#  In Press: Journal of Fuzzy Extension and Applications
A note on somewhat fuzzy continuity

Zanyar A. Ameen
Department of Mathematics, College of Science, University of Duhok, Duhok
42001, Iraq
<EMAIL_ADDRESS>

###### Abstract

Thangaraj and Balasubramanian introduced the so-called somewhat fuzzy
semicontinuous and somewhat fuzzy semiopen functions. Two years later, the
same authors defined two other types of functions called somewhat fuzzy
continuous and somewhat fuzzy open without indicating connections between
them. At first glance, we may easily conclude (from their definitions) that
every somewhat fuzzy continuous (resp. open) function is somewhat fuzzy
semicontinuous (resp. semiopen) but not conversely. In this note, we show that
they are equivalent. We further prove that somewhat fuzzy continuous functions
are weaker than fuzzy semicontinuous functions.

Key words and phrases: fuzzy continuous, somewhat continuous, somewhat fuzzy
continuous, somewhat fuzzy semicontinuous, somewhat fuzzy open
2020 MSC: 54A40, 03E72, 54A05, 54C99.

##  1 Introduction and Preliminaries

One of the fundamental concepts of all mathematics, particularly topology and
analysis, is continuity of functions, which means that ”small” changes in the
input result in ”small” changes in the output. After the introduction of the
fuzzy set, Chang defined the notion of fuzzy continuity [6] of functions
between fuzzy topological spaces in 1968. Following that, several
generalizations of fuzzy continuity were introduced. In 1981, Azad [5]
presented some classes of generalized fuzzy continuous functions. Fuzzy
semicontinuity is one of them that is weaker than fuzzy continuity. In
2001-2003, Thangaraj and Balasubramanian, respectively, introduced somewhat
fuzzy semicontinuous [8] and somewhat fuzzy continuous functions [9]. They
only discussed their relationship with fuzzy continuous functions. In the
present note, we show that somewhat fuzzy semicontinuity and somewhat fuzzy
continuity are the same, and somewhat fuzzy continuity is weaker than fuzzy
semicontinuity. It is worth stating somewhat continuity appeared in different
topological structures, (see, [2, 7, 9]). These classes of functions have a
great role in characterizing (soft) Baire spaces [3, 10] and weakly
equivalence spaces [7].

Let $X$ be a universe (domain). A mapping $\mu$ from $X$ to the unit interval
$\mathbb{I}$ is named a fuzzy set in $X$. The value $\mu(x)$ is called the
degree of the membership of $x$ in $\mu$ for each $x\in X$. The support of
$\mu$ is the set $\\{x\in X:\mu(x)>0\\}$. The complement of $\mu$ is, denoted
by $\mu^{c}$ (or $1-\mu$ if there is no confusion), given by
$\mu^{c}(x)=1-\mu(x)$ for all $x\in X$. Let $\\{\mu_{j}:j\in J\\}$ be a
collection of fuzzy sets in $X$, where $J$ is any index set. Then
$\bigvee\mu_{j}(x)=\sup\\{\mu_{j}(x):j\in J\\}$ and
$\bigwedge\mu_{j}(x)=\inf\\{\mu_{j}(x):j\in J\\}$ for each $x\in X$.

######  Definition 1.1.

[6] A collection $\mathcal{T}$ of fuzzy sets in $X$ is said to a fuzzy
topology on $X$ if

  1. (i)

$0,1\in\mathcal{T}$;

  2. (ii)

$\mu\wedge\lambda\in T$ whenever $\mu,\lambda\in\mathcal{T}$; and

  3. (iii)

$\bigvee\mu_{j}\in\mathcal{T}$ for any subcollection $\\{\mu_{j}:j\in J\\}$ of
$\mathcal{T}$.

The pair $(X,\mathcal{T})$ is called a fuzzy topological space. Members of
$\mathcal{T}$ are called fuzzy open subsets of $X$ and their complements are
called fuzzy closed sets.

Let $\lambda$ be a fuzzy set in $X$ and let $(X,\mathcal{T})$ be a fuzzy
topological space. The fuzzy interior of $\lambda$ is defined by
$\operatorname{Int}(\lambda):=\sup\\{\mu:\mu\leq\lambda,\mu\in T\\}$ and the
fuzzy closure of $\lambda$ is given by
$\operatorname{Cl}(\lambda):=\inf\\{\mu:\lambda\leq\mu,1-\mu\in T\\}$. The set
$\lambda$ is called fuzzy semiopen [5] if
$\lambda\leq\operatorname{Cl}(\operatorname{Int}(\lambda))$. The complement of
every fuzzy semiopen set is fuzzy semiclosed. Evidently, every fuzzy open set
is fuzzy semiopen but not the opposite (see, Example 2.4). The fuzzy semi-
interior of $\lambda$ is defined by
$\operatorname{Int}_{s}(\lambda):=\sup\\{\mu:\mu\leq\lambda,\mu\text{ is fuzzy
semiopen}\\}$ and the fuzzy semi-closure of $\lambda$ is given by
$\operatorname{Cl}_{s}(\lambda):=\inf\\{\mu:\lambda\leq\mu,\mu\text{ is fuzzy
semiclosed}\\}$.

######  Lemma 1.2.

[12] Let $\lambda$ be a fuzzy set in $X$ and let $(X,\mathcal{T})$ be a fuzzy
topological space. The following hold:

  1. (i)

$\operatorname{Int}(\lambda)\leq\operatorname{Int}_{s}(\lambda)$.

  2. (ii)

$\operatorname{Cl}_{s}(\lambda)\leq\operatorname{Cl}(\lambda)$.

To make our task easier, we introduce the following notions:

######  Definition 1.3.

Let $\lambda$ be a fuzzy set in $X$ and let $(X,\mathcal{T})$ be a fuzzy
topological space. Then

  1. (i)

$\lambda$ is called somewhat fuzzy open set if $\lambda=0$ or
$\operatorname{Int}(\lambda)\neq 0$.

  2. (ii)

$\lambda$ is called somewhat fuzzy semiopen set if $\lambda=0$ or
$\operatorname{Int}_{s}(\lambda)\neq 0$.

######  Definition 1.4.

A function $f$ from a fuzzy topological space $(X,\mathcal{T})$ into another
fuzzy topological space $(Y,\mathcal{U})$ is said to be

  1. (1)

fuzzy continuous [6] if $f^{-1}(\beta)\in\mathcal{T}$ for each
$\beta\in\mathcal{U}$.

  2. (2)

fuzzy semicontinuous [5] if $f^{-1}(\beta)$ is fuzzy semiopen for each
$\beta\in\mathcal{U}$.

  3. (3)

somewhat fuzzy continuous [9] if $f^{-1}(\beta)$ is somewhat fuzzy open for
each $\beta\in\mathcal{U}$.

  4. (4)

somewhat fuzzy semicontinuous [8] if $f^{-1}(\beta)$ is somewhat fuzzy
semiopen for each $\beta\in\mathcal{U}$.

######  Definition 1.5.

A function $f$ from a fuzzy topological space $(X,\mathcal{T})$ into another
fuzzy topological space $(Y,\mathcal{U})$ is said to be

  1. (1)

fuzzy open [11] if $f(\alpha)\in\mathcal{U}$ for each $\alpha\in\mathcal{T}$.

  2. (2)

fuzzy semiopen [5] if $f(\alpha)$ is fuzzy semiopen for each
$\alpha\in\mathcal{T}$.

  3. (3)

somewhat fuzzy open [9] if $f(\alpha)$ is somewhat fuzzy open for each
$\alpha\in\mathcal{T}$.

  4. (4)

somewhat fuzzy semiopen [8] if $f(\alpha)$ is somewhat fuzzy semiopen for each
$\alpha\in\mathcal{T}$.

For other undefined terminologies and notations stated in this note, we refer
the readers to [6, 11].

##  2 Main results

We start by proving several lemmas that help us achieve our goal.

######  Lemma 2.1.

Let $\lambda$ be a non-zero fuzzy set in a fuzzy topological space
$(X,\mathcal{T})$. Then $\lambda$ is fuzzy semiopen iff
$\operatorname{Cl}(\lambda)=\operatorname{Cl}(\operatorname{Int}(\lambda))$.

###### Proof.

Given a fuzzy semiopen set $\lambda$ such that $\lambda\neq 0$. Then
$\lambda\leq\operatorname{Cl}(\operatorname{Int}(\lambda))$ and so
$\operatorname{Cl}(\lambda)\leq\operatorname{Cl}(\operatorname{Int}(\lambda))$.
On the other hand, we always have $\operatorname{Int}(\lambda)\leq\lambda$.
Therefore
$\operatorname{Cl}(\operatorname{Int}(\lambda))\leq\operatorname{Cl}(\lambda)$.
Hence
$\operatorname{Cl}(\lambda)=\operatorname{Cl}(\operatorname{Int}(\lambda))$.

Conversely, assume that
$\operatorname{Cl}(\lambda)=\operatorname{Cl}(\operatorname{Int}(\lambda))$,
but $\lambda\leq\operatorname{Cl}(\lambda)$ always, so
$\lambda\leq\operatorname{Cl}(\operatorname{Int}(\lambda))$. Thus $\lambda$ is
fuzzy semiopen. ∎

######  Lemma 2.2.

Let $\lambda$ be a non-zero fuzzy set in the fuzzy topological space
$(X,\mathcal{T})$. If $\lambda$ is fuzzy semiopen, then
$\operatorname{Int}(\lambda)\neq 0$.

###### Proof.

Assume if possible that $\lambda$ is a fuzzy semiopen set for which
$\operatorname{Int}(\lambda)=0$, by Lemma 2.1, $\operatorname{Cl}(\lambda)=0$
implies that $\lambda=0$, a contradiction. ∎

######  Lemma 2.3.

Let $\lambda$ be a fuzzy set in a fuzzy topological space $(X,\mathcal{T})$.
Then $\lambda$ is somewhat fuzzy open iff it is somewhat fuzzy semiopen.

###### Proof.

Wolg we assume $\lambda\neq 0$, otherwise, the equivalence of these concepts
is clear. Since, by Lemma 1.2,
$\operatorname{Int}(\lambda)\leq\operatorname{Int}_{s}(\lambda)$, so this
concludes that somewhat fuzzy openness of $\lambda$ implies somewhat fuzzy
semiopenness.

Conversely, suppose that $\lambda$ is somewhat fuzzy semiopen. By definition,
for some $x\in\lambda$, there is a fuzzy semiopen set $\mu$ in $X$ such that
$x\in\mu\leq\lambda$. By Lemma 2.2, $\operatorname{Int}(\mu)\neq 0$.
Therefore, for some $x\in\lambda$, one can find a fuzzy open set $\beta$ such
that $x\in\beta\leq\mu\leq\lambda$. Hence, $\operatorname{Int}(\lambda)\neq
0$. This shows that $\lambda$ is somewhat fuzzy open. ∎

From the above lemmas, perhaps a relationship among the stated classes of
fuzzy open sets can be concluded.

fuzzy openfuzzy semiopensomewhat fuzzy opensomewhat fuzzy semiopen Diagram 1:
Relationships between generalized fuzzy open sets

None of the arrows in the diagram above can be reversed, as shown in the
following example:

######  Example 2.4.

Let $\mu,\lambda,\sigma,\alpha,\beta$ be fuzzy sets on $\mathbb{I}$ defined
as:

$\mu(x)=\begin{cases}0,&\text{ if }0\leq x\leq 1/2\\\ 2x-1,&\text{ if }1/2\leq
x\leq 1,\end{cases}$ $\lambda(x)=\begin{cases}1,&\text{ if }0\leq x\leq 1/4\\\
-4x+2,&\text{ if }1/4\leq x\leq 1/2\\\ 0,&\text{ if }1/2\leq x\leq
1,\end{cases}$ $\sigma(x)=\begin{cases}1,&\text{ if }0\leq x\leq 1/4\\\
-4x+2,&\text{ if }1/4\leq x\leq 1/2\\\ 2x-1,&\text{ if }1/2\leq x\leq
1,\end{cases}$ $\alpha(x)=\begin{cases}0,&\text{ if }0\leq x\leq 1/4\\\
4x/3-1/3,&\text{ if }1/4\leq x\leq 1,\end{cases}$

and

$\beta(x)=\begin{cases}2x,&\text{ if }0\leq x\leq 1/2\\\ 1,&\text{ if }1/2\leq
x\leq 1.\end{cases}$

The family $\mathcal{T}=\\{0,\mu,\lambda,\sigma,1\\}$ forms a fuzzy topology
on $\mathbb{I}$. The set $\alpha$ is fuzzy semiopen but not fuzzy open (see,
Example 4.5 in [5]). One can easily check that the set $\beta$ is somewhat
fuzzy open but not fuzzy semiopen.

Thangaraj and Balasubramanian showed that fuzzy continuity (openness) of a
function implies somewhat fuzzy continuity (openness), but in reality more
than that is true (from Diagram 1).

${\begin{tabular}[]{c}fuzzy continuous\\\ (fuzzy
open)\end{tabular}\implies}$${\begin{tabular}[]{c}fuzzy semicontinuous\\\
(fuzzy semiopen)\end{tabular}\implies}$ somewhat fuzzy continuous
---
(somewhat fuzzy open)
Diagram 2: The connections between the generalized fuzzy functions

######  Theorem 2.5.

For a function $f:(X,\mathcal{T})\rightarrow(Y,\mathcal{U})$, the following
statements hold:

  1. (1)

$f$ is somewhat fuzzy continuous iff it is somewhat fuzzy semicontinuous.

  2. (2)

$f$ is somewhat fuzzy open iff it is somewhat fuzzy semiopen.

###### Proof.

The Lemma 2.3 guarantees that $\operatorname{Int}(\mu)\neq 0$ iff
$\operatorname{Int}_{s}(\mu)\neq 0$ for every fuzzy set $\mu$ and the proof
can be concluded. ∎

We finish this note by pointing that the same result holds true for the
aforementioned functions in general and soft topology.

## References

  * [1] Z. A. Ameen, T. Al-shami, A. A. Azzam, A. Mhemdi, A novel fuzzy structure: Infra-fuzzy topological spaces, Journal of Function Spaces. 2022, Article ID 9778069, 2022, 1-11.
  * [2] Z. A. Ameen, B.A. Asaad, T.M. Al-shami, Soft somewhat continuous and soft somewhat open functions, TWMS Journal of Applied and Engineering Mathematics 13 (2) (2023), 792-806
  * [3] Z. A. Ameen, A. B. Khalaf, The invariance of soft Baire spaces under soft weak functions, Journal of Interdisciplinary Mathematics, 25 (5) (2022), 1295-1306.
  * [4] B. A. Asaad, T. M. Al-shami, Z. A. Ameen, On soft somewhere dense open functions and soft Baire spaces, Iraqi Journal of Science 64 (1) (2023), 373-384.
  * [5] K. K. Azad, On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82 (1981) 14–32.
  * [6] C. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24 (1) (1968) 182–190.
  * [7] K. R. Gentry, H. B. Hoyle III, Somewhat continuous functions, Czechoslovak Mathematical Journal, 21 (1) (1971), 5–12.
  * [8] G. Thangaraj, G. Balasubramanian, On somewhat fuzzy semicontinuous functions, Kybernetika, 37 (2) (2001) 165–170.
  * [9] G. Thangaraj, G. Balasubramanian, On somewhat fuzzy continuous functions, J. Fuzzy Math., 11 (2) (2003) 725–736.
  * [10] G. Thangaraj, R. Palani, Somewhat fuzzy continuity and fuzzy Baire spaces, Ann. Fuzzy. Math. and Info, 12 (1) (2016) 75–8.
  * [11] C. K. Wong, Fuzzy topology: Product and quotient theorems, Journal of Mathematical Analysis and Applications, 45 (1974) 512-521.
  * [12] T. H. Yalvaç, Semi-interior and semi-closure of a fuzzy set, Journal of Mathematical Analysis and Applications, 132 (1988) 356-36.


11footnotetext:  Supported by EPSRC grant EP/T028653/1

# On the rate of convergence in the Hall-Janson coverage theorem

Mathew D. Penrose1 and Xiaochuan Yang1
University of Bath and Brunel University

(September 3, 2024)

###### Abstract

Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with
$d\geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with
$r\geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with
finite $(2d)$-th moment. Let $A\subset\mathbb{R}^{d}$ be compact with a nice
boundary. Let $\alpha$ be the expected volume of a ball of radius $Y$, and
suppose $r=r(t)$ is chosen so that $\alpha tr^{d}-\log t-(d-1)\log\log t$ is a
constant indepenent of $t$. A classical result of Hall and of Janson
determines the (non-trivial) large-$t$ limit of the probability that $A$ is
fully covered by $Z$. In this paper we provide an $O((\log\log t)/\log t)$
bound on the rate of convergence in that result. With a slight adjustment to
$r(t)$, this can be improved to $O(1/\log t)$.

##  1 Introduction and statement of result

A classical problem in stochastic geometry [1, 3] is to determine the
probability of fully covering a region of Euclidean $d$-space with a union of
deterministic or random shapes centred on random points. There are
applications in topological data analysis, wireless communications, and also
in biology and chemistry, as discussed in e.g. [2, 3, 4, 9, 10]. An asymptotic
answer to this question was provided in the 1980s by Hall [2] and Janson [4].
In this paper we give a more quantitative version of that result.

Let $d\in\mathbb{N}$ and let $\mathbb{Q}$ be a probability measure on
$\mathbb{R}_{+}:=[0,\infty)$. Given $t>0$, let $\xi_{t}$ denote a Poisson
process in $\mathbb{R}^{d}\times\mathbb{R}_{+}$ with intensity measure
$t\lambda_{d}\otimes\mathbb{Q}$, where $\lambda_{d}$ denotes the Lebesgue
measure on $\mathbb{R}^{d}$. Given also $r>0$, define the occupied set
$Z(\xi_{t},r)$ and vacant set $V(\xi_{t},r)$ where for any simple counting
measure $\mu$ on $\mathbb{R}^{d}\times\mathbb{R}_{+}$ and $r\geq 0$, we define

$Z(\mu,r):=\cup_{(x,a)\in\mu}B(x,ra);\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ V(\mu,r)=\mathbb{R}^{d}\setminus Z(\mu,r),$

where for $x\in\mathbb{R}^{d}$ and $s>0$, we set
$B(x,s):=\\{y\in\mathbb{R}^{d}:\|y-x\|\leq s\\}$, and $\|\cdot\|$ is the
Euclidean norm. Then $Z(\xi_{t},r)$ is the occupied set of a spherical Poisson
Boolean model (SPBM) consisting of balls (known as grains) with independent
$\mathbb{Q}_{r}$-distributed random radii centred on the points of a
homogeneous Poisson process of intensity $t$ in $\mathbb{R}^{d}$. Here
$\mathbb{Q}_{r}$ denotes the distribution of $rY$, and $Y$ denotes a random
variable with distribution $\mathbb{Q}$.

Let $A$ denote a compact subset of $\mathbb{R}^{d}$, with interior denoted
$A^{o}$. Assume throughout that $\lambda_{d}(A)=\lambda_{d}(A^{o})>0$. This
paper is concerned with the asymptotic as $t\to\infty$ of the coverage
threshold $R_{t}$ defined by

$\displaystyle R_{t}:=\inf\\{r>0:A\subset Z(\xi_{t},r)\\}.$

Let $\theta_{d}:=\lambda_{d}(B_{1}(o))=\pi^{d/2}/\Gamma(1+d/2)$, the volume of
the $d$-dimensional unit ball. Here $o$ denotes the origin in
$\mathbb{R}^{d}$. The following result identifies the weak limit of $R_{t}$.

######  Theorem 1.1 (Hall, Janson).

Suppose $0<\mathbb{E}[Y^{d+\varepsilon}]<\infty$ for some $\varepsilon>0$. Set
$\alpha=\theta_{d}\mathbb{E}[Y^{d}]$. Let $\beta\in\mathbb{R}$ and define
$(r_{t})_{t>1}$ taking values in $\mathbb{R}_{+}$ by the relation

$\displaystyle\alpha tr_{t}^{d}=\left(\log t+(d-1)\log\log t+\beta\right)\vee
0,$ (1.1)

where we use $u\vee v$ for the maximum of two real numbers $u,v$. Then

$\displaystyle\lim_{t\to\infty}\mathbb{P}[R_{t}\leq
r_{t}]=\exp(-c_{d,Y}\lambda_{d}(A)e^{-\beta})$

where the constant $c_{d,Y}$ is given by

$\displaystyle
c_{d,Y}:=\frac{1}{d!}\Big{(}\frac{\sqrt{\pi}\Gamma(1+d/2)}{\Gamma((d+1)/2)}\Big{)}^{d-1}\frac{(\mathbb{E}[Y^{d-1}])^{d}}{(\mathbb{E}[Y^{d}])^{d-1}}.$
(1.2)

Theorem 1.1 is immediate from [2, Theorem 2] and can also be derived from the
result in [4]; see [9, Lemma 7.2] and [10, Lemma 3.2].

Another way to formulate this theorem is the convergence in distribution

$\alpha tR_{t}^{d}-\log t-(d-1)\log\log t\to G+\log(c_{d,Y}\lambda_{d}(A)),$

where $G$ is a standard Gumbel random variable.

Previous proofs of Theorem 1.1 do not provide any quantitative error bound for
finite $t$, or any rate of convergence. Our aim in this paper is to provide a
new proof (or at least, one with some new ingredients) that gives a rate of
convergence in the Hall-Janson theorem. We do so under the extra assumption
that $d\geq 2$, along with a stronger moment condition on $Y$ than in Theorem
1.1.

If $f(t)\in\mathbb{R}$ and $g(t)\in(0,\infty)$ are defined for all large
enough $t\in\mathbb{R}$, we say $f(t)=O(g(t))$ as $t\to\infty$ if
$\limsup_{t\to\infty}(|f(t)|/g(t))<\infty$.

######  Theorem 1.2.

Suppose $0<\mathbb{E}[Y^{2d}]<\infty$. Assume that $d\geq 2$, and let
$(r_{t})_{t>1}$ taking values in $\mathbb{R}_{+}$ be given by the relation

$\displaystyle\alpha tr_{t}^{d}=\left(\log t+(d-1)\log\log
t+\beta+\frac{(d-1)^{2}\log\log t}{\log t}\right)\vee 0.$ (1.3)

As $t\to\infty$, it holds that

$\displaystyle\mathbb{P}[R_{t}\leq
r_{t}]-\exp(-c_{d,Y}\lambda_{d}(A)e^{-\beta})=O\Big{(}\lambda_{d}(A^{\prime}\setminus
A)+\frac{1}{\log t}\Big{)},$ (1.4)

where $A^{\prime}:=A^{\prime}(t):=A\oplus B(o,\sqrt{r_{t}})$ and $\oplus$
denotes the Minkowski sum.

We note that if $A$ has a smooth or polytopal boundary, then the right hand
side of (1.4) is simply $O(1/(\log t))$.

We sketch the main ideas of our proof. Let us refer to the closure of each
connected component of $V(\xi_{t},r_{t})$ as a vacant region. A percolation
argument shows that with high probability, all vacant regions intersecting $A$
are confined to the slightly bigger region $A^{\prime}$. The lowest point of
each vacant region is a local minimum of the closure of $V(\xi_{t},r_{t})$,
and the mean number of local minima can be computed using the multivariate
Mecke formula. We can use a version of the Chen-Stein method from Penrose [8]
to show that the number of such local minima in $A$ or in $A^{\prime}$ is
approximately Poisson. It is this last step in the proof that takes the most
work; some of the earlier steps are similar to arguments in [4].

If we were to define $(r_{t})_{t>1}$ by the simpler relation (1.1) rather than
by (1.3), we would get (1.4) with $O((\log\log t)/\log t)$ instead of
$O(1/\log t)$ on the right. This can be done by following the proof of Theorem
1.2 with a few modifications, mainly in the proof of Proposition 3.1.

We now identify various possible extensions to our result that would be of
interest.

For fixed $k\in\mathbb{N}$, define $R_{t,k}$ to be the smallest $r$ such that
every point in $A$ is covered by at least $k$ of the balls
$\\{B(x,ra):(x,t)\in\xi_{t}\\}$ (so in particular $R_{t}=R_{t,1}$). There is a
more general version of Theorem 1.1 which provides a limiting distribution for
$\alpha tR_{t,k}^{d}-\log t-(d+k-2)\log\log t$. See e.g. [9, Lemma 7.2] and
[10, Lemma 3.2], which are based on [4]. Here we consider only the case with
$k=1$; our application of the Chen-Stein method requires us to find a coupling
of certain random variables, and (as we shall see) there is a natural coupling
in the case $k=1$. For $k\geq 2$, while it might be possible to find an
appropriate coupling, it would seem to make matters considerably more
complicated.

We restrict attention to grains that are Euclidean balls of possibly random
radius. More general distributions for the shapes of the grains in the Boolean
model are considered both by Hall [2] and by Janson [4]; for example, a
uniformly randomly rotated random polygon. It would be natural to try to
extend our result to these settings too.

Hall [2] shows that the number of vacant regions intersecting $A$ is
asymptotically Poisson with parameter $c_{d,Y}\lambda_{d}(A)e^{-\beta}$. It
may be possible to extend our method to give a bound on the rate of
convergence in total variation distance. Compared to the argument given here,
the main missing step would be to show that with high probability each vacant
region within $A^{\prime}$ has exactly one local minimum (for our result here,
all we need is the fact that each vacant region has at least one local
minimum).

Our method, in particular, the proof of Proposition 4.4, does not appear to
work for $d=1$.

In [9] and [10], analogous results to Theorem 1.1 are presented for the
restricted spherical Poisson Boolean model where one discards all grains
centred outside $A$, and considers coverage of $A$ by the remaining grains.
The proof of these results relies heavily on Theorem 1.1. Using Theorem 1.2,
it may be possible to obtain rates of convergence in those results too, at
least for $d\geq 3$. (The coverage process on the boundary of $A$ is a
$(d-1)$-dimensional SPBM, so we may need a rate of convergence for the
unrestrected SPBM in $d=1$ to derive a rate for the restricted SPBM in $d=2$.)
Let $B\subset\mathbb{R}^{d}$ be compact with $A\subset B^{o}$. Janson [4]
proves a variant of Theorem 1.1 where the Poisson point process $\xi_{t}$ is
replaced by a binomial point process in $B\otimes\mathbb{R}_{+}$, given by a
collection of $n$ random elements of $B\times\mathbb{R}_{+}$ with probability
distribution $\lambda_{d}(B)^{-1}\lambda_{d}|_{B}\otimes\mathbb{Q}$, with
$n\in\mathbb{N}$, as $n\to\infty$ with the scaling parameter $r$ now depending
on $n$. It would be of interest to provide a bound on the rate of convergence
for this result too.

It follows from our result that if $A$ has a nice boundary then there exist
positive finite constants $c^{\prime},t_{0}$ such that $|\mathbb{P}[R_{t}\leq
r_{t}]-\exp(-c_{d,Y}\lambda_{d}(A)e^{-\beta})|$ is bounded by $c^{\prime}/\log
t$ for all $t>t_{0}$. In principle, one could obtain numerical values for
$c^{\prime}$ and $t_{0}$ by going through our proof. For $d=2$, $Y\equiv 1$,
numerical upper and lower bounds for the probability of coverage can be found
in [3, 5]. Those bounds are not asymptotically sharp.

##  2 Reduction to a counting problem

We now embark on proving Theorem 1.2. We fix $d$, $\mathbb{Q}$, $A$ and
$\beta$, and assume from now on that $(r_{t})_{t>1}$ is given by (1.3).

We can and do assume without loss of generality that
$\mathbb{Q}(\\{0\\})=\mathbb{P}[Y=0]=0$. Indeed, the coverage capability of
$\mathrm{SPBM}(\xi_{t},r)$ in the case $\mathbb{P}[Y=0]>0$ is the same as
SPBM($\xi_{t}^{\prime},r$) where $\xi^{\prime}_{t}$ is a Poisson process with
intensity $\mathbb{P}[Y>0]tdx\mathbb{Q}[Y\in dy|Y>0]$.

For any $x,x^{\prime}\in\mathbb{R}^{d}$, we say $x$ is lower than $x^{\prime}$
if $\pi_{d}(x)<\pi_{d}(x^{\prime})$ where $\pi_{d}$ denotes the projection to
the $d$-th coordinate. Let $x_{1},\ldots,x_{d}$ be distinct vectors in
$\mathbb{R}^{d}$, $r>0$, and $a_{1},...,a_{d}\in\mathbb{R}_{+}$. For
$i\in[d]:=\\{1,\ldots,d\\}$, set $\mathbf{x}_{i}=(x_{i},a_{i})$. If
$\cap_{i\in[d]}\partial B(x_{i},ra_{i})$ contains exactly two points, write
these as
$p=p^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d}),q=q^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$
with $p$ lower than $q$, or with $p$ preceding $q$ lexicographically if
$\pi_{d}(p)=\pi_{d}(q)$. (Here $\partial D:=D\setminus D^{o}$ for any compact
$D\subset\mathbb{R}^{d}$.) Define the indicator function

$\displaystyle h^{(r)}(\mathbf{x}_{1},...,\mathbf{x}_{d}):=$
$\displaystyle{\bf 1}\\{\\#(\cap_{i\in[d]}\partial B(x_{i},ra_{i}))=2\\}$
$\displaystyle\times{\bf
1}\\{q^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\mbox{ is a local minimum of
\leavevmode\nobreak\
}\mathbb{R}^{d}\setminus\cup_{i\in[d]}B(x_{i},ra_{i})^{o}\\},$ (2.1)

where we say a point $x$ is a local minimum of a set $D\subset\mathbb{R}^{d}$
if $x\in D$, and for some neighbourhood $U$ of $x$, $x$ is lower than any
other point of $D\cap U$.

For any Borel-measurable set $D\subset\mathbb{R}^{d}$, $r>0$, and
$\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\in\mathbb{R}^{d}\times\mathbb{R}_{+}$,
define

$h^{(r)}_{D}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d}):=h^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d}){\bf
1}\\{q^{(r)}(\mathbf{x}_{1},...,\mathbf{x}_{d})\in D\\}.$

With $\sum_{\mathbf{x}_{1},...,\mathbf{x}_{d}\in\xi_{t}}$ denoting sum over
ordered $d$-tuples of distinct points $\mathbf{x}_{1},...,\mathbf{x}_{d}$
drawn from the Poisson process $\xi_{t}$, define the random variable

$\displaystyle
F_{D,t}:=\frac{1}{d!}\sum_{\mathbf{x}_{1},...,\mathbf{x}_{d}\in\xi_{t}}h^{(r_{t})}_{D}(\mathbf{x}_{1},...,\mathbf{x}_{d}){\bf
1}\\{q^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\in
V(\xi_{t}\setminus\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\\},r_{t})\\}.$ (2.2)

In this section we show that we can approximate $\mathbb{P}[R_{t}\leq r_{t}]$
from above with $\mathbb{P}[F_{A^{o},t}=0]$, and from below with
$\mathbb{P}[F_{A^{\prime},t}=0]$ minus a small correction.

Before starting on this, we give two alternative characterizations of the
‘local minimum’ condition in the definition of $h^{(r)}(\cdot)$ at (2.1).
Define the unit vector ${e}_{d}:=(0,0,\ldots,0,1)$,

Given distinct $q,x_{1},\ldots,x_{d}\in\mathbb{R}^{d}$, we define the
indicator function $g(q,x_{1},\ldots,x_{d})$ to be 1, if and only if the
following composite hyperplane condition holds:

(i) for any hyperplane $H$ that contains $q$ but not $q+{e}_{d}$ at least one
of $x_{1},\ldots,x_{d}$ is lower than $H$; and

(ii) there is no hyperplane $H$ containing both $q$ and $q+e_{d}$ such that
all points of $\\{x_{1},\ldots,x_{d}\\}\setminus H$ lie on the same side of
$H$.

Moreover, we define the indicator function $\tilde{g}(q,x_{1},\ldots,x_{d})$
to be 1 if and only if $e_{d}\in{\rm Cone}(q-x_{1},\ldots,q-x_{d})^{o}$.

######  Lemma 2.1.

Let $r,a_{1},\ldots,a_{d}\in(0,\infty)$. Then for all
$(x_{1},\ldots,x_{d})\in(\mathbb{R}^{d})^{d}$ such that
$\\#(\cap_{i=1}^{d}\partial B(x_{i},ra_{i}))=2$, setting
$\mathbf{x}_{i}:=(x_{i},a_{i})$ for each $i\in[d]$, we have that

$\displaystyle
h^{(r)}(x_{1},\ldots,x_{d})=g(q^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d}),x_{1},\ldots,x_{d})=\tilde{g}(q^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d}),x_{1},\ldots,x_{d}).$
(2.3)

###### Proof.

Suppose $x_{1},\ldots,x_{d}\in\mathbb{R}^{d}$ with $\\#\cap_{i=1}^{d}\partial
B(x_{i},ra_{i})=2$ and set $q:=q^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$.
We show first that $h^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\leq
g(q,x_{1},\ldots,x_{d})$. Suppose that $g(q,x_{1},\ldots,x_{d})=0$. Then
either (i) or (ii) in the definition of the hyperplane condition fails. We
claim that in either case $h^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})=0$.

Suppose there is a hyperplane $H$ containing $q$ but not $q+{e}_{d}$ with no
$x_{i}$’s below $H$. Let $e$ be a unit vector orthogonal to $H$ with $\langle
e,e_{d}\rangle<0$. Then for all $b>0$ and all $i\in[d]$, $q+be$ lies below $q$
and hence not on the same side of $H$ as $x_{i}$, so that $\|q+be-
x_{i}\|>\|q-x_{i}\|=ra_{i}$. Hence $q+be$ lies in
$\mathbb{R}^{d}\setminus\cup_{i=1}^{d}B(x_{i},ra_{i})$ and below $q$, and
therefore $q$ is not a local minimum of
$\mathbb{R}^{d}\setminus\cup_{i=1}^{d}B(x_{i},ra_{i})^{o}$ so
$h^{(r)}(x_{1},\ldots,x_{d})=0$.

Suppose instead that there exists a hyperplane $H$ with $q,q+e_{d}\in H$ and
all points of $\\{x_{1},\ldots,x_{d}\\}\setminus H$ on the same side of $H$.
Then taking $e$ orthogonal to $H$ and away from all $x_{i}$, we have for all
$b>0$ that $q+be\in\mathbb{R}^{d}\setminus\cup_{i=1}^{d}B(x_{i},ra_{i})$ and
hence $h^{(r)}(x_{1},\ldots,x_{d})=0$. Thus our earlier claim is justified,
and thus $h^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\leq
g(q,x_{1},\ldots,x_{d})$.

Conversely, if $h^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})=0$ then $q$ is
not a local minimum of
$\mathbb{R}^{d}\setminus\cup_{i\in[d]}B(x_{i},ra_{i})^{o}$, so there is a
sequence $(z_{n})_{n\in\mathbb{N}}$ taking values in
$\mathbb{R}^{d}\setminus\\{o\\}$ with $z_{n}\to o$ as $n\to\infty$, such that
$\langle z_{n},e_{d}\rangle\leq 0$ and
$q+z_{n}\notin\cup_{i=1}^{d}B(x_{i},ra_{i})$. Taking
$u_{n}:=\|z_{n}\|^{-1}z_{n}$ and taking a subsequence if needed, we have
$u_{n}\to u$ for some $u\in\mathbb{S}_{d-1}:=\partial B(o,1)$ with $\langle
u,e_{d}\rangle\leq 0$. Also we must have $\langle u,q-x_{i}\rangle\geq 0$ for
each $i\in[d]$, since otherwise there is $i$ such that for large $n$ we have
$q+z_{n}\in B(x_{i},ra_{i})$. Hence, writing $H$ for the hyperplane through
$q$ perpendicular to $u$, if $q+e_{d}\notin H$ we must have that no $x_{i}$
lies below $H$. If $q+e_{d}\in H$ then none of the $x_{i}$ lies strictly on
the same side of $H$ as $q+u$. Thus the hyperplane condition fails and
$g(q,x_{1},\ldots,x_{d})=0$. Thus $g(q,x_{1},\ldots,x_{d})\leq
h^{(r)}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$, and combined the earlier claim
this gives us the first equality of (2.3).

Next we show that $\tilde{g}(v,x_{1},\ldots,x_{d})\leq
g(v,x_{1},\ldots,x_{d})$. For $i\in[d]$ set $u_{i}:=q-x_{i}$. Suppose
$e_{d}\in{\rm Cone}(u_{1},\ldots,u_{d})^{o}$. Write
$e_{d}=\sum_{i=1}^{d}b_{i}u_{i}$ with $b_{i}>0$ for all $i\in[d]$. Then for
any hyperplane $H$ containing $q$ but not $q+e_{d}$, setting $f$ to be a
downward unit vector orthogonal to $H$ we have

$0>\langle f,e_{d}\rangle=\sum_{i=1}^{d}b_{i}\langle f,u_{i}\rangle$

so for at least one $i$ we have $\langle f,u_{i}\rangle<0$ and hence $x_{i}$
lies below $h$.

If there is a hyperplane $H$ containing $q$ and $q+e_{d}$ with all points of
$\\{x_{1},\ldots,x_{d}\\}\setminus H$ on the same side of $H$, let $f$ be
orthogonal to $H$ with $\langle f,u_{i}\rangle\leq 0$ for all $i\in[d]$. Then
$0=\langle f,e_{d}\rangle=\sum_{i=1}^{d}b_{i}\langle f,u_{i}\rangle$, so
$\langle f,u_{i}\rangle=0$ for al $i$. This would imply that
$x_{1},\ldots,x_{d}$ all lie in $H$ and $\cap_{i=1}^{d}\partial
B(x_{i},ra_{i})=\\{q\\}$, contrary to our assumption that
$\\#(\cap_{i=1}^{d}\partial B(x_{i},ra_{i}))=2$.

Thus the hyperplane condition holds so $g(q,x_{1},\ldots,x_{d})=1$. Therefore
$\tilde{g}(q,x_{1},\ldots,x_{d})\leq g(q,x_{1},\ldots,x_{d})$ as claimed.

To prove the reverse inequality, suppose $\tilde{g}(q,x_{1},\ldots,x_{d})=0$.

If $e_{d}\notin{\rm Cone}(u_{1},\ldots,u_{d})$, let $z$ be the nearest point
to $e_{d}$ of ${\rm Cone}(u_{1},\ldots,u_{d})$, and let $H$ be the hyperplane
through $z$ orthogonal to $z-e_{d}$. Then $o\in H$ and all of ${\rm
Cone}(u_{1},\ldots,u_{d})\setminus H$ is below $H$. Let $H^{\prime}:=H+q$.
Then for each $i\in[d]$, $q+u_{i}$ is on or below $H^{\prime}$ so
$x_{i}=q-u_{i}$ is not below $H^{\prime}$. Hence the hyperplane condition
fails and $g(q,x_{1},\ldots,x_{d})=0$.

If $e_{d}\in\partial{\rm Cone}(u_{1},\ldots,u_{d})$, let $H$ be a supporting
hyperplane to ${\rm Cone}(u_{1},\ldots,u_{d})$ with $e_{d}\in H$. Then $o\in
H$. Let $H^{\prime}=H+q$. Then $q,q+e_{d}\in H^{\prime}$ and all points of
$\\{x_{1},\ldots,x_{d}\\}\setminus H^{\prime}$ lie on the same side of
$H^{\prime}$. Thus the hyperplane condition fails so again
$g(q,x_{1},\ldots,x_{d})=0$. Thus we have verifed the second equality of
(2.3). ∎

######  Lemma 2.2.

For each $t>1$ with $r_{t}>0$ we have $\mathbb{P}[R_{t}\leq
r_{t}]\leq\mathbb{P}[F_{A^{o},t}=0].$

###### Proof.

Suppose $F_{A^{o},t}>0$. Then we can find distinct
$\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\in\xi_{t}$ with
$h^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})=1$ such that (setting
$p=p^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$ and
$q=q^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$) we have $q\in
V(\xi_{t}\setminus\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\\},r_{t})$ and $q\in
A^{o}$. Then we have for all all small enough $b>0$ that $q+b(q-p)$ lies in
$V(\xi_{t},r_{t})\cap A$. Therefore $R_{t}>r_{t}$, and the result follows. ∎

The other direction is more delicate, and will require some preparation. We
first prove a result about the geometry of intersections of spheres with
centres in general position (Corollary 2.4 below). A result along these lines
is also seen in [4, Lemma 4.1].

Given $k\in[d]$, we define a $k$-dimensional sphere to be a set of the form
$\partial B(x,r)\cap H$, for some $x\in\mathbb{R}^{d}$, some
$(k+1)$-dimensional affine subspace $H$ of $\mathbb{R}^{d}$ containing $x$,
and some $r>0$.

######  Lemma 2.3.

Suppose $a_{1},\dots,a_{d+1}\in(0,\infty)$. Let $k\in[d+1]$.

(i) If $k\leq d$ then for $\lambda_{d}^{k}$-almost all
$(x_{1},\ldots,x_{k})\in(\mathbb{R}^{d})^{k}$ the set
$S_{k}:=S_{k}(x_{1},\ldots,x_{k}):=\cap_{i=1}^{k}\partial B(x_{i},a_{i})$ is
either a $(d-k)$-dimensional sphere or the empty set.

(ii) For $\lambda_{d}^{d+1}$-almost all
$(x_{1},\ldots,x_{d+1})\in(\mathbb{R}^{d})^{d+1}$ we have
$S_{d+1}=\varnothing$.

###### Proof.

The result is clear for $k=1$. Suppose $k=2$. Set $D=\|x_{1}-x_{2}\|$.

Suppose also $d=2$. Then $S_{2}=\varnothing$ if $D>a_{1}+a_{2}$ or
$D<|a_{1}-a_{2}|$. If $|a_{1}-a_{2}|<D<a_{1}+a_{2}$ then $S_{2}$ has two
points and they are mirror to each other in the line containing $x_{1},x_{2}$;
this is geometrically obvious and can be verified in the special case where
$x_{1}=(0,0)$ and $x_{2}=(D,0)$ by solving the two simultaneous equations for
the two coordinates of any point of intersection.

Now suppose $d\geq 3$ and $k=2$. Suppose $|a_{1}-a_{2}|<D<a_{1}+a_{2}$, and
assume without loss of generality that $x_{1}=o$ and $x_{2}=De_{1}$, where we
set $e_{1}:=(1,0,\ldots,0)$. Given $u\in\mathbb{R}$ and
$z\in\mathbb{R}^{d-1}$, we have

$(u,z)\in S_{2}\Longleftrightarrow(u,\|z\|,0\ldots,0)\in S_{2}.$

Therefore picking $u_{0}\in\mathbb{R},v>0$ such that $(u_{0},v,0,\ldots,0)\in
S_{2}$ (by the case $d=2$ already considered there is just one such choice of
$u_{0},v$), we have that $S_{2}$ is the intersection of $\partial
B(u_{0}e_{1},v)$ with the hyperplane perpendicular to $e_{1}$ through
$u_{0}e_{1}$, and therefore is a $(d-2)$-dimensional sphere.

Now suppose $d\geq k=3$. Then $S_{3}=S_{2}\cap\partial B(x_{3},a_{3})$ and by
the case already considered, for almost all $(x_{1},x_{2})$, $S_{2}$ is either
empty or a $(d-2)$-dimensional sphere contained in a $(d-1)$-dimensional
affine subspace $H$ of $\mathbb{R}^{d}$. In the latter case, since $\partial
B(x_{3},a_{3})\cap H$ is (for almost all $x_{3}$) either empty or a
$(d-2)$-dimensional sphere in $H$, by the case $k=2$ its intersection with
$S_{2}$ is, for almost all choices of $x_{3}$, either a $(d-3)$-dimensional
sphere or the empty set.

Continuing in this way (inductively in $k$) we obtain the result (i). Part
(ii) then follows from the fact that $S_{d}$ has at most two points for
$\lambda_{d}^{d}$-almost all choices of $(x_{1},\ldots,x_{d})$. ∎

######  Corollary 2.4.

Let $r>0$, $k\in[d]$. Almost surely, for all $k$-tuples of distinct
$\mathbf{x}_{1}=(x_{1},a_{1}),\ldots,\mathbf{x}_{k}=(x_{k},a_{k})\in\xi_{t}$
and satisfying $S_{k}\neq\varnothing$, where we set

$S_{k}:=S_{k}(\mathbf{x}_{1},...,\mathbf{x}_{k}):=\cap_{i=1}^{k}\partial
B(x_{i},ra_{i}),$

the set $S_{k}$ is a $(d-k)$-dimensional sphere. Also, almost surely,
$S_{d+1}(\mathbf{x}_{1},...,\mathbf{x}_{d+1})=\varnothing$ for distinct
$\mathbf{x}_{1},...,\mathbf{x}_{d+1}\in\xi_{t}$.

###### Proof.

This follows from Lemma 2.3 and application of the multivariate Mecke formula
(see e.g. [6]) and Fubini’s theorem. ∎

Next we use a percolation argument to show that with high probability, all
components of $V(\xi_{t},r_{t})$ intersecting $A$ are confined to the set
$A^{\prime}:=A^{\prime}(t):=A\oplus B(o,\sqrt{r_{t}})$.

######  Lemma 2.5.

Given $t>0$, let $\mathscr{V}_{t}$ be the event that every connected component
of the vacant set $V(\xi_{t},r_{t})$ that intersects $A$ is contained in
$A^{\prime}$. Then there exists $c>0$ such that for all large enough $t$,
$\mathbb{P}[\mathscr{V}_{t}]\geq 1-\exp(-ct^{1/(3d)})$.

###### Proof.

Let $\delta>0$ be chosen so that $\mathbb{P}[Y>\sqrt{d}\delta]>\delta$.
Partition $\mathbb{R}^{d}$ into cubes of side length $\delta r_{t}$ indexed by
$\mathbb{Z}^{d}$ such that for each $z\in\mathbb{Z}^{d}$, the cube indexed by
$z$ is centred on $\delta r_{t}z$. We say two cubes are connected if the index
lattice points $x,y$ satisfy $\|x-y\|_{\infty}=1$. We call a connected
collection of cubes a lattice animal. If $\mathscr{V}_{t}$ does not occur,
then there exists a lattice animal such that

  * 1)

at least one constituting cube $Q$ of the animal intersects $A$;

  * 2)

the cardinality of the collection is at least $\sqrt{r_{t}}/(\sqrt{d}\delta
r_{t})=r_{t}^{-1/2}/(\sqrt{d}\delta)$;

  * 3)

each constituting cube $Q$ of the animal satisfies $Q\cap
V(\xi_{t},r_{t})\neq\varnothing$.

Item 3) implies that $\xi_{t}(Q\times[\sqrt{d}\delta,\infty)])=0$ because
otherwise $Q\cap V(\xi_{t},r_{t})=\varnothing$.

By e.g. [7, Lemma 9.3], there is a constant $c_{1}\in(1,\infty)$ such that for
all $n\in\mathbb{N}$ the number of animals of cardinality $n$ containing the
origin is at most $c_{1}^{n}$. Hence the number of such animals satisfying 1)
is at most $c_{2}r_{t}^{-d}c_{1}^{n}$ for some further constant $c_{2}$, and
hence for $t$ large is at most $tc_{1}^{n}$. Provided $t$ is also large enough
so that $\delta^{d+1}tr_{t}^{d}>2\log c_{1}$ and
$r_{t}^{-1/2}/(d^{1/2}\delta)\geq t^{1/(3d)}$, setting $c_{3}=\log c_{1}$ and
applying the union bound leads to

$\displaystyle\mathbb{P}[\mathscr{V}_{t}^{c}]$ $\displaystyle\leq t\sum_{n\geq
r_{t}^{-1/2}/(d^{1/2}\delta)}c_{1}^{n}\exp(-nt(\delta
r_{t})^{d}\mathbb{P}[Y>\sqrt{d}\delta])$ $\displaystyle\leq t\sum_{n\geq
t^{1/(3d)}}\exp(-c_{3}n)$ $\displaystyle\leq 2t\exp(-c_{3}t^{1/(3d)}),$

and the result follows for a suitable choice of $c$. ∎

We are now ready to provide an inequality in the opposite direction to lemma
2.2.

######  Lemma 2.6.

For all $t$, it holds that
$\mathbb{P}[\\{R_{t}>r_{t}\\}\cap\mathscr{V}_{t}]\leq\mathbb{P}[F_{A^{\prime},t}>0].$

###### Proof.

Suppose that event $\mathscr{V}_{t}$ occurs, and $R_{t}>r_{t}$. Then we can
and do choose $z\in V(\xi_{t},r_{t})\cap A$. Let $V_{z}$ denote the closure of
the connected component of the vacant set $V(\xi_{t},r_{t})$ containing $z$.
Then, since we assume $\mathscr{V}_{t}$ occurs, $V_{z}\subset A^{\prime}$.

Let $q$ denote a lowest point of $V_{z}$, i.e.
$\pi_{d}(q)\leq\pi_{d}(x^{\prime})$ for all $x^{\prime}\in V_{z}$. Then $q\in
A^{\prime}$.

For any simple counting measure $\mu$ on $\mathbb{R}^{d}\times\mathbb{R}_{+}$
and $r>0$, we define SPBM$(\mu,r)$ to be the collection of all balls of the
form $B(x,ra)$ with $(x,a)\in\mu$.

We claim that almost surely, $q$ lies on the boundary of exactly $d$ balls in
$\mathrm{SPBM}(\xi_{t},r_{t})$. It is clear that $q\not\in B^{o}$ for any
$B\in\mathrm{SPBM}(\xi_{t},r_{t})$ and $q\not\in V(\xi_{t},r_{t})$; otherwise
either $q$ lies in the interior of the occupied set $Z(\xi_{t},r_{t})$ or $q$
is not a lowest point of $V_{z}$. Hence $q$ has to lie on at least one sphere.

Suppose that $q$ lies on exactly $m$ spheres with $1\leq m\leq d-1$. In
particular, it lies on the hyperplanes tangent to the spheres at $q$. Let $I$
denote the intersection of these hyperplanes. Then the dimension of $I$ is at
least 1. For some small $\varepsilon>0$, any point in $(B(q,\varepsilon)\cap
I)\setminus\\{q\\}$ lies in the interior of $V_{z}$. Necessarily there exists
a point $q^{\prime}\in B(q,\varepsilon)\cap I$ that is as low as $q$ such that
for some $\varepsilon^{\prime}>0$, we have
$B(q^{\prime},\varepsilon^{\prime})\subset V_{z}$. It follows that the point
$q^{\prime}-\varepsilon^{\prime}e_{d}/2\in V_{z}$ is strictly lower than $q$,
a contradiction. Also, by Corollary 2.4 the intersection of $d+1$ distinct
spheres is empty. This completes the proof of the claim.

By Corollary 2.4 with $k=d$ and $r=r_{t}$, there exist distinct
$\mathbf{x}_{1},...,\mathbf{x}_{d}\in\xi_{t}$ with
$S_{d}(\mathbf{x}_{1},...,\mathbf{x}_{d})=\\{q,p\\}$ where $p$ is the mirror
of $q$ with respect to the affine space generated by
$\mathbf{x}_{1},...,\mathbf{x}_{d}$ by symmetry of balls. For $i\in[d]$ we
write $\mathbf{x}_{i}=(x_{i},a_{i})$ with $x_{i}\in\mathbb{R}^{d}$ and
$a_{i}\in\mathbb{R}_{+}$. Observe that $q\in
V(\xi_{t}\setminus\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\\})$, since $q$ lies
on exactly $d$ spheres.

We claim that $p$ is strictly lower than $q$. Indeed, by convexity of balls,
the segment $[p,q]$ is contained in the ball associated with $\mathbf{x}_{j}$
for all $j\in[d]$, and any other point on the line containing $[p,q]$ is not
covered by any of the $d$ balls. If $p$ were not strictly lower than $q$, then
for all sufficiently small $\delta>0$,
$q+\delta(q-p)\in\mathbb{R}^{d}\setminus Z(\xi_{t},r_{t})$ and taking a
further point just below $q+\delta(q-p)$ woud give us a point in $V_{z}$ that
is lower than $q$, and hence a contradiction.

Next we claim that $h^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})=1$.
Indeed, since $q\in V(\xi_{t}\setminus\cup_{i=1}^{d}\mathbf{x}_{i})$ and $q$
is a lowest point of $V_{z}$, it must be a local minimum of the set
$\mathbb{R}^{d}\setminus\cup_{i=1}^{d}B(x_{i},r_{t}a_{i})^{o}$. Here we use
the fact that for all small $\delta>0$ the set
$B(q,\delta)\setminus\cup_{i=1}^{d}B(x_{i},r_{t}a_{i})^{o}$ is connected.

It follows from all the above that $F_{A^{\prime},t}>0$, since the tuple
$(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$ contributes to the sum on the right
hand side of (2.2). ∎

##  3 Convergence of expectation

In this section we prove the convergence of the expectation of $F_{D,t}$
(defined at (2.2)), as $t\to\infty$, when either $D=A$ or $D=A^{\prime}(t)$.
Recall from Section 1 that we set $\alpha:=\theta_{d}\mathbb{E}[Y^{d}]$. Given
$(a_{1},a_{2},\ldots,a_{d})\in\mathbb{R}_{+}^{d}$, define

$\displaystyle
G(a_{1},\ldots,a_{d}):=\frac{1}{d!}\int_{\mathbb{R}^{d(d-1)}}h^{(1)}((o,a_{1}),(x_{2},a_{2}),\ldots,(x_{d},a_{d}))\lambda_{d}^{d-1}(d(x_{2},\ldots,x_{d})),$
(3.1)

and define the constant
$c_{0}:=\int_{\mathbb{R}_{+}^{d}}G(a_{1},\ldots,a_{d})\prod_{i=1}^{d}\mathbb{Q}(da_{i})$
(this constant depends on $d$ and $\mathbb{Q}$).

######  Proposition 3.1.

Suppose for all $t>1$ that $D(t)\subset\mathbb{R}^{d}$ is Borel-measurable
with $\lambda_{d}(D(t))=O(1)$ as $t\to\infty$. then as $t\to\infty$,

$\displaystyle\mathbb{E}[F_{D(t),t}]-c_{0}\alpha^{1-d}\lambda_{d}(D(t))e^{-\beta}=O\Big{(}\frac{1}{\log
t}\Big{)}.$ (3.2)

###### Proof.

Notice that for all $x\in\mathbb{R}^{d}$ we have $\xi_{t}(\\{(y,b):x\in
B(y,r_{t}b)\\})$ is Poisson distributed with mean
$t\int_{\mathbb{R}_{+}}\theta_{d}(br_{t})^{d}\mathbb{Q}(db)=\alpha
tr_{t}^{d}$, implying $\mathbb{P}[x\in V(\xi_{t})]=e^{-\alpha tr_{t}^{d}}$.
Therefore by (2.2) and the multivariate Mecke formula, writing just $D$ for
$D(t)$, we have

$\displaystyle\mathbb{E}[F_{D,t}]=\frac{t^{d}}{d!}\int_{\mathbb{R}_{+}^{d}}\prod_{i=1}^{d}\mathbb{Q}(da_{i})\int_{(\mathbb{R}^{d})^{d}}h_{D}^{(r_{t})}((x_{1},a_{1}),\ldots,(x_{d},a_{d}))e^{-\alpha
tr_{t}^{d}}\lambda_{d}^{d}(d(x_{1},\ldots,x_{d})).$ (3.3)

It is easy to see that for $a_{1},\dots,a_{d}$ fixed,

$\displaystyle\int_{(\mathbb{R}^{d})^{d}}h_{D}^{(r_{t})}((x_{1},a_{1}),\ldots,(x_{d},a_{d}))\lambda_{d}^{d}(d(x_{1},\ldots,x_{d}))$
$\displaystyle=\lambda_{d}(D)\int_{\mathbb{R}^{d(d-1)}}h^{(r_{t})}((o,a_{1}),(x_{2},a_{2}),\ldots,(x_{d},a_{d}))\lambda_{d}^{d-1}(d(x_{2},\ldots,x_{d})),$

and moreover

$\displaystyle
h^{(r_{t})}((o,a_{1}),(x_{2},a_{2}),\ldots,(x_{d},a_{d}))=h^{(1)}((o,a_{1}),(r_{t}^{-1}x_{2},a_{2}),...,(r_{t}^{-1}x_{d},a_{d})).$

Therefore a change of variables and (3.1) gives us

$\displaystyle\int_{(\mathbb{R}^{d})^{d}}h_{D}^{(r_{t})}((x_{1},a_{1}),\ldots,(x_{d},a_{d}))\lambda_{d}^{d}(d(x_{1},\ldots,x_{d}))=d!r_{t}^{d(d-1)}\lambda_{d}(D)G(a_{1},\ldots,a_{d}).$

Then using the definition at (3.1) of $c_{0}$, and first (3.3) and then (1.3)
we obtain that for $t$ large,

$\displaystyle\mathbb{E}[F_{D,t}]$ $\displaystyle=t^{d}e^{-\alpha
tr_{t}^{d}}r_{t}^{d(d-1)}\lambda_{d}(D)c_{0}$
$\displaystyle=c_{0}\lambda_{d}(D)e^{-\beta}\Big{(}\frac{t^{d-1}}{(\log
t)^{d-1}}\Big{)}\exp\Big{(}-\frac{(d-1)^{2}\log\log t}{\log t}\Big{)}$
$\displaystyle\quad\quad\times\Big{(}\frac{\log t+(d-1)\log\log
t+\beta+(d-1)^{2}(\log\log t)/\log t}{\alpha t}\Big{)}^{d-1}$
$\displaystyle=c_{0}\lambda_{d}(D)e^{-\beta}\alpha^{1-d}\Big{(}1+O\Big{(}\frac{1}{\log
t}\Big{)}\Big{)},$

as required. ∎

######  Corollary 3.2.

As $t\to\infty$, it holds that

$\mathbb{E}[F_{A^{o},t}]-c_{0}\alpha^{1-d}\lambda_{d}(A)e^{-\beta}=O\Big{(}\frac{1}{\log
t}\Big{)}$

and

$\displaystyle\mathbb{E}[F_{A^{\prime},t}]-c_{0}\alpha^{1-d}\lambda_{d}(A)e^{-\beta}=O\Big{(}\lambda_{d}(A^{\prime}\setminus
A)+\frac{1}{\log t}\Big{)}.$

The proof of Theorem 1.2 in the next section will in fact show that
$\mathbb{P}[R_{t}\leq
r_{t}]\to\exp(-c_{0}\alpha^{1-d}\lambda_{d}(A)e^{-\beta})$ as $t\to\infty$.
Thus for consistency with the known result of Theorem 1.1 the constant $c_{0}$
must satisfy $c_{0}\alpha^{1-d}=c_{d,Y},$ where $c_{d,Y}$ was given at (1.2).
i.e.

$\displaystyle
c_{0}=\frac{(\mathbb{E}[Y^{d-1}])^{d}\theta_{d}^{d-1}}{d!}\Big{(}\frac{\sqrt{\pi}\Gamma(1+d/2)}{\Gamma((1+d)/2)}\Big{)}^{d-1}=\frac{(\mathbb{E}[Y^{d-1}])^{d}\pi^{(d^{2}-1)/2}}{d!(\Gamma((1+d)/2))^{d-1}}.$
(3.4)

Nevertheless, to make this paper more self-contained, we sketch an argument to
verify (3.4) directly.

######  Lemma 3.3 (Computation of $c_{0}$).

Given $s_{1},\ldots,s_{d}\in\mathbb{R}_{+}$, it holds that

$\displaystyle
G(s_{1},\ldots,s_{d})=\frac{\pi^{(d^{2}-1)/2}}{d!(\Gamma((1+d)/2))^{d-1}}\prod_{i=1}^{d}s_{i}^{d-1},$
(3.5)

and (3.4) holds.

###### Proof.

Clearly it suffices to prove (3.5), which implies (3.4).

First suppose $d=2$. Fix $s_{1},s_{2}>0$. For $0<u<\pi/2,0<v<\pi/2$ consider
an infinitesimal portion of $\partial B(o,s_{1})$ at angle between $u$ and
$u+du$ clockwise from the top, and an infinitesimal portion of $\partial
B(x_{2},s_{2})$ at angle between $v$ and $v+dv$ anticlockwise from the top.

Since the two portions are at an angle $(u+v)$ to each other, the integral
over $\lambda_{2}(dx_{2})$ of the indicator that these portions cross (making
a local minimum of the set $\mathbb{R}^{2}\setminus(B(o,s_{1})^{o}\cup
B(x_{2},s_{2})^{o})$) is $s_{1}s_{2}\sin(u+v)dxdy$

Also for $0<v<u<\pi/2$ consider a portion of $\partial B(o,s_{1})$ at angle
between $u$ and $u+du$ clockwise from the bottom, and a portion of $\partial
B(x_{2},s_{2})$ at angle between $v$ and $v+dv$ clockwise from the top. The
integral over $\lambda_{2}(dx_{2})$ of the indicator that the portions cross
(again making a local minimum of the set
$\mathbb{R}^{2}\setminus(B(o,s_{1})^{o}\cup B(x_{2},s_{2})^{o})$) is
$s_{1}s_{2}\sin(u-v)$. Hence

$\displaystyle\int_{\mathbb{R}^{2}}h^{(1)}((o,s_{1}),(u,s_{2}))\lambda_{2}(du)$
$\displaystyle=2s_{1}s_{2}\int_{0}^{\pi/2}\int_{0}^{\pi/2}\sin(u+v)dvdu$
$\displaystyle\quad+4s_{1}s_{2}\int_{0}^{\pi/2}\int_{0}^{u}\sin(u-v)dvdu$
$\displaystyle=4\pi s_{1}s_{2}\Big{(}1+\int_{0}^{\pi/2}(1-\cos
u)du\Big{)}=2\pi s_{1}s_{2}.$

Thus using the definition (3.1) we otain that $G(s_{1},s_{2})=\pi s_{1}s_{2}$,
which gives us the case $d=2$ of (3.5)

The above argument uses infinitesimals but can be made rigorous by
approximating the two circles $\partial D_{1}$ and $\partial D_{2}(u)$ with
regular polygons with $m$ sides and taking the limit $m\to\infty$. We omit the
details.

Next we verify (3.5) in general dimension $d\geq 3$. Fix
$s_{1},\ldots,s_{d}\in(0,\infty)$. For $i\in[d]$ and $x_{i}\in\mathbb{R}^{d}$
consider a portion of $\partial B(x_{i},s_{i})$ at locations $x_{i}+s_{i}e$,
$e\in du_{i}$, where $du_{i}$ is an infinitesimal $(d-1)$-dimensional volume
element of the $(d-1)$-dimensional unit sphere $\mathbb{S}_{d-1}$.

The integral over $\lambda_{d}(dx_{2})\cdots\lambda_{d}(x_{d})$ of the
indicator that these portions have a mutual point of intersection with each
other comes to $|{\rm
Det}(u_{1},\ldots,u_{d})|\prod_{=1}^{d}(s_{i}^{d-1}du_{i})$, and the absolute
value of the determinant is unchanged if we replace any $u_{i}$ with $-u_{i}$.

By Lemma 2.1, for the point of intersection of these portions to make a local
minimum of the set $\mathbb{R}^{d}\setminus\cup_{i=1}^{d}B(x_{i},s_{i})^{o}$
it necessary and sufficient that $e_{d}\in{\rm Cone}(u_{1},\dots,u_{d})^{o}$.
If we fix $u_{1},\ldots,u_{d}$ in general position on $\mathbb{S}_{d-1}$ then
we can write $e_{d}$ as a linear combination of $u_{1},\ldots,u_{d}$, and if
we are free to reverse the direction of $u_{1},\ldots,u_{d}$ then for exactly
one out of the $2^{d}$ choices of the directions for $u_{1},\ldots,u_{d}$ this
linear combination will have positive coefficients. Thus

$\displaystyle\int_{\mathbb{R}^{d(d-1)}}h^{(1)}((o,s_{1}),(x_{2},s_{2}),\ldots,(x_{d},s_{d}))\lambda_{d}^{d-1}(d(x_{2},\ldots,x_{d}))$
$\displaystyle=2^{-d}\Big{(}\prod_{i=1}^{d}s_{i}^{d-1}\Big{)}\int_{\mathbb{S}_{d-1}}\cdots\int_{\mathbb{S}_{d-1}}|{\rm
Det}(u_{1},\ldots,u_{d})|du_{d}\cdots du_{1}.$

By [4, Lemma 9.3] the last expresion comes to

$\displaystyle
2^{-d}\Big{(}\prod_{i=1}^{d}s_{i}^{d-1}\Big{)}(d\theta_{d})^{d}\pi^{-1/2}\Gamma(d/2)^{d}\Gamma((d+1)/2)^{1-d}.$

Therefore using the formula for $\theta_{d}$ and (3.1) we obtain that

$\displaystyle
G(s_{1},\ldots,s_{d})=\frac{d^{d}\big{(}\prod_{i=1}^{d}s_{i}^{d-1}\big{)}\pi^{d^{2}/2}}{d!2^{d}\Gamma(1+d/2)^{d}}\times\frac{\pi^{-1/2}\Gamma(d/2)^{d}}{\Gamma((d+1)/2)^{d-1}}=\frac{\big{(}\prod_{i=1}^{d}s_{i}^{d-1}\big{)}\pi^{(d^{2}-1)/2}}{d!\Gamma((d+1)/2)^{d-1}},$

confirming (3.5). ∎

##  4 Proof of Theorem 1.2

The final ingredient in our proof is the Chen-Stein method, more precisely, a
coupling bound from [8], Lemma 4.1 below, for Poisson approximation of Poisson
functionals.

Let $\mathbb{X}:=\mathbb{R}^{d}\times[0,\infty)$ and let $\mathbf{N}$ be the
space of finite subsets of $\mathbb{X}$. For $k\in\mathbb{N}$ let
$\mathbf{N}_{k}$ be the space of subsets of $\mathbb{X}$ with $k$ elements.
See [8] for details of the relevant $\sigma$-algebras.

Given random variables $X,Y$, denote the distribution of $X$ by
$\mathcal{L}(X)$ and define the total variation distance between
$\mathcal{L}(X)$ and $\mathcal{L}(Y)$ by
$d_{\mathrm{TV}}(X,Y):=\sup_{A}|\mathbb{P}[X\in A]-\mathbb{P}[Y\in A]|,$ where
the supremum is over all Borel subsets $A$ of $\mathbb{R}$. For
$\mu\in\mathbb{R}$ let $\mathsf{Po}_{\mu}$ denote a Poisson random variable
with mean $\mu$.

######  Lemma 4.1 ([8, Theorem 3.1]).

Let $k\in\mathbb{N}$ and let $f:\mathbf{N}_{k}\times\mathbf{N}\to\\{0,1\\}$ be
measurable. For $\xi\in\mathbf{N}$, define

$\displaystyle
F(\xi)=\sum_{\\{\psi\subset\xi:|\psi|=k\\}}f(\psi,\xi\setminus\psi).$ (4.1)

Let $t>0$ and let $W:=F(\xi_{t})$. For
$\mathbf{x}_{1},...,\mathbf{x}_{k}\in\mathbb{X}$, set
$\pi(\mathbf{x}_{1},...,\mathbf{x}_{k})=\mathbb{E}[f(\\{\mathbf{x}_{1},...,\mathbf{x}_{k}\\},\xi_{t})].$

Suppose that for $(\lambda_{d}\otimes\mathbb{Q})^{k}$-a.e.
$\bar{\mathbf{x}}=(\mathbf{x}_{1},...,\mathbf{x}_{k})\in\mathbb{X}^{k}$ with
$\pi(\mathbf{x}_{1},...,\mathbf{x}_{k})>0$ we can find coupled random
variables $U_{\bar{\mathbf{x}}},V_{\bar{\mathbf{x}}}$ such that

  * •

$\mathcal{L}(U_{\bar{\mathbf{x}}})=\mathcal{L}(W)$;

  * •

$\mathcal{L}(V_{\bar{\mathbf{x}}})=\mathcal{L}(F(\\{\mathbf{x}_{1},...,\mathbf{x}_{k}\\}\cup\xi_{t})-1|f(\\{\mathbf{x}_{1},...,\mathbf{x}_{k}\\},\xi_{t})=1)$.

Set
$\gamma:=\mathbb{E}[W]=(t^{k}/k!)\int_{\mathbb{X}^{k}}\pi(\bar{\mathbf{z}})(\lambda_{d}\times\mathbb{Q})^{k}(d\bar{\mathbf{z}})$.
Then with $\wedge$ denoting minimum,

$\displaystyle
d_{\mathrm{TV}}(W,\mathsf{Po}_{\gamma})\leq\frac{1\wedge\gamma^{-1}}{k!}\int_{\mathbb{X}^{k}}\mathbb{E}[|U_{\bar{\mathbf{z}}}-V_{\bar{\mathbf{z}}}|]\pi(\bar{\mathbf{z}})t^{k}(\lambda_{d}\otimes\mathbb{Q})^{k}(d\bar{\mathbf{z}}).$

We shall use Lemma 4.1 to show that $F_{A^{o},t}$ and $F_{A^{\prime},t}$
converge to a Poisson random variable with mean given by the limiting
expectation of Corollary 3.2, with a bound on the rate of convergence in total
variation distance. Given $t>1$, for $D=A$ or $D=A^{\prime}$, the random
variable $F_{D,t}$ defined at (2.2) takes the form of (4.1) with $k=d$,
because both $h_{D}^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$ and
$q^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})$ are symmetric in
$\mathbf{x}_{1},\ldots,\mathbf{x}_{d}$. In this case, the function $f$
appearing in (4.1) is given by

$\displaystyle
f(\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\\},\xi)=h_{D}^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})){\bf
1}\\{q^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\in V(\xi,r_{t})\\},$
(4.2)

and the function $\pi(\bar{\mathbf{x}})$ appearing in the statement of Lemma
4.1 is given by

$\displaystyle\pi(\bar{\mathbf{x}}):=h^{(r_{t})}_{D}(\bar{\mathbf{x}})e^{-t\alpha
r_{t}^{d}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bar{\mathbf{x}}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\in\mathbb{X}^{d},$
(4.3)

where $\alpha:=\theta_{d}\mathbb{E}[Y^{d}]$ as before.

In our context, the coupling is easy to find. Indeed, the conditional
distribution of $\xi_{t}$ is
$\mathcal{L}(\xi_{t}\setminus\\{\mathbf{x}=(x,a):q^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{k})\in
B(x,r_{t}a)\\})$ which, by the Thinning theorem (see e.g. [6]), is still a
Poisson process. Bounding the $L^{1}$ distance of the coupled random
variables, however, is a delicate task.

Given $t>1$, for $x\in\mathbb{R}^{d}$ we set

$\displaystyle B(x):=\\{(y,b)\in\mathbb{R}^{d}\times\mathbb{R}_{+}:\|y-x\|\leq
r_{t}b\\}.$ (4.4)

Given
$\bar{\mathbf{x}}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\in(\mathbb{R}^{d}\times\mathbb{R}_{+})^{d}$
and $I\subset[d]$, set $\mathbf{x}_{I}=(\mathbf{x}_{i},i\in I)$. Then the
desired coupling is obtained by taking

$\displaystyle
U_{\bar{\mathbf{x}}}=\sum_{\\{\mathbf{y}_{1},...,\mathbf{y}_{d}\\}\subset\xi_{t}}h^{(r_{t})}_{A}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\\{\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\\})\cap
B(q_{\bar{\mathbf{y}}})=\varnothing\\}$ (4.5)

and

$\displaystyle V_{\bar{\mathbf{x}}}$
$\displaystyle=\sum_{\\{\mathbf{y}_{1},...,\mathbf{y}_{d}\\}\subset\xi_{t}\cap
B(q_{\bar{\mathbf{x}}})^{c}}h^{(r_{t})}_{A}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\mathbf{y}_{[d]})\cap B(q_{\bar{\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing\\}{\bf
1}\\{\mathbf{x}_{1},...,\mathbf{x}_{d}\not\in B(q_{\bar{\mathbf{y}}})\\}$
$\displaystyle\quad\quad+\sum_{\varnothing\neq
I\subsetneq[d]}\sum_{\\{\mathbf{y}_{1},\ldots,\mathbf{y}_{d-|I|}\\}\in\xi_{t}\cap
B(q_{\bar{\mathbf{x}}})^{c}}h^{(r_{t})}_{A}(\mathbf{x}_{I},\mathbf{y}_{1},\ldots,\mathbf{y}_{d-|I|})$
$\displaystyle\hskip 56.9055pt\times{\bf
1}\\{(\xi_{t}\setminus\mathbf{y}_{[d-|I|]}\\}\cap
B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing\\}{\bf 1}\\{\mathbf{x}_{[d]\setminus
I}\cap B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})=\varnothing\\},$ (4.6)

where $q_{\bar{\mathbf{y}}}$ denotes the usual $q^{(r_{t})}$ (higher point of
the intersection of $d$ spheres) generated by $\mathbf{y}_{i}$’s, and
$q_{\overline{\mathbf{x}_{I}\mathbf{y}}}$ denotes the $q^{(r_{t})}$ generated
by $d$-tuples composed of $\mathbf{x}_{i},i\in I$ and
$(\mathbf{y}_{1},\ldots,\mathbf{y}_{d-|I|})$, and for $J\subset\mathbb{N}$ we
set $\mathbf{x}_{J}=(\mathbf{x}_{i}:i\in J)$ and
$\mathbf{y}_{J}=\\{\mathbf{y}_{i}:i\in J\\}$.

######  Lemma 4.2.

Given $t>1$, let $D=A^{o}$ or $D=A^{\prime}(t)$. With $F_{D,t}$ and
$\pi(\bar{\mathbf{x}})$ defined at (2.2) and (4.3) respectively, setting
$\gamma=\mathbb{E}[F_{D,t}]$ we have

$\displaystyle
d_{\mathrm{TV}}(F_{D,t},\mathsf{Po}_{\gamma})\leq\frac{1}{d!}\int_{\mathbb{X}^{d}}\mathbb{E}[{S}^{(1)}_{\bar{\mathbf{x}}}+{S}^{(2)}_{\bar{\mathbf{x}}}+{S}^{(3)}_{\bar{\mathbf{x}}}+{S}^{(4)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}}),$

where for
$\bar{\mathbf{x}}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})\in\mathbb{X}^{d}$,
and $i=1,2,3,4$, we define
$S^{(i)}_{\bar{\mathbf{x}}}:=S^{(i)}_{\bar{\mathbf{x}}}(D,t)$ by

$\displaystyle{S}^{(1)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{(y_{1},b_{1}),\ldots,(y_{d},b_{d})\\}\subset\xi_{t}}h^{(r_{t})}_{D}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\bar{\mathbf{y}}})=\varnothing\\}{\bf
1}\\{q_{\bar{\mathbf{x}}}\in\cup_{i=1}^{d}B(y_{i},r_{t}b_{i})\\};$
$\displaystyle{S}^{(2)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\\}\subset\xi_{t}}h^{(r_{t})}_{D}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\bar{\mathbf{y}}})=\varnothing\\}{\bf
1}\\{q_{\bar{\mathbf{y}}}\in\cup_{i=1}^{d}B(x_{i},r_{t}a_{i})\\};$
$\displaystyle{S}^{(3)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{\mathbf{y}_{1},...,\mathbf{y}_{d}\\}\subset\xi_{t}}h^{(r_{t})}_{D}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\mathbf{y})\cap B(q_{\bar{\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing,(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\bar{\mathbf{y}}})\cap B(q_{\bar{\mathbf{x}}})\neq\varnothing\\}$
$\displaystyle\quad\quad\quad\quad\times{\bf
1}\\{\mathbf{x}_{1},...,\mathbf{x}_{d}\not\in
B(q_{\bar{\mathbf{y}}}),\mathbf{y}_{1},...,\mathbf{y}_{d}\not\in
B(q_{\bar{\mathbf{x}}})\\};$ $\displaystyle{S}^{(4)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\varnothing\neq
I\subsetneq[d]}\sum_{\mathbf{y}=\\{\mathbf{y}_{1},...,\mathbf{y}_{d-|I|}\\}\subset\xi_{t}\setminus
B(q_{\bar{\mathbf{x}}})}h^{(r_{t})}_{D}(\mathbf{x}_{I},\mathbf{y}_{1},...,\mathbf{y}_{d-|I|})$
$\displaystyle\hskip 56.9055pt\times{\bf 1}\\{(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing\\}{\bf 1}\\{\mathbf{x}_{[d]\setminus
I}\cap B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})=\varnothing\\}.$

###### Proof.

We apply Lemma 4.1, with $U_{\bar{\mathbf{x}}}$ and $V_{\bar{\mathbf{x}}}$
given by (4.5) and (4.6) respectively. We need to estimate the difference
$|U_{\bar{\mathbf{x}}}-V_{\bar{\mathbf{x}}}|$. There are three sources of
change induced by adding and removing points:

  * i)

existence of $\mathbf{y}_{1},...,\mathbf{y}_{d}\in\xi_{t}$ with
$h^{(r_{t})}_{A}(\bar{\mathbf{y}})=1$ which contributes to
$U_{\bar{\mathbf{x}}}$ but not to $V_{\bar{\mathbf{x}}}$;

  * ii)

existence of $\mathbf{y}_{1},...,\mathbf{y}_{d}\in\xi_{t}$ with
$h^{(r_{t})}_{A}(\bar{\mathbf{y}})=1$ which contributes to
$V_{\bar{\mathbf{x}}}$ but not to $U_{\bar{\mathbf{x}}}$;

  * iii)

existence of $I$ with $\varnothing\neq I\subsetneq[d]$ and
$\mathbf{y}_{1},...,\mathbf{y}_{d-|I|}\in\xi_{t}$ satisfying

$h^{(r_{t})}(\mathbf{x}_{I},\mathbf{y}_{1},...,\mathbf{y}_{d-|I|})=1$

which contributes solely to $V_{\bar{\mathbf{x}}}$.

For i), there are two possibilities. Either
$q_{\bar{\mathbf{x}}}\in\cup_{i=1}^{d}B(y_{i},r_{t}b_{i})$, (in which case
some of the $\mathbf{y}_{i}$’s are removed in the coupled conditional point
process therefore not contributing to $V_{\bar{\mathbf{x}}}$) or
$q_{\bar{\mathbf{y}}}\in\cup_{i=1}^{d}B(x_{i},r_{t}a_{i})$ (in which case
$q_{\bar{\mathbf{y}}}$ is not in the vacant set in the coupled process
therefore not contributing to $V_{\bar{\mathbf{x}}}$ as well). Here we are
writing $\mathbf{y}_{i}=(y_{i},b_{i})$ with $y_{i}\in\mathbb{R}^{d}$ and
$b_{i}\in\mathbb{R}_{+}$ for each $i\in[d]$ and
$\bar{\mathbf{y}}:=(\mathbf{y}_{1},\ldots,\mathbf{y}_{d})$. These two cases
give rise to ${S}_{\bar{\mathbf{x}}}^{(1)}$ and
${S}_{\bar{\mathbf{x}}}^{(2)}$, respectively. In both cases,
$(\xi_{t}\setminus\mathbf{y})\cap B(q_{\bar{\mathbf{y}}})=\varnothing$ must
hold to guarantee that the tuple $\bar{\mathbf{y}}$ contributes to
$U_{\bar{\mathbf{x}}}$.

For ii), it must hold that $(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\bar{\mathbf{y}}})\cap B(q_{\bar{\mathbf{x}}})\neq\varnothing$ and
$(\xi_{t}\setminus\mathbf{y})\cap B(q_{\bar{\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing$, as well as
$\mathbf{x}_{1},...,\mathbf{x}_{d}\not\in B(q_{\bar{\mathbf{y}}})$. Also,
$\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\not\in B(q_{\bar{\mathbf{x}}})$;
otherwise the $\mathbf{y}_{i}$ in $B(q_{\mathbf{x}})$ would have been removed
due to conditioning. This gives rise to ${S}_{\bar{\mathbf{x}}}^{(3)}$.

For iii), we refer to the last terms of $V_{\bar{\mathbf{x}}}$. ∎

It remains to estimate the four integrals arising from Lemma 4.2.

######  Proposition 4.3.

For $t>1$, set either $D=A^{o}$ or $D=A^{\prime}(t)$. Then

$\displaystyle\int_{(\mathbb{R}^{d}\times\mathbb{R}_{+})^{d}}\mathbb{E}[{S}^{(1)}_{\bar{\mathbf{x}}}(D,t)]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=O\big{(}\frac{\log
t}{t}\big{)}.$ (4.7)

and

$\displaystyle\int_{(\mathbb{R}^{d}\times\mathbb{R}_{+})^{d}}\mathbb{E}[{S}^{(2)}_{\bar{\mathbf{x}}}(D,t)]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=O\big{(}\frac{\log
t}{t}\big{)}.$ (4.8)

###### Proof.

By the multivariate Mecke equation and then(1.3),

$\displaystyle\mathbb{E}[{S}^{(1)}_{\bar{\mathbf{x}}}]$
$\displaystyle\leq\frac{t^{d}}{d!}\int_{(\mathbb{R}^{d}\times\mathbb{R}_{+})^{d}}e^{-\alpha
tr_{t}^{d}}h^{(r_{t})}(\bar{\mathbf{y}}){\bf
1}\\{\cup_{i=1}^{d}\\{\mathbf{y}_{i}\in
B(q_{\mathbf{x}})\\}\\}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{y}})$
$\displaystyle=O\Big{(}\frac{t^{d}}{t(\log
t)^{d-1}}\Big{)}\int_{B(q_{\mathbf{x}})}\int_{(\mathbb{R}^{d}\times\mathbb{R}_{+})^{d-1}}h^{(r_{t})}(\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{d})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
(\lambda_{d}\otimes\mathbb{Q})^{d-1}(d(\mathbf{y}_{2},\ldots,\mathbf{y}_{d}))(\lambda_{d}\otimes\mathbb{Q})(d\mathbf{y}_{1}).$

Next we use Lemma 3.3 (alternatively one can use [10, Lemma 3.4] which is more
elementary) in the last line of the following display, to argue that there is
a constant $c$ such that for fixed values of
$b_{1},\ldots,b_{d}\in\mathbb{R}_{+}$ setting $y_{i}=y_{1}+r_{t}z_{i}$ we have

$\displaystyle\int_{(\mathbb{R}^{d})^{d-1}}h^{(r_{t})}((y_{1},b_{1}),\ldots,(y_{d},b_{d}))\lambda_{d}^{d-1}(d(y_{2},\ldots,y_{d}))$
$\displaystyle=\int_{(\mathbb{R}^{d})^{d-1}}h^{(r_{t})}((o,b_{1}),(r_{t}z_{2},b_{2}),\ldots,(r_{t}z_{d},b_{d}))r_{t}^{d(d-1)}\lambda_{d}^{d-1}(d(z_{2},\ldots,z_{d}))$
$\displaystyle=r_{t}^{d(d-1)}\int_{(\mathbb{R}^{d})^{d-1}}h^{(1)}((o,b_{1}),(z_{2},b_{2}),\ldots,(z_{d},b_{d}))\lambda_{d}^{d-1}(d(z_{2},\ldots,z_{d}))$
$\displaystyle=cr_{t}^{d(d-1)}\prod_{i=1}^{d}b_{i}^{d-1}.$

Therefore since we assume $\mathbb{E}[Y^{2d}]<\infty$ so that
$\mathbb{E}[Y^{d-1}]<\infty$, using the definition (4.4) of
$B(z),z\in\mathbb{R}^{d}$, we have

$\displaystyle\mathbb{E}[{S}_{\bar{\mathbf{x}}}^{(1)}]$
$\displaystyle=O\Big{(}\frac{t^{d-1}r_{t}^{d(d-1)}}{(\log
t)^{d-1}}\Big{)}\int_{B(q_{\bar{\mathbf{x}}})}b_{1}^{d-1}(\lambda_{d}\otimes\mathbb{Q})(d(y_{1},b_{1}))$
$\displaystyle=O\Big{(}\frac{t^{d-1}r_{t}^{d(d-1)}}{(\log
t)^{d-1}}\Big{)}\int_{\mathbb{R}_{+}}b_{1}^{d-1}\theta_{d}(r_{t}b_{1})^{d}\mathbb{Q}(db_{1}).$

The first factor in the above expression is $O(1)$ by (1.3). Since our
assumption $\mathbb{E}[Y^{2d}]<\infty$ implies $\mathbb{E}[Y^{2d-1}]<\infty$,
this shows that $\mathbb{E}[{S}_{\bar{\mathbf{x}}}^{(1)}]=O(r_{t}^{d})=O((\log
t)/t)$, uniformly over $\bar{\mathbf{x}}$. Moreover, by the Mecke formula and
then Proposition 3.1,

$\displaystyle
t^{d}\int_{\mathbb{X}^{d}}\pi(\bar{\mathbf{x}})(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=d!\mathbb{E}[F_{D,t}]=O(1),$

and (4.7) follows.

For (4.8) we note that by the Mecke formula and (4.3),

$\displaystyle\int_{\mathbb{X}^{d}}\mathbb{E}[{S}_{\bar{\mathbf{x}}}^{(2)}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\times\mathbb{Q})^{d}(d\bar{\mathbf{x}})$
$\displaystyle=\frac{t^{2d}}{d!}\int_{\mathbb{X}^{d}}\Big{(}\int_{\mathbb{X}^{d}}h^{(r_{t})}_{D}(\bar{\mathbf{y}})e^{-\alpha
tr_{t}^{d}}{\bf
1}\\{q_{\bar{\mathbf{y}}}\in\cup_{i=1}^{d}B(x_{i},r_{t}a_{i})\\}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{y}})\Big{)}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \times
h_{D}^{(r_{t})}(\bar{\mathbf{x}})e^{-\alpha
tr_{t}^{d}}(\lambda_{d}\otimes\mathbb{Q})^{d}(d((x_{1},a_{1}),\ldots,(x_{d},a_{d}))),$
(4.9)

whereas

$\displaystyle\int_{\mathbb{X}^{d}}\mathbb{E}[{S}_{\bar{\mathbf{x}}}^{(1)}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\times\mathbb{Q})^{d}(d\bar{\mathbf{x}})=\frac{t^{2d}}{d!}\int_{\mathbb{X}^{d}}\Big{(}\int_{\mathbb{X}^{d}}h^{(r_{t})}_{D}(\bar{\mathbf{y}})e^{-\alpha
tr_{t}^{d}}{\bf
1}\\{q_{\bar{\mathbf{x}}}\in\cup_{i=1}^{d}B(y_{i},r_{t}b_{i})\\}$
$\displaystyle(\lambda_{d}\otimes\mathbb{Q})^{d}(d((y_{1},b_{1}),\ldots,(y_{d},b_{d})))\Big{)}h_{D}^{(r_{t})}(\bar{\mathbf{x}})e^{-\alpha
tr_{t}}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}}).$ (4.10)

An application of Fubini’s theorem shows that the integrals in (4.9) and
(4.10) are the same, and therefore (4.8) follows from (4.7). ∎

The integral of ${S}_{\bar{\mathbf{x}}}^{(3)}$ is more delicate. The
$(\lambda_{d}\otimes\mathbb{Q})$-measure of $B(q_{\bar{\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})$ can be so small that the event of vacancy in
${S}^{(3)}_{\bar{\mathbf{x}}}$ does not provide much decay. We make a change
of variables and a geometric observation which gives the desired decay.

######  Proposition 4.4.

For $t>1$, set $D=A^{o}$ or $D=A^{\prime}$. As $t\to\infty$, it holds that

$\int_{\mathbb{X}^{d}}\mathbb{E}[{S}^{(3)}_{\bar{\mathbf{x}}}(D,t)]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=O((\log
t)^{-1}).$

###### Proof.

We write
${S}^{(3)}_{\bar{\mathbf{x}}}={S}^{(3a)}_{\bar{\mathbf{x}}}+{S}^{(3b)}_{\bar{\mathbf{x}}}$,
where we set

$\displaystyle{S}^{(3a)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\\}\subset\xi_{t}}h^{(r_{t})}_{D}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\mathbf{y})\cap B(q_{\bar{\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing,(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\bar{\mathbf{y}}})\cap B(q_{\bar{\mathbf{x}}})\neq\varnothing\\}$
$\displaystyle\quad\quad\quad\quad\times{\bf
1}\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\not\in
B(q_{\bar{\mathbf{y}}}),\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\not\in
B(q_{\bar{\mathbf{x}}}),\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\leq
2r_{t}\\};$ $\displaystyle{S}^{(3b)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\\}\subset\xi_{t}}h^{(r_{t})}_{D}(\bar{\mathbf{y}}){\bf
1}\\{(\xi_{t}\setminus\mathbf{y})\cap B(q_{\bar{\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing,(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\bar{\mathbf{y}}})\cap B(q_{\bar{\mathbf{x}}})\neq\varnothing\\}$
$\displaystyle\quad\quad\quad\quad\times{\bf
1}\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\not\in
B(q_{\bar{\mathbf{y}}}),\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\not\in
B(q_{\bar{\mathbf{x}}}),\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|>2r_{t}\\}.$

Notice that there exists $c_{1}>0$ independent of $t$ such that for
$x,y\in\mathbb{R}^{d}$,

$\displaystyle(\lambda_{d}\otimes\mathbb{Q})(B(y)\setminus B(x))\geq
c_{1}\|y-x\|r_{t}^{d-1},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{whenever}\leavevmode\nobreak\
\leavevmode\nobreak\ \|y-x\|\leq 2r_{t}.$ (4.11)

Therefore applying the Mecke equation, (4.3) and (1.3) leads to

$\displaystyle\int\mathbb{E}[{S}^{(3a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})$
$\displaystyle\leq\frac{t^{2d}}{d!t(\log t)^{d-1}}\int
h^{(r_{t})}_{D}(\bar{\mathbf{x}})h^{(r_{t})}_{D}(\bar{\mathbf{y}})\exp(-c_{1}t\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|r_{t}^{d-1})$
$\displaystyle\quad\quad\quad{\bf
1}\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{d}\not\in
B(q_{\bar{\mathbf{y}}}),\mathbf{y}_{1},\ldots,\mathbf{y}_{d}\not\in
B(q_{\bar{\mathbf{x}}}),\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\leq
2r_{t}\\}(\lambda_{d}\otimes\mathbb{Q})^{2d}(d(\bar{\mathbf{x}},\bar{\mathbf{y}})).$

For $i\in[d]$ let $\mathbf{x}_{i}=(x_{i},a_{i})$ and
$\mathbf{y}_{i}=(y_{i},b_{i})$ with $x_{i},y_{i}\in\mathbb{R}^{d}$ and
$a_{i},b_{i}\in\mathbb{R}_{+}$. Write
$x_{i}=q_{\bar{\mathbf{x}}}+r_{t}a_{i}u_{i}$ and
$y_{i}=q_{\bar{\mathbf{y}}}+r_{t}b_{i}u_{d+i}$, $i\in[d]$, where

$u_{i}:=\Big{(}\cos\phi_{i1},\cos\phi_{i2}\sin\phi_{i1},...,\cos\phi_{i(d-1)}\prod_{j=1}^{d-2}\sin\phi_{ij},\prod_{j=1}^{d-1}\sin\phi_{ij}\Big{)}$

with $\phi_{ij}\in[0,\pi]$ for $j\in[d-2]$ and $\phi_{i(d-1)}\in[0,2\pi)$,
$i\in[2d]$. For fixed values of the marks $(a_{1},\ldots,a_{d})$ (for
$x_{1},\ldots,x_{d})$ respectively) and $(b_{1},\ldots,b_{d})$ (for
$y_{1},\ldots,y_{d})$ respectively, let $J,J^{\prime}$ denote the Jacobians in
the changes of variables

$\displaystyle\lambda_{d}^{d}(d(x_{1},\ldots,x_{d}))$
$\displaystyle=J(q_{\bar{\mathbf{x}}},\phi_{[d],.})\lambda_{d}(dq_{\bar{\mathbf{x}}})\prod_{i\in[d],j\in[d-1]}d\phi_{ij};$
$\displaystyle\lambda^{d}_{d}(d(y_{1},\ldots,y_{d}))$
$\displaystyle=J^{\prime}(q_{\bar{\mathbf{y}}},\phi_{[2d]\setminus[d],.})\lambda_{d}(dq_{\bar{\mathbf{y}}})\prod_{i\in[2d]\setminus[d],j\in[d-1]}d\phi_{ij}.$

Here, we use $\phi_{I,.}$ to denote the set of variables $\phi_{ij},i\in
I,j\in[d-1]$. We see that $J$ is bounded by
$c_{2}r_{t}^{d(d-1)}\prod_{i=1}^{d}a_{i}^{d-1}$ and $J^{\prime}$ is bounded by
$c_{2}r_{t}^{d(d-1)}\prod_{i=1}^{d}b_{i}^{d-1}$ for some universal constant
$c_{2}<\infty$. Moreover, when we integrate out each factor of $b_{i}^{d-1}$
with respect to the measure $\mathbb{Q}$ we get a value of
$\mathbb{E}[Y^{d-1}]$, which is finite since we assume
$\mathbb{E}[Y^{2d}]<\infty$.

Let $u$ be the unit directional vector of
$q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}$. We claim that if
$h^{(r_{t})}(\bar{\mathbf{x}})=h^{(r_{t})}(\bar{\mathbf{y}})=1$, and
$\mathbf{x}_{i}\notin B(q_{\bar{\mathbf{y}}})$ and $\mathbf{y}_{i}\notin
B(q_{\bar{\mathbf{x}}})$ for all $i\in[d]$, then there exists $i\in[2d]$ such
that

$\displaystyle|\langle
r_{t}u_{i},u\rangle|\leq\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|/2.$
(4.12)

To prove this, suppose that $q_{\bar{\mathbf{y}}}$ is lower than
$q_{\bar{\mathbf{x}}}$. Let $L$ denote the hyperplane such that
$q_{\bar{\mathbf{y}}}$ and $q_{\bar{\mathbf{x}}}$ are mirror to each other
with respect to $L$. For $i\in[d]$ we have
$\|x_{i}-q_{\bar{\mathbf{x}}}\|=a_{i}r_{t}<\|x_{i}-q_{\bar{\mathbf{y}}}\|$, so
that $x_{1},\ldots,x_{d}$ all lie above $L$. Let $L^{+}$ (resp. $L^{-}$)
denote the hyperplane parallel to $L$ that contains $q_{\bar{\mathbf{x}}}$
(resp. $q_{\bar{\mathbf{y}}}$). From
$h^{(r_{t})}(\mathbf{x}_{1},\ldots,\mathbf{x}_{d})=1$, we see that at least
one of $x_{1},...,x_{d}$, say $x_{1}$, lies below $L^{+}$ (but above $L$). As
a consequence, the projection of $x_{1}-q_{\bar{\mathbf{x}}}$ to $u$ is
bounded by $\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|/2$, as required. The
case where $q_{\bar{\mathbf{x}}}$ is lower than $q_{\bar{\mathbf{y}}}$ is
completely analogous, with $x_{i}$ replaced by $y_{i}$, so that (4.12) holds
for some $u_{i},i\in[2d]\setminus[d]$.

In summary, we have proved that

$\displaystyle\int\mathbb{E}[{S}^{(3a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})\leq$
$\displaystyle\frac{c_{2}^{2}(\mathbb{E}[Y^{d-1}])^{2d}}{d!}\sum_{i=1}^{2d}\frac{t^{2d-1}r_{t}^{2d(d-1)}}{(\log
t)^{d-1}}$ $\displaystyle\times\int\cdots\int
e^{-c_{1}t\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|r_{t}^{d-1}}{\bf
1}\\{\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\leq
2r_{t},q_{\bar{\mathbf{x}}}\in D\\}$ $\displaystyle\times{\bf 1}\\{|\langle
r_{t}u_{i},u\rangle|\leq\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\\}\lambda^{2}_{d}(d(q_{\bar{\mathbf{x}}},q_{\bar{\mathbf{y}}}))\prod_{i\in[2d],j\in[d-1]}d\phi_{ij}.$

Let $v$ denote the surface measure on the unit sphere $\mathbb{S}_{d-1}$.
There is a finite constant $c_{3}$ such that
$v(\\{u_{1}\in\mathbb{S}_{d-1}:|\langle
r_{t}u_{1},u\rangle|\leq\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\\})\leq
c_{3}\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|/r_{t}$ for all $t$ and all
$q_{\bar{\mathbf{x}}},q_{\bar{\mathbf{y}}}$ with
$\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\leq 2r_{t}$. Hence, for some
finite $c_{4}$,

$\displaystyle\int\mathbb{E}[{S}^{(3a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})$
$\displaystyle\leq\frac{c_{4}t^{2d-1}r_{t}^{2d(d-1)}}{(\log
t)^{d-1}}\int\exp(-c_{1}t\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|r_{t}^{d-1})(\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|/r_{t})$
$\displaystyle\hskip 56.9055pt\times{\bf
1}\\{\|q_{\bar{\mathbf{x}}}-q_{\bar{\mathbf{y}}}\|\leq
2r_{t},q_{\bar{\mathbf{x}}}\in
D\\}\lambda^{2}_{d}(d(q_{\bar{\mathbf{x}}},q_{\bar{\mathbf{y}}})).$

By changing variables from $q_{\bar{\mathbf{y}}}$ to
$z=q_{\bar{\mathbf{y}}}-q_{\bar{\mathbf{x}}}$ and then to $w=tr_{t}^{d-1}z$,
we see that the second factor inside the last integral becomes
$\|w\|/(tr_{t}^{d})$ and the last integral as a whole is
$O((tr_{t}^{d})^{-1}(tr_{t}^{d-1})^{-d}).$ Hence

$\displaystyle\int\mathbb{E}[{S}^{(3a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=O((tr_{t}^{d})^{d-2}(\log
t)^{1-d})=O((\log t)^{-1}).$ (4.13)

For ${S}_{\bar{\mathbf{x}}}^{(3b)}$, note that there is a constant $c_{5}>0$
such that for $x,y\in\mathbb{R}^{d}$ we have

$\displaystyle\mathbb{P}[\xi_{t}(B(y)\setminus B(x))=0]\leq
e^{-c_{5}tr_{t}^{d}}\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{whenever}\leavevmode\nobreak\ \leavevmode\nobreak\ \|y-x\|\geq 2r_{t},$
(4.14)

and also for any $b>0$, $\lambda_{d}(B(y,r_{t}b)\cap
B(x,r_{t}b))\leq\theta_{d}(r_{t}b)^{d}{\bf 1}\\{\|y-x\|\leq 2r_{t}b\\}$ so
that by Markov’s inequality

$\displaystyle\mathbb{P}[\xi_{t}(B(y)\cap B(x))\geq 1]$ $\displaystyle\leq
t(\lambda_{d}\otimes\mathbb{Q})(B(y)\cap B(x))$ $\displaystyle\leq
t\int_{(\|y-x\|/(2r_{t}),\infty)}\theta_{d}(br_{t})^{d}\mathbb{Q}(db)$
$\displaystyle=\theta_{d}tr_{t}^{d}\mathbb{E}[Y^{d}{\bf
1}\\{Y\geq\|y-x\|/(2r_{t})\\}].$

We apply the Mecke equation, (4.3) and (1.3) to deduce that

$\displaystyle\int\mathbb{E}[{S}^{(3b)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})$
$\displaystyle\leq\frac{t^{2d}\theta_{d}(tr_{t}^{d})}{t(\log t)^{d-1}}\int
h^{(r_{t})}_{D}(\bar{\mathbf{x}})h^{(r_{t})}_{D}(\bar{\mathbf{y}})e^{-c_{5}tr_{t}^{d}}\mathbb{E}[Y^{d}{\bf
1}\\{Y>\|q_{\bar{\mathbf{y}}}-q_{\bar{\mathbf{x}}}\|/(2r_{t})\\}]$
$\displaystyle\quad\quad\quad\hskip 142.26378pt\times{\bf
1}\\{\|q_{\mathbf{x}}-q_{\mathbf{y}}\|>2r_{t}\\}(\lambda_{d}\otimes\mathbb{Q})^{2d}(d(\bar{\mathbf{x}},\bar{\mathbf{y}})).$

Going into polar coordinates as before once more gives us a factor of
$r_{t}^{2d(d-1)}$. Writing $z=q_{\bar{\mathbf{y}}}-q_{\bar{\mathbf{x}}}$ and
using our assumption that $\mathbb{E}[Y^{2d}]<\infty$, we obtain that the last
expression is

$\displaystyle O\Big{(}\frac{t^{2d}r_{t}^{2d^{2}-d}e^{-c_{5}tr_{t}^{d}}}{(\log
t)^{d-1}}\Big{)}\int_{\mathbb{R}^{d}}\mathbb{E}[Y^{d}{\bf
1}\\{Y\geq\|z\|/(2r_{t})\\}]\lambda_{d}(dz)$
$\displaystyle=O\Big{(}\frac{t^{2d}r_{t}^{2d^{2}-d}e^{-c_{5}tr_{t}^{d}}}{(\log
t)^{d-1}}\Big{)}\int_{\mathbb{R}^{d}}\int_{(\|z\|/(2r_{t}),\infty)}b^{d}\mathbb{Q}(db)\lambda_{d}(dz)$
$\displaystyle=O\Big{(}\frac{t^{2d}r_{t}^{2d^{2}-d}e^{-c_{5}tr_{t}^{d}}}{(\log
t)^{d-1}}\Big{)}\int_{(0,\infty)}b^{d}\theta_{d}(2r_{t}b)^{d}\mathbb{Q}(db)$
$\displaystyle=O\Big{(}\frac{(tr_{t}^{d})^{2d}e^{-c_{5}tr_{t}^{d}}}{(\log
t)^{d-1}}\Big{)}=O(e^{-(c_{5}/2)\log t}).$

Combined with (4.13) this gives us the result. ∎

The integral of ${S}_{\bar{\mathbf{x}}}^{(4)}$ is handled as follows.

######  Proposition 4.5.

For $t>1$ let $D=A^{o}$ or $D=A^{\prime}(t)$. As $t\to\infty$ it holds that

$\int_{\mathbb{X}^{d}}\mathbb{E}[{S}^{(4)}_{\bar{\mathbf{x}}}(D,t)]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=O((\log
t)^{-1}).$

###### Proof.

For each non-empty strict subset $I$ of $[d]$, let
${S}_{\bar{\mathbf{x}}}^{(I)}$ be the contribution of $I$ to
${S}_{\bar{\mathbf{x}}}^{(4)}$. Also define

$\displaystyle{S}^{(I,a)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{\mathbf{y}_{1},...,\mathbf{y}_{d-|I|}\\}\subset\xi_{t}\setminus
B(q_{\bar{\mathbf{x}}})}h^{(r_{t})}_{D}(\mathbf{x}_{I},\mathbf{y}_{1},...,\mathbf{y}_{d-|I|}){\bf
1}\\{\|q_{\overline{\mathbf{x}_{I}\mathbf{y}}}-q_{\bar{\mathbf{x}}}\|\leq
2r_{t}\\}$ $\displaystyle\hskip 56.9055pt\times{\bf
1}\\{(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing\\}{\bf 1}\\{\mathbf{x}_{[d]\setminus
I}\cap B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})=\varnothing\\};$
$\displaystyle{S}^{(I,b)}_{\bar{\mathbf{x}}}$
$\displaystyle:=\sum_{\mathbf{y}=\\{\mathbf{y}_{1},...,\mathbf{y}_{d-|I|}\\}\subset\xi_{t}\setminus
B(q_{\bar{\mathbf{x}}})}h^{(r_{t})}_{D}(\mathbf{x}_{I},\mathbf{y}_{1},...,\mathbf{y}_{d-|I|}){\bf
1}\\{\|q_{\overline{\mathbf{x}_{I}\mathbf{y}}}-q_{\bar{\mathbf{x}}}\|>2r_{t}\\}$
$\displaystyle\hskip 56.9055pt\times{\bf 1}\\{(\xi_{t}\setminus\mathbf{y})\cap
B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})\setminus
B(q_{\bar{\mathbf{x}}})=\varnothing\\}{\bf 1}\\{\mathbf{x}_{[d]\setminus
I}\cap B(q_{\overline{\mathbf{x}_{I}\mathbf{y}}})=\varnothing\\}.$

Then
${S}^{(4)}_{\bar{\mathbf{x}}}=\sum_{I}({S}_{{\bar{\mathbf{x}}}}^{(I)}=\sum_{I}({S}_{{\bar{\mathbf{x}}}}^{(I,a)}+{S}_{{\bar{\mathbf{x}}}}^{(I,b)})$.

Set $j=|I|$. By the Mecke equation, (4.3), (4.11) and (1.3), there exists
$c_{6}>0$ such that

$\displaystyle\int\mathbb{E}[{S}^{(I,a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})$
$\displaystyle\leq\frac{c_{6}t^{2d-j}}{t(\log t)^{d-1}}\int
h^{(r_{t})}_{D}(\bar{\mathbf{x}})h^{(r_{t})}_{D}(\mathbf{x}_{I},\mathbf{y}_{1},...,\mathbf{y}_{d-j})\times\exp(-c_{1}t\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|r_{t}^{d-1})$
$\displaystyle\times{\bf 1}\\{\mathbf{x}_{[d]\setminus I}\cap
B(q_{\mathbf{x}_{I}\mathbf{y}})=\varnothing,\mathbf{y}_{[d-j]}\cap
B(q_{\mathbf{x}})=\varnothing\\}(\lambda_{d}\otimes\mathbb{Q})^{2d-j}(d(\bar{\mathbf{x}},\mathbf{y}_{1},\ldots,\mathbf{y}_{d-j})),$
(4.15)

where $\mathbf{y}_{[d-j]}:=\\{\mathbf{y}_{1},\ldots,\mathbf{y}_{d-j}\\}$.

For each $i\in[d]$ write $\mathbf{x}_{i}=(x_{i},a_{i})$, and if $i\leq d-j$
write $\mathbf{y}_{i}=(y_{i},b_{i})$. Given the values of
$q_{\bar{\mathbf{x}}}$ and $q_{\overline{\mathbf{x}_{I}\mathbf{y}}}$, by
definition for each $i\in I$ we have
$\|x_{i}-q_{\bar{\mathbf{x}}}\|=\|x_{i}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|=r_{t}a_{i}$.
In particular
$\|q_{{\bar{\mathbf{x}}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|\leq
2r_{t}\min_{i\in I}a_{i}$. Set
$z:=q_{\overline{\mathbf{x}_{I}\mathbf{y}}}-q_{\bar{\mathbf{x}}}$. For each
$i\in I$ the vector $x_{i}$ lies on the surface of the sphere centred on
$q_{\bar{\mathbf{x}}}+\frac{1}{2}z$ of radius
$\sqrt{(a_{i}r_{t})^{2}-(\|z\|/2)^{2}}$ and also on the hyperplane through
$q_{\bar{\mathbf{x}}}+\frac{1}{2}z$ perpendicular to $z$. Hence for each $i\in
I$ we can write

$\displaystyle
x_{i}=q_{\bar{\mathbf{x}}}+(1/2)z+((a_{i}r_{t})^{2}-(\|z\|/2)^{2})^{1/2}\Theta_{\|z\|^{-1}z}((\omega_{i},0))$

for some $\omega_{i}\in\mathbb{S}_{d-2}$, where for any unit vector $e$ in
$\mathbb{R}^{d}$ we denote by $\Theta_{e}$ the rotation that sends ${e}_{d}$
to $e$, and $\mathbb{S}_{k}$ denotes the $k$-dimensional unit sphere for all
$k\in\mathbb{N}$.

Write $x_{i}=q_{\bar{\mathbf{x}}}+r_{t}a_{i}u_{i}$ for $i\in[d]\setminus I$
and
$y_{i}=q_{\overline{\mathbf{x}_{I}\mathbf{y}}}+r_{t}b_{i}u^{\prime}_{i},i\in[d-j]$.
Write $v_{k}$ for the $k$-dimensional surface measure on $\mathbb{S}_{k}$; as
in the proof of Proposition 4.4, the $k$-dimensional surface integral can be
written in terms of $k$ real-valued polar coordinates. There is a finite
constant $c_{7}$ such that for any fixed
$(a_{1},\ldots,a_{d},b_{1},\ldots,b_{d-j})$ the Jacobian $J$ in

$\displaystyle\lambda_{d}^{2d-j}(d(x_{1},\ldots,x_{d},y_{1},\ldots,y_{d-j}))=$
$\displaystyle
J(q_{\bar{\mathbf{x}}},q_{\overline{\mathbf{x}_{I}\mathbf{y}}},\omega_{I},u_{[d]\setminus
I},u^{\prime}_{[d-j]})$ $\displaystyle\times
v_{d-2}^{j}(d\omega_{I})v_{d-1}^{d-j}(du_{[d]\setminus
I})v_{d-1}^{d-j}(du^{\prime}_{[d-j]})dzdq_{{\bar{\mathbf{x}}}}$

satisfies

$\displaystyle J$ $\displaystyle\leq
c_{7}r_{t}^{j(d-2)+2(d-j)(d-1)}\prod_{i\in
I}a_{i}^{d-2}\prod_{i\in[d]\setminus
I}a_{i}^{d-1}\prod_{i\in[d-j]}b_{i}^{d-1}$
$\displaystyle=c_{7}r_{t}^{2d^{2}-d(2+j)}\prod_{i\in
I}a_{i}^{d-2}\prod_{i\in[d]\setminus
I}a_{i}^{d-1}\prod_{i\in[d-j]}b_{i}^{d-1}.$

Therefore using (4.15) and the assumption $\mathbb{E}[Y^{2d}]<\infty$, writing
$q$ for $q_{\bar{\mathbf{x}}}$ we obtain that

$\displaystyle\int\mathbb{E}[{S}^{(I,a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})\leq$
$\displaystyle
c_{6}c_{7}(\mathbb{E}[Y^{d-2}])^{j}(\mathbb{E}[Y^{d-1}])^{2d-j}\Big{(}\frac{t^{2d-j-1}r_{t}^{2d^{2}-d(2+j)}}{(\log
t)^{d-1}}\Big{)}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\times\int_{D}\int_{\mathbb{R}^{d}}\exp(-c_{1}tr_{t}^{d-1}\|z\|)\lambda_{d}(dz)\lambda_{d}(dq).$

Changing variable to $w=tr_{t}^{d-1}z$ gives us a further factor of
$(tr_{t}^{d-1})^{-d}$. Therefore

$\displaystyle\int\mathbb{E}[{S}^{(I,a)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}(\lambda_{d}\otimes\mathbb{Q})^{d}(d\bar{\mathbf{x}})=O((\log
t)^{1-d}t^{d-j-1}r_{t}^{d^{2}-d-jd})=O((\log t)^{-j}).$ (4.16)

Now consider ${S}^{(I,b)}$. Using (4.14) we obtain a similar bound to (4.15)
but with ${S}^{(I,b)}$ replacing ${S}^{(I,a)}$ on the left and a factor
$e^{-c_{5}tr_{t}^{d}}$ replacing the factor of
$\exp(-c_{1}t\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|r_{t}^{d-1})$
on the right.

We make the same change of variables as before, noting that for each $i\in I$
we require
$r_{t}a_{i}\geq\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|/2$.
Hence for the integral with respect to
$\mathbb{Q}(da_{i})v_{d-2}(d\omega_{i})$ we obtain a factor of

$\displaystyle\int_{(\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|/(2r_{t}),\infty)}((ar_{t})^{2}-(\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|/2)^{2})^{(d-2)/2}\mathbb{Q}(da)$
$\displaystyle\leq\int_{(\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|/(2r_{t}),\infty)}(ar_{t})^{d-2}\mathbb{Q}(da)=r_{t}^{d-2}\mathbb{E}[Y^{d-2}{\bf
1}\\{Y\geq\|q_{\bar{\mathbf{x}}}-q_{\overline{\mathbf{x}_{I}\mathbf{y}}}\|/(2r_{t})\\}].$

The first factor of $r_{t}^{d-2}$ in the above is the same as before. For the
integrals with respect to $v_{d-1}^{d-j}(du_{[d]\setminus
I})v_{d-1}^{d-j}(du^{\prime}_{[d-j]})$ we obtain a factor bounded by
$cr_{t}^{2(d-j)(d-1)}$ as before. Therefore integrating out everything except
the variables $q_{\bar{\mathbf{x}}}$ and
$q_{\overline{\mathbf{x}_{I}\mathbf{y}}}$ and writing $z$ for
$q_{\overline{\mathbf{x}_{I}\mathbf{y}}}-q_{\bar{\mathbf{x}}}$ and then
$w=(2r_{t})^{-1}z$ we obtain similarly to before that there is a finite
constant $c_{9}$ such that

$\displaystyle\int\mathbb{E}[{S}^{(I,b)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}\lambda^{d}_{d}(d\bar{\mathbf{x}})$
$\displaystyle\leq\frac{c_{9}t^{2d-j-1}r_{t}^{2d^{2}-d(2+j)}}{(\log t)^{d-1}}$
$\displaystyle\times\int_{D}\int_{\mathbb{R}^{d}}e^{-c_{5}tr_{t}^{d}}(\mathbb{E}[Y^{d-2}{\bf
1}\\{Y\geq\|z\|/(2r_{t})\\}])^{j}\lambda_{d}(dz)\lambda_{d}(dq)$
$\displaystyle\leq\frac{2^{d}c_{9}\lambda_{d}(D)t^{2d-j-1}r_{t}^{2d^{2}-d(1+j)}e^{-c_{5}tr_{t}^{d}}}{(\log
t)^{d-1}}\int_{\mathbb{R}^{d}}\Big{(}\int_{[\|w\|,\infty)}a^{d-2}\mathbb{Q}(da)\Big{)}^{j}dw.$

Since we assume $\mathbb{E}[Y^{2d}]<\infty$ the integral on the last line is
finite. Indeed, for $j=1$, by Fubini’s theorem it comes to
$\theta_{d}\int_{\mathbb{R}_{+}}a^{2d-2}\mathbb{Q}(da),$ which is finite. For
general $j\geq 1$, choosing $K\in\mathbb{R}_{+}$ such that
$\mathbb{E}[Y^{d-2}{\bf 1}\\{Y\geq K\\}]\leq 1$, we find that the contribution
to the integral from $(w,a)$ with $\|w\|>K$ is no greater for $j>1$ than it
would be for $j=1$ and therefore is finite, while the contribution from
$(w,a)$ with $\|w\|\leq K$ is also finite. Thus

$\int\mathbb{E}[{S}^{(I,b)}_{\bar{\mathbf{x}}}]\pi(\bar{\mathbf{x}})t^{d}\lambda^{d}_{d}(d\bar{\mathbf{x}})=O((tr_{t}^{d})^{2d-j-1}(\log
t)^{1-d}e^{-c_{5}tr_{t}^{d}})=O(e^{-(c_{5}/2)tr_{t}^{d}}),$

and combined with (4.16) this gives us the result. ∎

###### Proof of Theorem 1.2.

By Lemma 4.2 and Propositions 4.3, 4.4 and 4.5, for either $D=A^{o}$ or
$D=A^{\prime}$ we have as $t\to\infty$ that

$d_{\mathrm{TV}}(F_{D},\mathsf{Po}_{\mathbb{E}[F_{D,t}]})=O((\log t)^{-1}).$

By Lemma 2.2,

$\displaystyle\mathbb{P}[R_{t}\leq r_{t}]\leq\mathbb{P}[F_{A^{o},t}=0]\leq
e^{-\mathbb{E}[F_{A^{o},t}]}+d_{\mathrm{TV}}(F_{A^{o},t},\mathsf{Po}_{\mathbb{E}[F_{A^{o},t}]})$

By rearranging the inequality given in Lemma 2.6, using the union bound and
Lemma 2.5 we obtain for some constant $c$ that

$\displaystyle\mathbb{P}[R_{t}\leq r_{t}]$
$\displaystyle\geq\mathbb{P}[F_{A^{\prime},t}=0]-\mathbb{P}[\mathscr{V}_{t}^{c}]$
$\displaystyle\geq\mathbb{P}[F_{A^{\prime},t}=0]-\exp(-ct^{1/(3d)})$
$\displaystyle\geq
e^{-\mathbb{E}[F_{A^{\prime},t}]}-d_{\mathrm{TV}}(F_{A^{\prime},t},\mathsf{Po}_{\mathbb{E}[F_{A^{\prime},t}]})-\exp(-ct^{1/(3d)}).$

Putting together the preceding inequalities, along with Corollary 3.2, yields
the result. ∎

Acknowledgements. This research was supported by EPSRC grant EP/T028653/1. For
the purpose of open access, the authors have applied a Creative Commons
Attribution (CC-BY) licence to any Author Accepted Manuscript version arising.

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