

Then $P_{n}^{+}\subset\cup_{i=1}^{k_{n}}B(x_{n,i},10n^{\zeta}r_{n})\cap
A_{n}^{*}$, and $B(x_{n,i},10n^{\zeta}r_{n})$ can be covered by
$O(n^{d{\zeta}})$ balls of radius $r_{n}$. Hence by taking
$\varepsilon={\zeta}/2$ in Lemma 3.13, and using (2.5), we have

$\displaystyle\mathbb{P}[F_{n}(P^{+}_{n},{\cal U}^{\prime}_{n,W_{n}})^{c}]$
$\displaystyle\leq\sum_{i=1}^{k_{n}}\mathbb{P}[F_{n}(B(x_{n,i},10n^{\zeta}r_{n})\cap
A_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})^{c}]$
$\displaystyle=O(r_{n}^{1-d}n^{-{\zeta}-d{\zeta}}n^{d{\zeta}}n^{\varepsilon-(d-1)/d})=O(n^{-{\zeta}/2}),$

which tends to zero. ∎

######  Lemma 3.16.

Suppose $n\geq n_{0}$ and $i<j\leq\lambda(n)$, $i,j\in\mathbb{N}$. Then
$(S_{n,i}^{+})^{o}\cap(S_{n,j}^{+})^{o}=\varnothing$.

###### Proof.

Suppose $(S_{n,i}^{+})^{o}\cap(S_{n,j}^{+})^{o}\neq\varnothing$; we shall
obtain a contradiction. Let $x\in(S_{n,i}^{+})^{o}\cap(S_{n,j}^{+})^{o}$. Let
$y$ be the closest point in $I^{+}_{n,i}$ to $x$, and $y^{\prime}$ the closest
point in $I^{+}_{n,j}$ to $x$. Choose $\ell,m$ such that $I^{+}_{n,i}\subset
H_{n,\ell}$ and $I^{+}_{n,j}\subset H_{n,m}$. Then $\ell\neq m$ since if
$\ell=m$ then clearly we would have
$(S_{n,i}^{+})^{o}\cap(S_{n,j}^{+})^{o}=\varnothing$.

Let $J_{n,\ell}\subset\mathbb{R}^{d-1}$ be the image of $H_{n,\ell}$ under
projection onto the first $d-1$ coordinates, and write $y=(u,\phi_{n}(u))$
with $u\in J_{n,\ell}$. Let $v\in\partial J_{n,\ell}$. By (3.37) and (3.39),

$\displaystyle|\phi_{n}(v)-\phi_{n}(u)|\leq|\phi(v)-\phi(u)|+4Kn^{18{\zeta}}r_{n}^{2}\leq(1/9)\|v-u\|+4Kn^{18{\zeta}}r_{n}^{2}.$

Since $y\in I^{+}_{n,i}$ we have $\|y-(v,\phi_{n}(v))\|\geq\,{\rm
dist}(y,\partial H_{n,\ell})\geq n^{4{\zeta}}r_{n}$, so that

$\displaystyle
n^{4{\zeta}}r_{n}\leq\|u-v\|+|\phi_{n}(u)-\phi_{n}(v)|\leq(10/9)\|u-v\|+4Kn^{18{\zeta}}r_{n}^{2},$

and hence provided $n$ is large enough, $\|u-v\|\geq n^{4{\zeta}}r_{n}/2$, so
that writing $y^{\prime}:=(u^{\prime},\phi_{n}(u^{\prime}))$ we have

$\|y-y^{\prime}\|\geq\|u-u^{\prime}\|\geq\,{\rm dist}(u,\partial
J_{n,\ell})\geq n^{4{\zeta}}r_{n}/2.$

But also $\|y-y^{\prime}\|\leq\|y-x\|+\|y^{\prime}-x\|\leq
2n^{3{\zeta}}r_{n}$, and we have our contradiction. ∎

Denote the union of the boundaries (relative to
$\mathbb{R}^{d-1}\times\\{0\\}$) of the lower faces of the blocks making up
the strip/slab $S_{n}$, by $C_{n}^{0}$, and the
$(9n^{\zeta}r_{n})$-neighbourhood in $\mathbb{H}$ of this region by $C_{n}$
(the $C$ can be viewed as standing for ‘corner region’), i.e.

$\displaystyle C_{n}^{0}:=\cup_{i=1}^{\lambda(n)}\theta_{n,i}(\partial
I^{+}_{n,i}),~{}~{}~{}~{}~{}~{}~{}C_{n}:=(C_{n}^{0}\oplus
B(o,9n^{\zeta}r_{n}))\cap\mathbb{H}.$ (3.51)

Here $\partial I^{+}_{n,i}$ denotes the relative boundary of $I^{+}_{n,i}$
(relative to the face $H_{n,j}$ containing $I^{+}_{n,i}$).

The next lemma says that the corner region $C_{n}$ is covered with high
probability. It is needed because locations in $C_{n}$ lie near the boundaries
of the blocks assembled to make the induced coverage process, so that coverage
of these locations in the induced coverage process does not necessarily
correspond to coverage of their pre-images in the original coverage process.

######  Lemma 3.17.

It is the case that $\lim_{n\to\infty}\mathbb{P}[F_{n}(C_{n},{\cal
U}^{\prime}_{n,\mathbb{H}})]=1$.

###### Proof.

For each of the $(d-1)$ dimensional hypercubes
$\theta_{n,i}(S_{n,i}^{+}),1\leq i\leq\lambda(n)$, the number of balls of
radius $n^{\zeta}r_{n}$ required to cover the boundary is
$O((n^{3{\zeta}}/n^{\zeta})^{d-2})$. Also
$\lambda(n)=O((n^{3{\zeta}}r_{n})^{1-d})$, so we can take points
$x_{n,1},\ldots,x_{n,m_{n}}\in L_{n}$, with
$m_{n}=O(n^{-{\zeta}-d{\zeta}}r_{n}^{1-d})$, such that
$C_{n}^{0}\subset\cup_{i=1}^{m_{n}}B(x_{n,i},n^{\zeta}r_{n})$. Then
$C_{n}\subset\cup_{i=1}^{m_{n}}B(x_{n,i},10n^{\zeta}r_{n})$, and
$B(x_{i},10n^{\zeta}r_{n})$ can be covered by $O(n^{d{\zeta}})$ balls of
radius $r_{n}$. Hence by (3.43) from Lemma 3.13, taking
$\varepsilon={\zeta}/2$, we obtain the estimate

$\mathbb{P}[F_{n}(C_{n},{\cal
U}^{\prime}_{n,\mathbb{H}})^{c}]=O(n^{d{\zeta}}n^{-{\zeta}-d{\zeta}}r_{n}^{1-d}n^{\varepsilon-(d-1)/d})=O(n^{-{\zeta}/2}),$

which tends to zero. ∎

######  Lemma 3.18 (Limiting coverage probabilities for the induced coverage
process).

With $c_{d,Y}$ given at (3.41),

$\displaystyle\lim_{n\to\infty}\mathbb{P}[F_{n}(S_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})]=\lim_{n\to\infty}\mathbb{P}[F_{n}(L_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})]=\exp(-c_{d,Y}|\Gamma|e^{-\beta/2}).$ (3.52)

###### Proof.

The second equality of (3.52) is easily obtained using (3.16) from Lemma 3.5,
together with Lemma 3.3.

Recall that $L_{n}=D_{n}\times\\{0\\}$. Also $\partial L_{n}\subset
C_{n}^{0}$, so that $(\partial D_{n}\oplus
B_{(d-1)}(o,n^{\zeta}r_{n}))\times[0,2n^{\zeta}r_{n}]\subset C_{n}$, and
therefore by (3.17) from Lemma 3.5,

$\displaystyle\mathbb{P}[(F_{n}(L_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})\setminus F_{n}(S_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}}))\cap F_{n}(C_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})]\to 0.$

Therefore using also Lemma 3.17 shows that $\mathbb{P}[F_{n}(L_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})\setminus F_{n}(S_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})]\to 0$, and this gives us the rest of (3.52). ∎

###### Proof of Proposition 3.10.

We shall approximate the event $F_{n}(W^{*}_{n},{\cal U}^{\prime}_{n,W_{n}})$
by events $F_{n}(L_{n},\tilde{\cal U}^{\prime}_{n,\mathbb{H}})$ and
$F_{n}(S_{n},\tilde{\cal U}^{\prime}_{n,\mathbb{H}})$, and apply Lemma 3.18.
The sets $Q_{n}^{+}$, $P_{n}^{+}$ and $S_{n}$ and were defined at (3.45),
(3.47) and (3.49) respectively.

Suppose $F_{n}(Q^{+}_{n}\cup P_{n}^{+},{\cal U}^{\prime}_{n,W_{n}})\setminus
F_{n}(W_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})$ occurs, and choose $x\in
W_{n}^{*}\setminus(Q_{n}^{+}\cup P_{n}^{+})\setminus Z_{n}({\cal
U}^{\prime}_{n,W_{n}})$. Let $y\in\Gamma_{n}$ with $\|y-x\|=\,{\rm
dist}(x,\Gamma_{n})$. Then $\|y-x\|\leq 2n^{\zeta}r_{n}$, and since $x\notin
Q_{n}^{+}$, we have $\,{\rm dist}(x,\partial_{d-2}\Gamma_{n})\geq
8dn^{4{\zeta}}r_{n}$, and hence $\,{\rm dist}(y,\partial_{d-2}\Gamma_{n})\geq
8dn^{4{\zeta}}r_{n}-2n^{\zeta}r_{n}\geq 3n^{4{\zeta}}r_{n}$, provided $n$ is
large enough. Therefore $y$ lies in the interior of the face $H_{n,i}$ for
some $i$ and $x-y$ is perpendicular to $H_{n,i}$ (if $y\neq x$). Also, since
$x\notin P_{n}^{+}$, $\,{\rm dist}(x,P_{n})\geq 9n^{{\zeta}}r_{n}$, so $\,{\rm
dist}(y,P_{n})\geq 7n^{{\zeta}}r_{n}$. Therefore $y\in I_{n,j}$ for some $j$,
and $x$ lies in the block $S_{n,j}$. Hence
$B(\theta_{n,j}(x),n^{\zeta}r_{n})\cap\mathbb{H}\subset\theta_{n,j}(S^{+}_{n,j})$,
and hence by (3.50),

$\\#(\\{(z,s)\in\tilde{\cal U}^{\prime}_{n,\mathbb{H}}:\theta_{n,j}(x)\in
B(z,r_{n}s)\\})=\\#(\\{(w,t)\in{\cal U}^{\prime}_{n,W_{n}}:x\in
B(w,r_{n}t)\\})<k,$

so the event $F_{n}(S_{n},\tilde{\cal U}^{\prime}_{n,\mathbb{H}})$ does not
occur. Hence

$F_{n}(S_{n},\tilde{\cal U}^{\prime}_{n,\mathbb{H}})\setminus
F_{n}(W_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})\subset F_{n}(Q^{+}_{n}\cup
P^{+}_{n},{\cal U}^{\prime}_{n,W_{n}})^{c},$

so by Lemmas 3.14 and 3.15, $\mathbb{P}[F_{n}(S_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})\setminus F_{n}(W^{*}_{n},{\cal
U}^{\prime}_{n,W_{n}})]\to 0$, and hence using (3.52) we have

$\displaystyle\liminf_{t\to\infty}\mathbb{P}[F_{n}(W_{n}^{*},{\cal
U}^{\prime}_{n,W_{n}})]\geq\exp(-c_{d,Y}|\Gamma|e^{-\beta/2}).$ (3.53)

Suppose $F_{n}(W^{*}_{n},{\cal U}^{\prime}_{n,W_{n}})\setminus
F_{n}(L_{n},\tilde{\cal U}^{\prime}_{n,\mathbb{H}})$ occurs, and choose $y\in
L_{n}\setminus Z_{n}(\tilde{\cal U}^{\prime}_{n,\mathbb{H}})$. Take
$i\in\\{1,\ldots,\lambda(n)\\}$ such that $y\in\theta_{n,i}(I_{n,i}^{+})$.
Then $\,{\rm dist}(y,\theta_{n,i}(\partial I_{n,i}))\leq n^{\zeta}r_{n}$,
since otherwise $\theta_{n,i}^{-1}(y)$ would be a location in
$A^{**}_{n}\setminus Z_{n}({\cal U}^{\prime}_{n,W_{n}})$. Thus $y\in C_{n}$ by
(3.51), and therefore using Lemma 3.17 yields that

$\mathbb{P}[F_{n}(W^{*}_{n},{\cal U}^{\prime}_{n,W_{n}})\setminus
F_{n}(L_{n},\tilde{\cal
U}^{\prime}_{n,\mathbb{H}})]\leq\mathbb{P}[F_{n}(C_{n},{\cal
U}^{\prime}_{n,W_{n}})^{c}]\to 0.$

Combining this with (3.52) and (3.53) completes the proof. ∎

###  3.6 Proof of Theorem 2.2

Proposition 3.10 gives the limiting probability of coverage of a polytopal
approximation to a region near part of $\partial A$. The next two lemmas show
that $\mathbb{P}[F_{n}(W^{*}_{n},{\cal U}^{\prime}_{n,W_{n}})]$ approximates
$\mathbb{P}[F_{n}(A^{*}_{n},{\cal U}^{\prime}_{n,A_{n}})]$ (recall the
definitions at (3.40)). From this we can deduce that we get the same limiting
probability even after dispensing with the polytopal approximation.

######  Lemma 3.19.

Let $E^{(1)}_{n}:=F_{n}(W_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})\setminus
F_{n}(A^{*}_{n},{\cal U}^{\prime}_{n,A_{n}})$. Then
$\mathbb{P}[E^{(1)}_{n}]\to 0$ as $n\to\infty$.

###### Proof.

Let $\varepsilon\in(0,(1/(2d))-9{\zeta})$. Suppose $E^{(1)}_{n}\cap
F_{n}(Q_{n}^{+},{\cal U}^{\prime}_{n,A_{n}})$ occurs. Then since ${\cal
U}^{\prime}_{n,W_{n}}\subset{\cal U}^{\prime}_{n,A_{n}}$,
$\mathbb{R}^{d}\setminus Z_{n}({\cal U}^{\prime}_{n,A_{n}})$ intersects with
$A^{*}_{n}\setminus W^{*}_{n}$, and therefore by (3.39), includes locations
distant at most $2Kn^{18{\zeta}}r_{n}^{2}$ from $\Gamma_{n}$. Also
$\Gamma_{n}\subset Z_{n}({\cal U}^{\prime}_{n,A_{n}})$, since
$\Gamma_{n}\subset W^{*}_{n}$.

Pick a location $x\in\overline{A_{n}^{*}\setminus Z_{n}({\cal
U}^{\prime}_{n,A_{n}})}$ of minimal distance from $\Gamma_{n}$. Then $x\notin
Q_{n}^{+}$, so the nearest point in $\Gamma_{n}$ to $x$ lies in the interior
of $H_{n,i}$ for some $i$. We claim that $x$ lies at the intersection of the
boundaries of $d$ of the balls making up $Z_{n}({\cal U}^{\prime}_{n,A_{n}})$;
this is proved similarly to the similar claim concerning $w$ in the proof of
Lemma 3.5. Moreover, $x$ lies in at most $k-1$ of the other balls making up
$Z_{n}({\cal U}^{\prime}_{n,A_{n}})$. Also $x$ does not lie in the interior of
$W_{n}^{*}$.

Thus if $E^{(1)}_{n}\cap F_{n}(Q_{n}^{+},{\cal U}^{\prime}_{n,A_{n}})$ occurs,
there must exist $d$ points $(x_{1},s_{1}),(x_{2},s_{2}),\ldots,$
$(x_{d},s_{d})$ of ${\cal U}_{n,A_{n}}$ such that $\cap_{i=1}^{d}\partial
B(x_{i},r_{n}s_{i})$ includes a point in $A_{n}^{*}$ but outside the interior
of $W_{n}^{*}$, within distance $2Kn^{18{\zeta}}r_{n}^{2}$ of $\Gamma_{n}$ and
in $B(x,r_{n}s)$ for at most $k-1$ of the other points $(x,s)$ of ${\cal
U}_{n,A_{n}}$. Hence by Markov’s inequality and the Mecke formula, we obtain
that

$\displaystyle\mathbb{P}[E^{(1)}_{n}\cap F_{n}(Q_{n}^{+},{\cal
U}^{\prime}_{n,A_{n}})]\leq I_{n,1}+I_{n,2}$ (3.54)

where, taking $Y_{1},\ldots,Y_{d}$ to be independent random variables with the
distribution of $Y$, and writing $f_{n,A}(x)$ for $\mathbb{P}[x\in Z_{n}({\cal
U}^{\prime}_{n,A})]$ for all $x\in\mathbb{R}^{d}$, we set

$\displaystyle I_{n,1}:=$ $\displaystyle
n^{d}\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}\mathbb{E}\,[h_{n}((x_{1},Y_{1}),(x_{1}+y_{2},Y_{2}),\ldots,(x_{1}+y_{d},Y_{d}))$
$\displaystyle\times{\bf
1}\\{q_{n}((x_{1},Y_{1}),(x_{1}+y_{2},Y_{2}),\ldots,(x_{1}+y_{d},Y_{d}))\in
A_{n}^{*}\cap(\Gamma_{n}\oplus B(o,2Kn^{18{\zeta}}r_{n}))\\}$
$\displaystyle\times(1-f_{n,A}(q_{n}((x_{1},Y_{1}),(x_{1}+y_{2},Y_{2}),\ldots,(x_{1}+y_{d},Y_{d}))))]dy_{d}\cdots
dy_{2}dx_{1},$

and $I_{n,2}$ is defined similarly with $p_{n}(\cdot)$ replacing
$q_{n}(\cdot)$. Then changing variables $y_{i}\mapsto r_{n}^{-1}y_{i}$ we have

$\displaystyle I_{n,1}=$ $\displaystyle
n^{d}r_{n}^{d(d-1)}\mathbb{E}\,\Big{[}\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}h_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d},Y_{d}))$
$\displaystyle\times{\bf
1}\\{x_{1}+q_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d}Y_{d}))\in
A_{n}^{*}\cap(\Gamma_{n}\oplus B(o,2Kn^{18{\zeta}}r_{n}))\\}$
$\displaystyle\times(1-f_{n,A}(x_{1}+q_{n}((o,Y_{1}),(r_{n}y_{2},Y_{2}),\ldots,(r_{n}y_{d},Y_{d}))))dy_{d}\cdots
dy_{2}dx_{1}\Big{]}.$

By Lemma 3.13, the last factor of $1-f_{n,A}(\cdot)$ is
$O(n^{\varepsilon-1+1/d})$ whenever the indicator function in the previous
factor is 1. Also using Fubini’s theorem we can take the integral over $x_{1}$
inside all the other integrals and integrate it out, getting a factor of
$O(n^{18{\zeta}}r_{n}^{2})$ by Lemma 3.11(a). Thus we obtain for a suitable
constant $c$ that

$\displaystyle I_{n,1}\leq
cn^{d}r_{n}^{d(d-1)}n^{18{\zeta}+\varepsilon-1+1/d}r_{n}^{2}\mathbb{E}\,\Big{[}\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}h((o,r_{n}Y_{1}),(r_{n}y_{2},r_{n}Y_{2}),\ldots,(r_{n}y_{d},r_{n}Y_{d}))$
$\displaystyle dy_{d}\cdots dy_{2}\Big{]}$
$\displaystyle=c(nr_{n}^{d})^{d-1+2/d}n^{18{\zeta}+\varepsilon-1/d}\mathbb{E}\,\Big{[}\int_{\mathbb{R}^{d}}\cdots\int_{\mathbb{R}^{d}}h((o,Y_{1}),(y_{2},Y_{2}),\ldots,(y_{d},Y_{d}))dy_{d}\cdots
dy_{2}\Big{]}.$

In the last line the expectation is finite by Lemma 3.4 and our moment
condition on $Y$. Also $(nr_{n}^{d})^{d}=O(n^{\varepsilon})$ by (2.5). Thus
$I_{n,1}=O(n^{2\varepsilon+18{\zeta}-1/d})$ so $I_{n,1}\to 0$ as $n\to\infty$,
and by an identical argument the same holds for $I_{n,2}$. Also
$\mathbb{P}[F_{n}(Q_{n}^{+},{\cal U}^{\prime}_{n,A_{n}})]\to 1$ by Lemma 3.14,
so using (3.54) we obtain that $\mathbb{P}[E^{(1)}_{n}]\to 0$, as required. ∎

######  Lemma 3.20.

Let $E^{(2)}_{n}:=F_{n}(A_{n}^{*},{\cal U}^{\prime}_{n,A_{n}})\setminus
F_{n}(W_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})$. Then
$\lim_{n\to\infty}\mathbb{P}[E^{(2)}_{n}]=0$.

###### Proof.

If the event $E^{(2)}_{n}$ occurs, then since $W^{*}_{n}\subset A_{n}^{*}$,
the set $W^{*}_{n}\cap Z_{n}({\cal U}^{\prime}_{n,A_{n}})\setminus Z_{n}({\cal
U}^{\prime}_{n,W_{n}})$ is nonempty. Hence there exists $(x,s)\in{\cal
U}^{\prime}_{n,A_{n}}\setminus{\cal U}^{\prime}_{n,W_{n}}$ with
$B(x,r_{n}s)\cap W^{*}_{n}\setminus Z_{n}({\cal
U}^{\prime}_{n,W_{n}})\neq\varnothing$. Therefore

$\displaystyle E^{(2)}_{n}\subset F_{n}(Z_{n}({\cal
U}^{\prime}_{n,A_{n}}\setminus{\cal U}^{\prime}_{n,W_{n}})\cap W_{n}^{*},{\cal
U}^{\prime}_{n,W_{n}})^{c}.$ (3.55)

Let $\varepsilon\in(0,(1/(2d))-(9+d/2){\zeta})$. Let ${\cal Q}_{n}:={\cal
U}^{\prime}_{n,A_{n}}\setminus{\cal U}^{\prime}_{n,W_{n}}$. Then ${\cal
U}^{\prime}_{n,W_{n}}$ and ${\cal Q}_{n}$ are independent Poisson processes
with intensity measures $n{\rm Leb}_{d}\otimes\mu_{Y}$ in
$W_{n}\times[0,n^{\zeta}]$, $(A_{n}\setminus W_{n})\times[0,n^{\zeta}]$
respectively. By Lemma 3.13 and the union bound, there is a constant $c$ such
that for any $m\in\mathbb{N}$ and any set of $m$ points
$(x_{1},t_{1}),\ldots,(x_{m},t_{m})$ in $\mathbb{R}^{d}\times[0,n^{\zeta}]$,
we have

$\displaystyle\mathbb{P}\left[F_{n}(\cup_{i=1}^{m}B(x_{i},r_{n}t_{i})\cap
W_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})^{c}\right]\leq
cmn^{d{\zeta}}n^{\varepsilon-(d-1)/d}.$

Let $N_{n}:={\cal Q}_{n}(\mathbb{R}^{d}\times[0,n^{\zeta}])$. By Lemma
3.11(a), $\mathbb{E}\,[N_{n}]=O(n^{1+18{\zeta}}r_{n}^{2})$, so that by
conditioning on ${\cal Q}_{n}$ we have

$\displaystyle\mathbb{P}[F_{n}(\cup_{(x,t)\in{\cal Q}_{n}}B(x,r_{n}t)\cap
W_{n}^{*},{\cal U}^{\prime}_{n,W_{n}})^{c}]$ $\displaystyle\leq
cn^{\varepsilon+d{\zeta}-1+1/d}\mathbb{E}\,[N_{n}]$
$\displaystyle=O(n^{(18+d){\zeta}+2\varepsilon-1/d}),$

which tends to zero by the choice of $\varepsilon$. Hence by (3.55),
$\mathbb{P}[E^{(2)}_{n}]\to 0$. ∎

To complete the proof of Theorem 2.10, we shall break $\partial A$ into
finitely many pieces, with each piece contained in a single chart. We would
like to write the probability that all of $\partial A$ is covered as the
product of probabilities for each piece, but to achieve the independence
needed for this, we need to remove a region near the boundary of each piece.
By separate estimates we can show the removed regions are covered with high
probability, and this is the content of the next lemma.

With $\Gamma$ and $\partial\Gamma$ as in Section 3.4, define the sets
${\Delta}_{n}:=\partial\Gamma\oplus B(o,n^{29{\zeta}}r_{n})$ and
${\Delta}_{n}^{+}:=\partial\Gamma\oplus B(o,n^{49{\zeta}}r_{n}).$

######  Lemma 3.21.

It is the case that $\lim_{t\to\infty}F_{n}({\Delta}_{n}^{+}\cap A,{\cal
U}^{\prime}_{n,A})=1$.

###### Proof.

Let $\varepsilon\in(0,(1/d)-98{\zeta})$. Since we assume
$\kappa(\partial\Gamma,r)=O(r^{2-d})$ as $r\downarrow 0$, for each $n$ we can
take $x_{n,1},\ldots,x_{n,k(n)}\in\mathbb{R}^{d}$ with
$\partial\Gamma\subset\cup_{i=1}^{k(n)}B(x_{n,i},n^{49{\zeta}}r_{n})$, and
with $k(n)=O((n^{49{\zeta}}r_{n})^{2-d})$. Then
${\Delta}_{n}^{+}\subset\cup_{i=1}^{k(n)}B(x_{n,i},2n^{49{\zeta}}r_{n})$. For
each $i\in\\{1,\ldots,k(n)\\}$, we can cover the ball
$B(x_{n,i},2n^{49{\zeta}}r_{n})$ with $O(n^{49d{\zeta}})$ smaller balls of
radius $r_{n}$. Then we end up with balls of radius $r_{n}$, denoted
$B_{n,1},\ldots,B_{n,m(n)}$ say, such that
${\Delta}_{n}^{+}\subset\cup_{i=1}^{m(n)}B_{n,i}$ and
$m(n)=O(r_{n}^{2-d}n^{98{\zeta}})$. By (3.44) from Lemma 3.13, and the union
bound,

$\displaystyle\mathbb{P}[\cup_{i=1}^{m(n)}(F_{n}(B_{n,i}\cap A,{\cal
U}^{\prime}_{n,A})^{c})]=O(r_{n}^{2-d}n^{98{\zeta}+\varepsilon-1+1/d})=O(n^{98{\zeta}+\varepsilon-1/d}),$

which tends to zero. ∎

Given $n>0$, define the sets
$\Gamma^{(n^{29{\zeta}}r_{n})}:=\Gamma\setminus{\Delta}_{n}$ and

$\Gamma^{(n^{29{\zeta}}r_{n})}_{n^{\zeta}r_{n}}:=(\Gamma^{(n^{29{\zeta}}r_{n})}\oplus
B(o,n^{\zeta}r_{n}))\cap
A;~{}~{}~{}~{}~{}\Gamma_{n^{\zeta}r_{n}}:=(\Gamma\oplus
B(o,n^{\zeta}r_{n}))\cap A,$

and define the event
$F_{n}^{\Gamma}:=F_{n}(\Gamma^{(n^{29{\zeta}}r_{n})}_{n^{\zeta}r_{n}},{\cal
U}^{\prime}_{n,A})$.

Note that the definition of $F_{n}^{\Gamma}$ does not depend on the choice of
chart. This fact will be needed for the last stage of the proof of Theorem
2.10. Lemma 3.24 below shows that $\mathbb{P}[F_{n}^{\Gamma}]$ is well
approximated by $\mathbb{P}[F_{n}(A_{n}^{*},{\cal U}^{\prime}_{n,A_{n}})]$ and
we have already determined the limiting behaviour of the latter. We prepare
for the proof of Lemma 3.24 with two geometrical lemmas.

######  Lemma 3.22.

For all large enough $n$, it is the case that
$\Gamma_{n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}\subset A_{n}^{*}$.

###### Proof.

Let $x\in\Gamma^{(n^{29{\zeta}}r_{n})}_{n^{\zeta}r_{n}}$, and take
$y\in\Gamma^{(n^{29{\zeta}}r_{n})}$ with $\|x-y\|\leq n^{\zeta}r_{n}$. Writing
$y=(u,\phi(u))$ with $u\in U$, we claim that $\,{\rm dist}(u,\partial
U)\geq(1/2)n^{29{\zeta}}r_{n}$. Indeed, if we had $\,{\rm dist}(u,\partial
U)<(1/2)n^{29{\zeta}}r_{n}$, then we could take $w\in\partial U$ with
$\|u-w\|<(1/2)n^{29{\zeta}}r_{n}$. Then $(w,\phi(w))\in\partial\Gamma$ and by
(3.37), $|\phi(w)-\phi(u)|\leq(1/4)n^{29{\zeta}}r_{n}$, so

$\|(u,\phi(u))-(w,\phi(w))\|\leq\|u-w\|+|\phi(u)-\phi(w)|\leq(3/4)n^{29{\zeta}}r_{n},$

contradicting the assumption that $y\in\Gamma^{(n^{29{\zeta}}r_{n})}$, so the
claim is justified.

Writing $x=(v,s)$ with $v\in\mathbb{R}^{d-1}$, and $s\in\mathbb{R}$, we have
$\|v-u\|\leq\|x-y\|\leq n^{\zeta}r_{n}$, so $\,{\rm dist}(v,\partial
U)\geq(1/2)n^{29{\zeta}}r_{n}-n^{\zeta}r_{n}$, and hence $v\in U_{n}^{-}$,
provided $n$ is big enough ($U_{n}^{-}$ was defined shortly after (3.38).)
Also $|\phi(v)-\phi(u)|\leq n^{\zeta}r_{n}/4$ by (3.37), so
$|\phi_{n}(v)-\phi(u)|\leq n^{\zeta}r_{n}/2$, provided $n$ is big enough, by
(3.39). Also $|s-\phi(u)|\leq\|x-y\|\leq n^{\zeta}r_{n}$, so
$|s-\phi_{n}(v)|\leq(3/2)n^{\zeta}r_{n}.$ Therefore $x\in A^{*}_{n}$ by
(3.40). ∎

######  Lemma 3.23.

For all large enough $n$, we have (a) $[A_{n}^{*}\oplus
B(o,4n^{\zeta}r_{n})]\cap A\subset A_{n}$, and (b) $[A_{n}^{*}\oplus
B(o,4n^{\zeta}r_{n})]\cap\partial A\subset\Gamma$, and (c)
$[\Gamma_{n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}\oplus
B(o,4n^{\zeta}r_{n})]\cap\partial A\subset\Gamma$.

###### Proof.

Let $x\in A_{n}^{*}$. Write $x=(u,z)$ with $u\in U_{n}^{-}$ and
$\phi_{n}(u)-3n^{\zeta}r_{n}/2\leq z\leq\phi(u)$.

Let $y\in B(x,4n^{\zeta}r_{n})\cap A$, and write $y=(v,s)$ with
$v\in\mathbb{R}^{d-1}$ and $s\in\mathbb{R}$. Then $\|v-u\|\leq
4n^{\zeta}r_{n}$ so provided $n$ is big enough, $v\in U_{n}$. Also $|s-z|\leq
4n^{\zeta}r_{n}$, and $|\phi(v)-\phi(u)|\leq n^{\zeta}r_{n}$ by (3.37), so

$|s-\phi(v)|\leq|s-z|+|z-\phi(u)|+|\phi(u)-\phi(v)|\leq
4n^{\zeta}r_{n}+2n^{\zeta}r_{n}+n^{\zeta}r_{n},$

and since $y\in A$, by (3.35) and (3.36) we must have $0\leq s\leq\phi(v)$,
provided $n$ is big enough. Therefore $y=(v,s)\in A_{n}$, which gives us (a).

If also $y\in\partial A$, then $\phi(v)=s$, so $y\in\Gamma$. Hence we have
part (b). Then by Lemma 3.22 we also have part (c). ∎

######  Lemma 3.24.

It is the case that $\mathbb{P}[F_{n}^{\Gamma}\triangle F_{n}(A_{n}^{*},{\cal
U}^{\prime}_{n,A_{n}})]\to 0$ as $n\to\infty$.

###### Proof.

Since $\Gamma_{n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}\subset A_{n}^{*}$ by
Lemma 3.22, and moreover ${\cal U}^{\prime}_{n,A_{n}}\subset{\cal
U}^{\prime}_{n,A}$, it follows that $F_{n}(A_{n}^{*},{\cal
U}^{\prime}_{n,A_{n}})\subset
F_{n}(\Gamma^{(n^{29{\zeta}}r_{n})}_{n^{\zeta}r_{n}},{\cal
U}^{\prime}_{n,A})=F_{n}^{\Gamma}$. Therefore it suffices to prove that

$\displaystyle\mathbb{P}[F_{n}(\Gamma^{(n^{29\xi}r_{n})}_{n^{\zeta}r_{n}},{\cal
U}^{\prime}_{n,A})\setminus F_{n}(A_{n}^{*},{\cal U}^{\prime}_{n,A_{n}})]\to
0.$ (3.56)

Let $\varepsilon>0$. Suppose event
$F_{n}(\Gamma^{(n^{29\xi}r_{n})}_{n^{\zeta}r_{n}},{\cal U}^{\prime}_{n,A})\cap
F_{n}({\Delta}_{n}^{+}\cap A,{\cal U}^{\prime}_{n,A})\setminus
F_{n}(A_{n}^{*},{\cal U}^{\prime}_{n,A_{n}})$ occurs. Choose $x\in
A_{n}^{*}\setminus Z_{n}({\cal U}^{\prime}_{n,A_{n}})$. Then by Lemma 3.23(a),
$B(x,n^{\zeta}r_{n})\cap A\subset A_{n}$. Hence ${\cal
U}^{\prime}_{n,A}\cap(B(x,n^{\zeta}r_{n})\times\mathbb{R}_{+})\subset{\cal
U}^{\prime}_{n,A_{n}}$, and therefore $x\notin Z_{n}({\cal
U}^{\prime}_{n,A})$.

Since we are assuming
$F_{n}(\Gamma^{(n^{29{\zeta}}r_{n})}_{n^{\zeta}r_{n}},{\cal
U}^{\prime}_{n,A})$ occurs, we therefore have $\,{\rm
dist}(x,\Gamma^{(n^{29{\zeta}}r_{n})})>n^{\zeta}r_{n}$. Since we also assume
$F_{n}({\Delta}_{n}^{+}\cap A,{\cal U}^{\prime}_{n,A})$, we also have $\,{\rm
dist}(x,\partial\Gamma)\geq n^{49{\zeta}}r_{n}$ and therefore $\,{\rm
dist}(x,{\Delta}_{n})=\,{\rm dist}(x,\partial\Gamma\oplus
B(o,n^{29{\zeta}}r_{n}))\geq n^{29{\zeta}}r_{n}$. Hence

$\,{\rm dist}(x,\Gamma)\geq\min(\,{\rm
dist}(x,\Gamma^{(n^{29{\zeta}}r_{n})}),\,{\rm dist}(x,\partial\Gamma\oplus
B(o,n^{29{\zeta}}r_{n}))>n^{\zeta}r_{n}.$

Moreover, by Lemma 3.23(b), $\,{\rm dist}(x,(\partial
A)\setminus\Gamma)>n^{\zeta}r_{n}$. Thus $\,{\rm dist}(x,\partial
A)>n^{\zeta}r_{n}$. Moreover, $\,{\rm dist}(x,\partial A)\leq\,{\rm
dist}(x,\Gamma)\leq 2n^{\zeta}r_{n}$ because $x\in A_{n}^{*}$, and therefore
$x\notin A^{[\varepsilon]}$ (provided $n$ is large enough) since
$\overline{A^{[\varepsilon]}}$ is compact and contained in $A^{o}$ (the set
$A^{[\varepsilon]}$ was defined in Section 2.) Therefore the event
$F_{n}(A^{(n^{\zeta}r_{n})}\setminus A^{[\varepsilon]},{\cal
U}^{\prime}_{n,A})^{c}$ occurs. Thus, for large enough $n$ we have the event
inclusion

$\displaystyle F_{n}(\Gamma^{(n^{29{\zeta}}r_{n})}_{r_{n}},{\cal
U}^{\prime}_{n,A})\cap F_{n}({\Delta}_{n}^{+}\cap A,{\cal
U}^{\prime}_{n,A})\setminus F_{n}(A_{n}^{*},{\cal
U}^{\prime}_{n,A_{n}})\subset F_{n}(A^{(n^{\zeta}r_{n})}\setminus
A^{[\varepsilon]},{\cal U}^{\prime}_{n,A})^{c}.$ (3.57)

By (2.5),

$\displaystyle\lim_{n\to\infty}(\omega_{d}\mathbb{E}\,[Y^{d}]nr_{n}^{d}-\log
n-(d+k-2)\log\log n)=\begin{cases}\beta&{\rm if}~{}d=2,k=1\\\ +\infty&{\rm
otherwise.}\end{cases}$ (3.58)

Hence by Lemma 3.2 applied to the set $A\setminus A^{[\varepsilon]}$, and
Lemma 3.3 we have that

$\displaystyle\liminf_{n\to\infty}\mathbb{P}[F_{n}(A^{(n^{\zeta}r_{n})}\setminus
A^{[\varepsilon]},{\cal
U}^{\prime}_{n,A})]=\liminf_{n\to\infty}\mathbb{P}[F_{n}(A^{(n^{\zeta}r_{n})}\setminus
A^{[\varepsilon]},{\cal U}^{\prime}_{n,\mathbb{R}^{d}})]$
$\displaystyle\geq\liminf_{n\to\infty}\mathbb{P}[F_{n}(A\setminus
A^{[\varepsilon]},{\cal U}_{n,\mathbb{R}^{d}})]$
$\displaystyle=\begin{cases}\exp\Big{(}-c_{d}\Big{(}\frac{(\mathbb{E}\,[Y^{d-1}])^{d}}{(\mathbb{E}\,[Y^{d}])^{d-1}}\Big{)}|A\setminus
A^{[\varepsilon]}|e^{-\beta}\Big{)}&{\rm if~{}}d=2,k=1\\\ 1&{\rm
otherwise.}\end{cases}$ (3.59)

Therefore since $\varepsilon$ can be arbitrarily small and $|A\setminus
A^{[\varepsilon]}|\to 0$ as $\varepsilon\downarrow 0$, the event displayed on
the left hand side of (3.57) has probability tending to zero. Then using Lemma
3.21 and Lemma 3.3, we have (3.56), which completes the proof. ∎

######  Corollary 3.25.

It is the case that
$\lim_{n\to\infty}\mathbb{P}[F_{n}^{\Gamma}]=\exp(-c_{d,Y}|\Gamma|e^{-\beta/2}).$

###### Proof.

By Lemmas 3.19 and 3.20, $\mathbb{P}[F_{n}(W^{*}_{n},{\cal
U}^{\prime}_{n,W_{n}})\triangle F_{n}(A_{n}^{*},{\cal
U}^{\prime}_{n,A_{n}})]\to 0$. Then by Lemma 3.24,
$\mathbb{P}[F_{n}^{\Gamma}\triangle F_{n}(A^{**}_{n},{\cal
U}^{\prime}_{n,W_{n}})]\to 0$, and now the result follows by Proposition 3.10.
∎

###### Proof of Theorem 2.2.

Let $x_{1},\ldots,x_{J}$ and $r(x_{1}),\ldots,r(x_{J})$ be as described at
(3.34). Set $\Gamma_{1}:=B(x_{1},r(x_{1}))\cap\partial A$, and for
$j=2,\ldots,J$, let

$\Gamma_{j}:=\overline{B(x_{j},r(x_{j}))\cap\partial
A\setminus\cup_{i=1}^{j-1}B(x_{i},r(x_{i}))},$

and $\partial\Gamma_{i}:=\Gamma_{i}\cap\overline{\partial
A\setminus\Gamma_{i}}$. Then $\Gamma_{1},\ldots,\Gamma_{J}$ comprise a finite
collection of closed sets in $\partial A$ with disjoint interiors, each of
which satisfies $\kappa(\partial\Gamma_{i},r)=O(r^{2-d})$ as $r\downarrow 0$,
and is contained in a single chart $B(x_{j},r(x_{j}))$, and with union
$\partial A$. For $1\leq i\leq J$, define $F_{n}^{\Gamma_{i}}$ analogously to
$F_{n}^{\Gamma}$ that is,
$F_{n}^{\Gamma_{i}}:=F_{n}(\Gamma_{i,n^{{\zeta}}r_{n}}^{(n^{29{\zeta}}r_{n})},{\cal
U}^{\prime}_{n,A})$ with

$\Gamma_{i,n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}:=\left(\left[\Gamma_{i}\setminus((\partial\Gamma_{i})\oplus
B(o,n^{29{\zeta}}r_{n}))\right]\oplus B(o,n^{\zeta}r_{n})\right)\cap A.$

First we claim that the following event inclusion holds:

$\displaystyle\cap_{i=1}^{J}F_{n}^{\Gamma_{i}}\cap
F_{n}(A^{(n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})\setminus F_{n}(A,{\cal
U}^{\prime}_{n,A})\subset\left(\cap_{i=1}^{J}F_{n}([(\partial\Gamma_{i})\oplus
B(o,n^{49{\zeta}}r_{n})]\cap A,{\cal U}^{\prime}_{n,A})\right)^{c}.$

Indeed, suppose $\cap_{i=1}^{J}F_{n}^{\Gamma_{i}}\cap
F_{n}(A^{(n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})\setminus F_{n}(A,{\cal
U}^{\prime}_{n,A})$ occurs, and choose $x\in A\setminus Z_{n}({\cal
U}^{\prime}_{n,A})$. Then $\,{\rm dist}(x,\partial A)\leq n^{\zeta}r_{n}$
since we assume $F_{n}(A^{(n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})$ occurs.
Then for some $i\in\\{1,\ldots,J\\}$ and some $y\in\Gamma_{i}$ we have
$\|x-y\|\leq n^{\zeta}r_{n}$. Since we assume $F_{n}^{\Gamma_{i}}$ occurs, we
have $x\notin\Gamma_{i,n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}$, and hence
$\,{\rm dist}(y,\partial\Gamma_{i})\leq n^{29{\zeta}}r_{n}$, so $\,{\rm
dist}(x,\partial\Gamma_{i})<n^{49{\zeta}}r_{n}$. Therefore
$F_{n}([(\partial\Gamma_{i})\oplus B(o,n^{49{\zeta}}r_{n})]\cap A,{\cal
U}^{\prime}_{n,A})$ fails to occur, justifying the claim.

By the preceding claim and and the union bound,

$\displaystyle\mathbb{P}[F_{n}(A,{\cal U}^{\prime}_{n,A})]$
$\displaystyle\leq\mathbb{P}[\cap_{i=1}^{J}F_{n}^{\Gamma_{i}}\cap
F_{n}(A^{(n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})]$
$\displaystyle\leq\mathbb{P}[F_{n}(A,{\cal
U}^{\prime}_{n,A})]+\sum_{i=1}^{J}\mathbb{P}[F_{n}([(\partial\Gamma_{i})\oplus
B(o,n^{49{\zeta}}r_{n})]\cap A,{\cal U}^{\prime}_{n,A})^{c}].$

By Lemma 3.21, $\mathbb{P}[F_{n}([(\partial\Gamma_{i})\oplus
B(o,n^{49{\zeta}}r_{n})]\cap A,{\cal U}^{\prime}_{n,A})]\to 1$ for each $i$.
Therefore

$\displaystyle\lim_{n\to\infty}\mathbb{P}[F_{n}(A,{\cal
U}^{\prime}_{n,A})]=\lim_{n\to\infty}\mathbb{P}[\cap_{i=1}^{J}F_{n}^{\Gamma_{i}}\cap
F_{n}(A^{(n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})],$ (3.60)

provided the last limit exists. By Corollary 3.25, we have for each $i$ that

$\displaystyle\lim_{t\to\infty}(\mathbb{P}[F^{\Gamma_{i}}_{n}])=\exp(-c_{d,Y}|\Gamma_{i}|e^{-\beta/2}).$
(3.61)

Also, we claim that for large enough $n$ the events $F_{n}^{\Gamma_{1}}$, …,
$F_{n}^{\Gamma_{J}}$ are mutually independent. Indeed, given distinct
$i,j\in\\{1,\ldots,J\\}$, if
$x\in\Gamma_{i,n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}$ and
$y\in\Gamma_{j,n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}$, then we can take
$y^{\prime}\in\Gamma_{j}\setminus(\partial\Gamma_{j}\oplus
B(o,n^{29{\zeta}}r_{n}))$ with $\|y^{\prime}-y\|\leq n^{\zeta}r_{n}$. If
$\|x-y\|\leq 3n^{\zeta}r_{n}$ then by the triangle inequality
$\|x-y^{\prime}\|\leq 4n^{\zeta}r_{n}$, but since
$y^{\prime}\notin\Gamma_{i}$, this would contradict Lemma 3.23(c). Therefore
$\|x-y\|>3n^{\zeta}r_{n}$, and hence the $n^{\zeta}r_{n}$-neighbourhoods of
$\Gamma_{i,n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}$ and of
$\Gamma_{j,n^{\zeta}r_{n}}^{(n^{29{\zeta}}r_{n})}$ are disjoint. This gives us
the independence claimed.

Now observe that $F_{n}(A^{(n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})\subset
F_{n}(A^{(4n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})$. We claim that

$\displaystyle\mathbb{P}[F_{n}(A^{(4n^{\zeta}r_{n})},{\cal
U}^{\prime}_{n,A})\setminus F_{n}(A^{(n^{\zeta}r_{n})},{\cal
U}^{\prime}_{n,A})]\to 0~{}~{}~{}{\rm as}~{}n\to\infty.$ (3.62)

Indeed, given $\varepsilon>0$, for large $n$ the probability on the left side
of (3.62) is bounded by $\mathbb{P}[F_{n}(A^{(n^{\zeta}r_{n})}\setminus
A^{[\varepsilon]},{\cal U}^{\prime}_{n,A})^{c}]$, and by (3.59) the limsup of
the latter probability can be made arbitrarily small by the choice of
$\varepsilon$. Hence by Lemma 3.2 and (3.58), Lemma 3.3 and the fact that
$c_{2}=1$,

$\displaystyle\lim_{n\to\infty}\mathbb{P}[F_{n}(A^{(4n^{\zeta}r_{n})},{\cal
U}^{\prime}_{n,A})]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}[F_{n}(A^{(n^{\zeta}r_{n})},{\cal
U}^{\prime}_{n,A})]$
$\displaystyle=\lim_{n\to\infty}\mathbb{P}[F_{n}(A^{(n^{\zeta}r_{n})},{\cal
U}^{\prime}_{n,\mathbb{R}^{d}})]$
$\displaystyle=\begin{cases}\exp\Big{(}-\Big{(}\frac{(\mathbb{E}\,[Y])^{2}}{\mathbb{E}\,[Y^{2}]}\Big{)}|A|e^{-\beta}\Big{)}&{\rm
if}~{}d=2,k=1\\\ 1&{\rm otherwise}.\end{cases}$ (3.63)

Moreover, by (3.60) and (3.62),

$\displaystyle\lim_{n\to\infty}\mathbb{P}[F_{n}(A,{\cal
U}^{\prime}_{n,A})]=\lim_{n\to\infty}\mathbb{P}[\cap_{i=1}^{J}F_{n}^{\Gamma_{i}}\cap
F_{n}(A^{(4n^{\zeta}r_{n})},{\cal U}^{\prime}_{n,A})],$ (3.64)

provided the last limit exists. However, the events in the right hand side of
(3.64) are mutually independent, so using (3.61), (3.41) and (3.63), we obtain
that

$\displaystyle\lim_{n\to\infty}\mathbb{P}[F_{n}(A,{\cal
U}^{\prime}_{n,A})]=\exp\Big{(}-c_{d,k}\Big{(}\frac{(\mathbb{E}\,[Y^{d-1}])^{d-1}}{(\mathbb{E}\,[Y^{d}])^{d-2+1/d}}\Big{)}|\partial
A|e^{-\beta/2}$
$\displaystyle-\Big{(}\frac{\mathbb{E}\,[Y]^{2}}{\mathbb{E}\,[Y^{2}]}\Big{)}|A|e^{-\beta}{\bf
1}_{\\{(d,k)=(2,1)\\}}\Big{)}.$

By Lemma 3.3, if we replace ${\cal U}^{\prime}_{n,A}$ with ${\cal U}_{n,A}$ on
the left, we get the same limit, i.e. (2.6) if $d\geq 3$ and (2.1) if d=2. ∎

##  4 Proof of results for the J-M model

######  Lemma 4.1 (Scaling lemma).

Suppose $L>0$ and $\rho=L^{d+1}$. Then $L\tilde{T}_{\rho}$ has the same
distribution as $\tilde{\tau}_{L}$. Moreover $LT_{\rho}$ has the same
distribution as $\tau_{L}$.

###### Proof.

For any $t>0$, setting $t^{\prime}=Lt$, we have

$\displaystyle A\subset\cup_{(x,s)\in{\cal
H}_{\rho}\cap(\mathbb{R}^{d}\times[0,t])}B(x,t-s)\Leftrightarrow
A_{L}\subset\cup_{(y,u)\in L{\cal
H}_{\rho}\cap(\mathbb{R}^{d}\times[0,t^{\prime}])}B(y,t^{\prime}-u)$

so that by (2.8),

$\displaystyle L\tilde{T}_{\rho}=\inf\\{t^{\prime}:A_{L}\subset\cup_{(y,u)\in
L{\cal
H}_{\rho}\cap(\mathbb{R}^{d}\times[0,t^{\prime}])}B(y,t^{\prime}-u)\\}.$

By our choice of $\rho$ and the Mapping theorem (see e.g. [6]), $L{\cal
H}_{\rho}$ is a homogeneous Poisson process of intensity $1$ on
$\mathbb{R}^{d}\times\mathbb{R}_{+}$, and therefore by the definition (2.10),
$L\tilde{T}_{\rho}$ has the same distribution as $\tilde{\tau}_{L}$.

The proof that $LT_{\rho}$ has the same distribution as $\tau_{L}$ is similar.
∎

###### Proof of Proposition 2.8.

Suppose for some $\beta\in\mathbb{R}$ that $(t_{\rho})_{\rho>0}$ satisfies

$\displaystyle\lim_{\rho\to\infty}\left(\omega_{d}\rho
t_{\rho}^{d+1}-d\log\rho-d^{2}\log\log\rho\right)=\beta,$ (4.1)

i.e.
$t_{\rho}=\left(\left(d\log\rho+d^{2}\log\log\rho+\beta+o(1)\right)/(\omega_{d}\rho)\right)^{1/(d+1)}$.

Write ${\cal
H}_{\rho}\cap(\mathbb{R}^{d}\times[0,t_{\rho}])=\\{(X_{i},S_{i})\\}_{i\in\mathbb{N}}$.
Then the point process ${\cal P}_{\rho
t_{\rho}}:=\\{X_{i}\\}_{i\in\mathbb{N}}$ is a homogeneous Poisson process of
intensity $\rho t_{\rho}$ in $\mathbb{R}^{d}$.

The balls $B(X_{i},t_{\rho}-S_{i})$, $i\in\mathbb{N},$ form a SPBM in
$\mathbb{R}^{d}$, and in the notation of Lemma 3.2, here we have
$\delta(\lambda)=t_{\rho}$, and $Y$ uniformly distributed over $[0,1]$, so
that $\alpha=\omega_{d}/(d+1)$, and $\lambda=\rho t_{\rho}$, so that

$\log\lambda=\left(\frac{d}{d+1}\right)\log\rho+\frac{\log\log\rho+\log(d/\omega_{d})}{d+1}+o(1),$

and $\log\log\lambda=\log\log\rho+\log(d/(d+1))+o(1)$. Hence,

$\displaystyle\alpha\delta(\lambda)^{d}\lambda-\log\lambda-(d-1)\log\log\lambda$
$\displaystyle=\frac{\omega_{d}\rho
t_{\rho}^{d+1}}{d+1}-\frac{d\log\rho}{d+1}-\frac{d^{2}\log\log\rho}{d+1}-\frac{\log(d/\omega_{d})}{d+1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-(d-1)\log\left(\frac{d}{d+1}\right)+o(1),$

and by (4.1), as $\rho\to\infty$ this tends to

$\frac{\beta}{d+1}-\log\left(\left(\frac{d^{d^{2}}}{\omega_{d}(d+1)^{d^{2}-1}}\right)^{1/(d+1)}\right).$

Hence we have the condition (3.2) from Lemma 3.2 (with $\beta$ on the right
hand side replaced by the last displayed expression). Also
$\mathbb{E}\,[Y^{d-1}]=1/d$ and $\mathbb{E}\,[Y^{d}]=1/(d+1)$.

We have $\tilde{T}_{\rho}\leq t_{\rho}$ if and only if
$A\subset\cup_{i\in\mathbb{N}}B(X_{i},t_{\rho}-S_{i})$. Therefore by taking
$k=1$ and $B_{\lambda}=A$ for all $\lambda$ in Lemma 3.2, we obtain that

$\mathbb{P}[\tilde{T}_{\rho}\leq
t_{\rho}]\to\exp\left(-c_{d}\left(\frac{(d+1)^{d-1}}{d^{d}}\right)\left(\frac{d^{d^{2}}}{\omega_{d}(d+1)^{d^{2}-1}}\right)^{1/(d+1)}|A|e^{-\beta/(d+1)}\right).$

This yields (2.11).

For (2.12), take $\rho=L^{d+1}$, so that $\log\log L=\log\log\rho-\log(d+1)$.
By Lemma 4.1, $L\tilde{T}_{\rho}$ has the same distribution as
$\tilde{\tau}_{L}$. Therefore

$\displaystyle\mathbb{P}\left[\omega_{d}\tilde{\tau}_{L}^{d+1}-d(d+1)\log
L-d^{2}\log\log L\leq\beta\right]$
$\displaystyle=\mathbb{P}\left[\omega_{d}\rho\tilde{T}_{\rho}^{d+1}-d\log\rho-d^{2}\log\log\rho+d^{2}\log(d+1)\leq\beta\right],$

and by (2.11), this tends to
$\exp(-c_{d}(d^{d}\omega_{d})^{-1/(d+1)}(d+1)^{d^{2}/(d+1)}|A|e^{-\beta/(d+1)}),$
and then (2.12) follows. ∎

We now derive Theorems 2.9 and 2.10 from Theorems 2.1 and 2.2, respectively.

###### Proof of Theorem 2.9.

We take $Y$ to be uniformly distributed on $[0,1]$. If the J-M model with
intensity $\rho$ runs for time $t$, the intensity of the resulting restricted
SPBM is $n=t\rho$, and the distribution of radii is that of $tY$. Let
$\beta\in\mathbb{R}$, and define

$\displaystyle t_{\rho}:=\Big{(}\frac{(2\log\rho+4\log\log\rho+\beta)\vee
0}{\pi\rho}\Big{)}^{1/3},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\rho>1,$

so that $\pi\rho t_{\rho}^{3}-2\log\rho-4\log\log\rho=\beta,$ for all large
enough $\rho$. Then as $\rho\to\infty$,

$\displaystyle\log(\rho t_{\rho})$
$\displaystyle=(2/3)\log\rho+(1/3)\log\log\rho+\log((2/\pi)^{1/3})+o(1),$

and $\log\log(\rho t_{\rho})=\log\log\rho+\log(2/3)+o(1).$ Therefore since
$\mathbb{E}\,[Y^{2}]=1/3$, we have

$\displaystyle\pi\rho t_{\rho}^{3}\mathbb{E}\,[Y^{2}]-\log(\rho
t_{\rho})-\log\log(\rho t_{\rho})=$
$\displaystyle\beta/3+(2/3)\log\rho+(4/3)\log\log\rho$
$\displaystyle-(2/3)\log\rho-(1/3)\log\log\rho-\log((2/\pi)^{1/3})$
$\displaystyle-\log\log\rho-\log(2/3)+o(1)$ $\displaystyle=$
$\displaystyle(\beta/3)-\log((16/(27\pi))^{1/3})+o(1),$

so applying Theorem 2.1 with $n=\rho t_{\rho}$, $r_{n}=t_{\rho}$, $k=1$ and
$Y$ uniform on $[0,1]$ gives us

$\displaystyle\mathbb{P}[\pi\rho
T_{\rho}^{3}-2\log\rho-4\log\log\rho\leq\beta]=\mathbb{P}[T_{\rho}\leq
t_{\rho}]=\mathbb{P}[A\subset Z_{n}(n,r_{n}Y)]$
$\displaystyle\to\exp\Big{(}-\Big{(}\frac{3}{4}\Big{)}|A|\Big{(}\frac{16}{27\pi}\Big{)}^{1/3}e^{-\beta/3}-\Big{(}\frac{3}{4\pi}\Big{)}^{1/2}|\partial
A|\Big{(}\frac{16}{27\pi}\Big{)}^{1/6}e^{-\beta/6}\Big{)},$

and hence (2.13).

For (2.14), let $\rho=L^{3}$, so that $\log\log\rho=\log\log L+\log(3)$. Then
by Lemma 4.1,

$\displaystyle\mathbb{P}[\pi\tau_{L}^{3}-6\log L-4\log\log
L\leq\beta]=\mathbb{P}[\pi\rho T_{\rho}^{3}-2\log\rho-4\log\log\rho+\log
81\leq\beta],$

and by (2.13), this converges to
$\exp(-(4\pi)^{-1/3}|A|3^{4/3}e^{-\beta/3}-(2\pi^{2})^{-1/3}|\partial
A|3^{4/6}e^{-\beta/6}),$ and hence (2.14). ∎

###### Proof of Theorem 2.10.

Let $\beta\in\mathbb{R}$ and for $\rho>1$ define

$\displaystyle
t_{\rho}:=\Big{(}\frac{(2(d-1)\log\rho+2d(d-1)\log\log\rho+\beta)\vee
0}{\omega_{d}\rho}\Big{)}^{1/(d+1)},$ (4.2)

so that $\omega_{d}\rho
t_{\rho}^{d+1}-2(d-1)\log\rho-2d(d-1)\log\log\rho=\beta,$ for all large
$\rho$. Then as $\rho\to\infty$,

$\displaystyle\log(\rho t_{\rho})=$
$\displaystyle\frac{d\log\rho+\log\log\rho}{d+1}+\log\Big{(}\Big{(}\frac{2(d-1)}{\omega_{d}}\Big{)}^{1/(d+1)}\Big{)}+o(1),$

and $\log\log(\rho t_{\rho})=\log\log\rho+\log(d/(d+1))+o(1).$ Therefore
taking $Y$ to be uniformly distributed on $[0,1]$, so that
$\mathbb{E}\,[Y^{d}]=1/(d+1)$, using (4.2) we obtain that

$\displaystyle\rho
t_{\rho}\omega_{d}t_{\rho}^{d}\mathbb{E}\,[Y^{d}]-(2-2/d)\log(\rho
t_{\rho})-2(d-2+1/d)\log\log(\rho t_{\rho})$
$\displaystyle=\frac{2(d-1)\log\rho+2d(d-1)\log\log\rho}{d+1}+\frac{\beta}{d+1}$
$\displaystyle-\frac{(2-2/d)(d\log\rho+\log\log\rho)}{d+1}-(2-2/d)\log\Big{(}\Big{(}\frac{2(d-1)}{\omega_{d}}\Big{)}^{1/(d+1)}\Big{)}$
$\displaystyle-2(d-2+1/d)\log\log\rho-2(d-2+1/d)\log(d/(d+1))+o(1)$
$\displaystyle=\frac{2d(d-1)-(2-2/d)-2(d-2+1/d)(d+1)}{d+1}\log\log\rho+\frac{\beta}{d+1}-c_{d}^{\prime\prime\prime}+o(1)$
$\displaystyle=\frac{\beta}{d+1}-c_{d}^{\prime\prime\prime}+o(1),$

where we set

$\displaystyle
c^{\prime\prime\prime}_{d}:=\log\Big{(}\Big{(}\frac{2(d-1)}{\omega_{d}}\Big{)}^{(2-2/d)/(d+1)}\Big{(}\frac{d}{d+1}\Big{)}^{2(d-2+1/d)}\Big{)}.$

Therefore applying Theorem 2.2 with $k=1$, $n=\rho t_{\rho}$ and
$r_{n}=t_{\rho}$, we obtain that if $d\geq 3$ then

$\displaystyle\mathbb{P}[\omega_{d}\rho
T_{\rho}^{d+1}-2(d-1)\log\rho-2d(d-1)\log\log\rho\leq\beta]=\mathbb{P}[T_{\rho}\leq
t_{\rho}]=\mathbb{P}[A\subset Z_{n}(n,r_{n}Y)]$
$\displaystyle\to\exp\Big{(}-\Big{(}\frac{c_{d,1}(d+1)^{d-2+1/d}|\partial
A|}{d^{d-1}}\Big{)}\Big{(}\frac{2(d-1)}{\omega_{d}}\Big{)}^{(1-1/d)/(d+1)}$
$\displaystyle\times\Big{(}\frac{d}{d+1}\Big{)}^{d-2+1/d}e^{-\beta/(2d+2)}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{c_{d-1}\omega_{d-1}^{2d-3}(d-1)^{d-2+(1/d)+((1-1/d)/(d+1))}|\partial
A|e^{-\beta/(2d+2)}}{\omega_{d-2}^{d-1}\omega_{d}^{d-2+(1/d)+((d-1)/(d(d+1)))}2^{1-(1/d)-(1-1/d)/(d+1)}d^{d-1}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{c_{d-1}\omega_{d-1}^{2d-3}(d-1)^{d(d-1)/(d+1)}|\partial
A|e^{-\beta/(2d+2)}}{\omega_{d-2}^{d-1}\omega_{d}^{d(d-1)/(d+1)}2^{(d-1)/(d+1)}d^{d-1}}\Big{)},$