thm prop defi lma cor re

#  CONVERGENCE RATE FOR GEOMETRIC STATISTICS OF POINT PROCESSES WITH FAST
DECAY DEPENDENCE

Tianshu Cong email<EMAIL_ADDRESS>Work supported by a
Research Training Program Scholarship, a faculty of science postgraduate
writing-up award and a Xing Lei Cross-Disciplinary PhD Scholarship in
Mathematics and Statistics at the University of Melbourne.. School of
Mathematics and Statistics, the University of Melbourne, Parkville VIC 3010,
Australia  Aihua Xia email<EMAIL_ADDRESS>Work supported by the
Australian Research Council Grant No DP190100613. School of Mathematics and
Statistics, the University of Melbourne, Parkville VIC 3010, Australia

(August 27, 2024)

###### Abstract

[Błaszczyszyn, Yogeshwaran and Yukich (2019)] established central limit
theorems for geometric statistics of point processes having fast decay
dependence. As limit theorems are of limited use unless we understand their
errors involved in the approximation, in this paper, we consider the rates of
a normal approximation in terms of the Wasserstein distance for statistics of
point processes on $\mathbb{R}^{d}$ satisfying fast decay dependence. We
demonstrate the use of the theorems for statistics arising from two families
of point processes: the rarified Gibbs point processes and the determinantal
point processes with fast decay kernels.

Key words and phrases: Wasserstein distance; Stein’s method; fast decay
dependence; Gibbs point process; determinantal point process.

AMS 2020 Subject Classification: primary 60F05; secondary 60D05, 60G55, 62E20,
05C80.

##  1 Introduction

Random events in space and time can be represented as point processes and
statistics arising from such random events are often of the form
$\sum_{x\in\Xi\cap A}\eta(x,\Xi)$, where $\Xi$ is a point process on
$\mathbb{R}^{d}$, $A\subset\mathbb{R}^{d}$ is a bounded Borel set, and
$\eta(x,\Xi)$, known as a score function, is a functional representing the
interaction between the point at $x$ and the point process $\Xi$. The form
dates back to [Beardwood, Halton and Hammersley (1959), Steele (1981)] with
$\Xi$ as a binomial point process and [Avram and Bertsimas (1993)] with $\Xi$
being a Poisson point process. As the density of points or the size of the
observation window $A$ increases, various limit theorems of the functionals
have been established since then, see, e.g., [Schulte (2012), Schulte (2016),
Lachièze-Rey, Schulte and Yukich (2019)] and references therein. If
$\eta(x,\Xi)=\bm{1}_{x\in\Xi}$ and $\Xi$ is an $\alpha-$determinantal point
process with some condition on $\alpha$, [Soshnikov (2002), Shirai and
Takahashi (2003), Nazarov and Sodin (2012)] proved that the counting
statistics are asymptotically normal. The asymptotic normality also holds when
the input process $\Xi$ is a Gibbsian point process [Schreiber and Yukich
(2013), Xia and Yukich (2015)]. The common feature leading to these limit
theorems is the local dependence [Chen and Shao (2004), p. 133] in the sense
that each summand is affected by a small neighbourhood only and hence it makes
a nearly independent contribution to the statistics of interest. More
generally, when the underline point process $\Xi$ is a stationary point
process and has fast decay dependence, and the score function $\eta(x,\Xi)$ is
determined by points of $\Xi$ not too far away from $x$, it is possible to
establish central limit theorems for such statistics [Błaszczyszyn,
Yogeshwaran and Yukich (2019)]. In this paper, we aim to quantify the errors
associated with the limit theorems in [Błaszczyszyn, Yogeshwaran and Yukich
(2019)] because the limit theorems are of limited practical value unless we
understand the magnitude of the errors involved in the approximation of these
statistics.

The local influence of the score function $\eta$ can be controlled through the
concept of stabilisation [Barbour and Xia (2001), Penrose and Yukich (2001),
Penrose and Yukich (2003), Penrose and Yukich (2005), Penrose (2007a), Penrose
(2007b), Schreiber and Yukich (2013), Xia and Yukich (2015)] and the fast
decay of dependence of the input point process can be represented through its
correlation functions [Lachièze-Rey, Schulte and Yukich (2019), (1.2)] being
controlled by a correlation decay function [Lachièze-Rey, Schulte and Yukich
(2019), Definition 1.2]. The limit theorems in [Lachièze-Rey, Schulte and
Yukich (2019)] are formulated in terms of the Palm distributions of all orders
[Kallenberg (1983), p. 110] of the point process, and their proofs hinge on
the Marcinkiewicz theorem [Soshnikov (2002), Lemma 3] and the method of
cumulants. The Marcinkiewicz theorem is a handy tool to prove the central
limit theorems, but when we aim for the errors of approximation, it seems
impractical to use the cumulants to control the errors because these
quantities are hard to obtain in applications. For this reason, we impose the
condition of the decay of dependence through the $\beta$-mixing coefficient in
A2.0 Exponentially Decay Dependence (EDD). We will show in Section 3 that the
Gibbs point processes with nearly finite range potentials, a class of the
determinantal point processes with fast decay kernels, the $r$-dependent point
processes and the Boolean models all possess the EDD property. We also
consider a wider class of score functions with different types of edge
effects. In Section 4, we demonstrate the use of the theorems for statistics
arising from the rarified Gibbs point processes and the determinantal point
processes with fast decay kernels. For ease of reading, we postpone the proofs
of the main results to Section 5.

The asymptotic normality depends on the lower bound of the variance of the
statistics, and showing the order of the variance is itself an interesting but
hard topic [Lachièze-Rey, Schulte and Yukich (2019), (1.26) and Remark (iii)].
In an attempt to recover the volume order of the variances, we obtained
Theorem 2.11, which is an analogue of [Lachièze-Rey, Schulte and Yukich
(2019), Theorem 1.15].

##  2 General results

To start with, we recall the definition of the marked point processes on
$(\mathbb{R}^{d},\mathscr{B}(\mathbb{R}^{d}))$ with marks in a measurable
space $(T,\mathscr{T})$, where $\mathbb{R}^{d}$ is equipped with the Euclidean
norm $\|\cdot\|$ and the Borel $\sigma$-algebra $\mathscr{B}(\mathbb{R}^{d})$,
and $\mathscr{T}$ is a $\sigma$-algebra on $T$. Let
$\bm{S}:=\mathbb{R}^{d}\times T$ be equipped with the product $\sigma$-field
$\mathscr{S}:=\mathscr{B}(\mathbb{R}^{d})\times\mathscr{T}$. We use
$\bm{C}_{\bm{S}}$ to denote the space of all locally finite (with respect to
the first coordinate in $\mathbb{R}^{d}$) non-negative integer-valued measures
$\xi$, often called configurations, on $\bm{S}$ such that $\xi(\\{{x}\\}\times
T)\leq 1$ for all ${x}\in\mathbb{R}^{d}$. The space $\bm{C}_{\bm{S}}$ is
endowed with the $\sigma$-field $\mathscr{C}_{\bm{S}}$ generated by the vague
topology [Kallenberg (1983), p. 169]. A marked point process $\Xi$ on
$\mathbb{R}^{d}$ is a measurable mapping from
$(\Omega,\mathscr{F},\mathbb{P})$ to $(\bm{C}_{\bm{S}},\mathscr{C}_{\bm{S}})$
[Kallenberg (2017), p. 49]. The induced simple point process
$\overline{\Xi}(\cdot):=\Xi(\cdot\times T)$ is called the ground process
[Daley and Vere-Jones (2008), p. 3] or projection [Kallenberg (2017), p. 17]
of the marked point process $\Xi$ on $\mathbb{R}^{d}$. We use $M_{x}$ to
denote the mark of $\Xi$ at $x$ for $x\in\overline{\Xi}$.

For a marked point process $\Xi$, let $\Xi_{A}$ be the restriction of $\Xi$ to
$A\times T$ defined as $\Xi_{A}(B\times D):=\Xi((A\cap B)\times D)$, $\Xi^{x}$
be the shifted point process of $\Xi$ by $x$ defined as $\Xi^{x}(B\times
D):=\Xi((B+x)\times D)$ for all $x\in\mathbb{R}^{d}$, $D\in\mathscr{T}$ and
$A,\leavevmode\nobreak\ B\in\mathscr{B}(\mathbb{R}^{d})$. We say that the
marked point process $\Xi$ is stationary if $\Xi\overset{d}{=}\Xi^{x}$ for all
$x\in\mathbb{R}^{d}$, where $\overset{d}{=}$ stands for ‘equal in
distribution’. To avoid using the Palm distributions of all orders and all
cumulants in the approximation bounds, in this paper, the fast decay
dependence of the marked point process $\Xi$ is quantified through its
$\beta$-mixing coefficient [Volkonskiǐ and Rozanov (1959), Rio (2017)]: for
$A_{1},A_{2}\in\mathscr{B}(\mathbb{R}^{d})$,

$\beta_{A_{1},A_{2}}:=\frac{1}{2}\int_{\zeta_{1}\in\bm{C}_{A_{1}\times
T}}\int_{\zeta_{2}\in\bm{C}_{A_{2}\times
T}}\left|\mathbb{P}\left(\Xi_{A_{1}}\in d\zeta_{1},\Xi_{A_{2}}\in
d\zeta_{2}\right)-\mathbb{P}\left(\Xi_{A_{1}}\in
d\zeta_{1}\right)\mathbb{P}\left(\Xi_{A_{2}}\in d\zeta_{2}\right)\right|,$

where and in the following, $\bm{C}_{A_{i}\times T}$ and
$\mathscr{C}_{A_{i}\times T}$ are defined in the same way as $\bm{C}_{\bm{S}}$
and $\mathscr{C}_{\bm{S}}$ with $\bm{S}$ replaced by $A_{i}\times T$, $i=1,2.$

To define the decay of dependence, we set ${\rm
diam}(A):=\sup\\{\|x-y\|;\leavevmode\nobreak\ x,y\in A\\}$,
$d(A,B):=\inf\\{\|x-y\|;x\in A,\leavevmode\nobreak\ y\in B\\}$ for
$A,B\in\mathscr{B}(\mathbb{R}^{d})$, where we used the convention
$\sup\\{\emptyset\\}=0$ and $\inf\\{\emptyset\\}=\infty$. We use $\vee$ to
stand for the maximum.

A2.0 Exponentially Decay Dependence We say that the marked point process $\Xi$
has the exponentially decay dependence (EDD) if there exist constants
$\theta_{0}\in\mathbb{R}_{0}:=[0,\infty)$,
$\theta_{i}\in\mathbb{R}_{+}:=(0,\infty)$, $1\leq i\leq 4$, such that for any
$A$, $B\in\mathscr{B}(\mathbb{R}^{d})$ with $d(A,B)\geq\theta_{3}\ln({\rm
diam}(A)\vee{\rm diam}(B)\vee\theta_{4})$,

$\beta_{A,B}\leq\theta_{1}({\rm diam}(A)^{\theta_{0}}\vee 1)({\rm
diam}(B)^{\theta_{0}}\vee 1)e^{-\theta_{2}d(A,B)}.$ (2.1)

The idea of the EDD is that the total variation distance between the law of
$(\Xi_{A},\Xi_{B})$ and the law of the independent union of $\Xi_{A}$ and
$\Xi_{B}$ decays exponentially fast as the distance between $A$ and $B$
becomes large.

The following lemma says that, in applications, it is sometimes more
convenient to verify the EDD via the volumes of the sets. To this end, let
$\mathrm{Vol}(A)$ denote the volume of the set
$A\in\mathscr{B}(\mathbb{R}^{d})$.

######  Lemma 2.1.

If there exist constants $\theta_{0}^{\prime}\in\mathbb{R}_{0}$,
$\theta_{i}^{\prime}\in\mathbb{R}_{+}$, $1\leq i\leq 4$, such that for any
$A$, $B\in\mathscr{B}(\mathbb{R}^{d})$ with
$d(A,B)\geq\theta_{3}^{\prime}\ln(\mathrm{Vol}(A)\vee\mathrm{Vol}(B)\vee\theta_{4}^{\prime})$,

$\beta_{A,B}\leq\theta_{1}^{\prime}(\mathrm{Vol}(A)^{\theta_{0}^{\prime}}\vee
1)(\mathrm{Vol}(B)^{\theta_{0}^{\prime}}\vee
1)e^{-\theta_{2}^{\prime}d(A,B)},$

then $\Xi$ satisfies the EDD.

Proof. Given the dimension $d$,
$\text{Vol}(A)\leq\frac{\pi^{d/2}}{2^{d}\Gamma(\frac{d}{2}+1)}{\rm
diam}(A)^{d}$, hence (2.1) follows immediately.

######  Remark 2.2.

The constants $\theta_{4}$ in the definition of the EDD and
$\theta_{4}^{\prime}$ in Lemma 2.1 are not essential and they can be replaced
by any positive constants because the definition of $\beta$-mixing coefficient
ensures that $\beta$ is non-decreasing in the sense of inclusion, i.e.,
$\beta_{A,B}\leq\beta_{A^{\prime},B^{\prime}}$ for all
$A,A^{\prime},B,B^{\prime}\in\mathscr{B}(\mathbb{R}^{d})$ such that $A\subset
A^{\prime}$ and $B\subset B^{\prime}$.

Let $\Xi$ be a stationary marked point process on $\bm{S}$ with independent
and identically distributed (${i.i.d.}$) marks that satisfies the EDD. Writing
the law of $\Xi$ as $\mathscr{P}$ and the law of the independent marks as
$\mathscr{L}_{T}$. The functionals we study in the paper are defined on
$\Gamma_{\alpha}:=\left[-\frac{1}{2}\alpha^{\frac{1}{d}},\frac{1}{2}\alpha^{\frac{1}{d}}\right]^{d}$,
the cube with volume $\alpha$ on $\mathbb{R}^{d}$ centred at $\bm{0}$, having
the forms

$W_{\alpha}:={\sum_{(x,m)\in\Xi_{\Gamma_{\alpha}}}\eta(\left(x,m\right),\Xi)}$

and

$\bar{W}_{\alpha}:={\sum_{{(x,m)\in\Xi_{\Gamma_{\alpha}}}}\eta(\left(x,m\right),\Xi_{\Gamma_{\alpha}},\Gamma_{\alpha})=\sum_{{(x,m)\in\Xi_{\Gamma_{\alpha}}}}\eta(\left(x,m\right),\Xi,\Gamma_{\alpha})}.$

The function $\eta$ is called a score function (resp. restricted score
function), i.e., a measurable function on
$\left(\bm{S}\times\bm{C}_{\bm{S}},\mathscr{S}\times\mathscr{C}_{\bm{S}}\right)$
to $\left(\mathbb{R},\mathscr{B}\left(\mathbb{R}\right)\right)$ (resp. a
function mapping $\bm{S}\times\bm{C}_{\Gamma_{\alpha}\times
T}\times\mathscr{B}(\mathbb{R}^{d})$ to $\mathbb{R}$ which is
$\left(\bm{S}\times\bm{C}_{\Gamma_{\alpha}\times
T},\mathscr{S}\times\mathscr{C}_{\Gamma_{\alpha}\times T}\right)$ to
$\left(\mathbb{R},\mathscr{B}\left(\mathbb{R}\right)\right)$ measurable when
the third coordinate is fixed) and it represents the interaction between a
point with its mark and the configuration of the point process. The class of
score functions considered here is broader than that considered in
[Błaszczyszyn, Yogeshwaran and Yukich (2019)]. More precisely, if the score
function for the restricted case does not depend on the third argument, it
reduces to that in [Błaszczyszyn, Yogeshwaran and Yukich (2019)]. Because the
interest is in the values of the score function of the points in a
configuration, for convenience,
$\eta\left((x,m),\mathscr{X}\right){\leavevmode\nobreak\
(\mbox{resp.}\leavevmode\nobreak\
\eta\left((x,m),\mathscr{X},\Gamma_{\alpha}\right))}$ is understood as $0$ for
all $x\in\mathbb{R}^{d}$ and $\mathscr{X}\in\bm{C}_{\bm{S}}$ such that
$(x,m)\notin\mathscr{X}$.

We need Palm processes and reduced Palm processes as the tools for stating the
conditions and constructing proofs. For ease of reading, we briefly recall
their definitions. Let $H$ be a Polish space with Borel $\sigma$-algebra
$\mathscr{B}\left(H\right)$ and configuration space
$\left(\bm{C}_{H},\mathscr{C}_{H}\right)$, let $\Upsilon$ be a point process
on $\left(H,\mathscr{B}\left(H\right)\right)$ and write the mean measure of
$\Upsilon$ as $\mathbb{E}\Upsilon$. The point processes $\\{\Upsilon_{x}:x\in
H\\}$ are said to be the reduced Palm processes associated with $\Upsilon$ if
for any measurable function
$f:(H\times\bm{C}_{H},\mathscr{B}\left(H\right)\times\mathscr{C}_{H})\rightarrow(\mathbb{R}_{0},\mathscr{B}(\mathbb{R}_{0}))$,

$\mathbb{E}\left[\int_{H}f(x,\Upsilon)\Upsilon(dx)\right]=\int_{H}\mathbb{E}f(x,\Upsilon_{x}+\delta_{x})\mathbb{E}\Upsilon(dx),$
(2.2)

[Kallenberg (1983), § 10.1], where $\delta_{x}$ is the Dirac measure at $x$.
The distributions of $\Upsilon_{x}$ and $\Upsilon_{x}+\delta_{x}$ are
respectively called the reduced Palm distribution and the Palm distribution of
$\Upsilon$ at $x$. When the point process $\Upsilon$ is simple, i.e.,
$\mathbb{P}(\Upsilon(\\{x\\})\in\\{0,1\\}\mbox{ for all }x\in H)=1$, the Palm
distribution of $\Upsilon_{x}+\delta_{x}$ can be interpreted as the
conditional distribution of $\Upsilon$ given $\Upsilon(\\{x\\})=1$ [Kallenberg
(1983), § 10.1].

For the marked point process $\Xi\sim\mathscr{P}$, since the marks are
independent of each other and independent of the ground process, we can adapt
(2.2) to the marked point process $\Xi$. For
$f:\left(\bm{S}\times\bm{C}_{\bm{S}},\mathscr{S}\times\mathscr{C}_{\bm{S}}\right)\to(\mathbb{R}_{0},\mathscr{B}(\mathbb{R}_{0}))$,
recalling that $M_{x}$ is the mark of $\Xi$ at the point $x\in\overline{\Xi}$,
we have

$\mathbb{E}\left[\int_{\mathbb{R}^{d}}f((x,M_{x}),\Xi)\overline{\Xi}(dx)\right]=\int_{\mathbb{R}^{d}}\mathbb{E}f((x,M),\Xi_{x}+{\delta}_{(x,M)})\lambda
dx,$ (2.3)

where $M\sim\mathscr{L}_{T}$,
$\lambda:=\mathbb{E}\left(\overline{\Xi}(d\bm{0})\right)/d\bm{0}$, $\Xi_{x}$
is the point process obtained by attaching the reduced Palm process
$\overline{\Xi}_{x}$ of $\overline{\Xi}$ with i.i.d. marks following
$\mathscr{L}_{T}$, and $M$ is independent of $\Xi_{x}$. Without loss, we use
the convention that $M_{x}$ is independent of $\Xi_{x}$ throughout the paper.
Hence, when we need to emphasise the location of the mark, we can replace $M$
by $M_{x}$ at the right hand side of (2.3).

The following assumptions are adapted from those in [Cong and Xia (2020)]
which were initiated in [Penrose and Yukich (2001)] and further refined in
[Xia and Yukich (2015)].

A2.1 Stabilisation

For a locally finite configuration $\mathscr{X}$ and
$z\in\bm{S}\cup\\{\emptyset\\}$, write $\mathscr{X}^{\lbag
z\rbag}=\mathscr{X}$ if $z=\emptyset$ and $\mathscr{X}^{\lbag
z\rbag}=\mathscr{X}\cup\\{z\\}$ otherwise. We use $B(x,r)$ to stand for the
ball with centre $x$ and radius $r\geq 0$.

######  Definition 2.3.

(unrestricted case) A score function $\eta$ on $\bm{S}$ is range-bound (resp.
exponentially stabilising) with respect to $\mathscr{P}$ if for all
$x\in\mathbb{R}^{d}$, $z\in\bm{S}\cup\\{\emptyset\\}$, and almost all
realisations $\mathscr{X}$ of the marked point process $\Xi_{x}$, there exists
a radius of stabilisation $R:=R(x):=R((x,m_{x}),\mathscr{X}^{\lbag
z\rbag})\in(0,\infty)$, such that for all locally finite
$\mathscr{Y}\subset(\mathbb{R}^{d}\backslash B(x,R))\times T$, we have

$\eta\left(\left(x,m_{x}\right),\left[\mathscr{X}^{\lbag
z\rbag}\cap\left(B(x,R)\times
T\right)\right]\cup\mathscr{Y}\right)=\eta\left(\left(x,m_{x}\right),\mathscr{X}^{\lbag
z\rbag}\cap\left(B(x,R)\times T\right)\right)$

and the tail probability

$\tau(t):=\sup_{(x,m_{x})\in\mathbb{R}^{d}\times\rm{supp}(\mathscr{L}_{T})}\sup_{z\in\mathbf{S}\cup\\{\emptyset\\}}\mathbb{P}\left(R((x,m_{x}),\Xi_{x}^{\lbag
z\rbag}+\delta_{(x,m_{x})})\geq t\right)$

satisfies $\tau(t)=0\mbox{ for some }t>0$ (resp. $\tau(t)\leq
C_{1}e^{-C_{2}t}{\leavevmode\nobreak\ \mbox{for all }t>0}$, where $C_{1}$ and
$C_{2}$ are positive constants independent of $t$).

For the functionals with the input of a restricted marked point process, we
have the following counterpart of stabilisation. Note that the score function
for the restricted input is not affected by points outside $\Gamma_{\alpha}$.

######  Definition 2.4.

(restricted case) We say that the score function $\eta$ is range-bound (resp.
exponentially stabilising) with respect to $\mathscr{P}$ if for
$\alpha\in\mathbb{R}_{+}$, $x\in\Gamma_{\alpha}$, and
$z\in(\Gamma_{\alpha}\times T)\cup\\{\emptyset\\}$, almost all realisations
$\mathscr{X}$ of the marked point process $\Xi_{x}$, there exists a radius of
stabilisation
$\bar{R}:=\bar{R}(x,\alpha):=\bar{R}((x,m_{x}),\alpha,\mathscr{X}^{\lbag
z\rbag})\in(0,\infty)$ such that for all locally finite
$\mathscr{Y}\subset(\Gamma_{\alpha}\backslash B(x,R))\times T$, we have

$\displaystyle\eta\left(\left(x,m_{x}\right),\left[\mathscr{X}_{\Gamma_{\alpha}}^{{\lbag
z\rbag}}\cap\left(B(x,\bar{R})\times
T\right)\right]\cup\mathscr{Y},\Gamma_{\alpha}\right)=\eta\left(\left(x,m_{x}\right),\mathscr{X}_{\Gamma_{\alpha}}^{{\lbag
z\rbag}}\cap\left(B(x,\bar{R})\times T\right),\Gamma_{\alpha}\right)$

and the tail probability

$\bar{\tau}(t):=\sup_{(x,m_{x})\in\mathbb{R}^{d}\times\rm{supp}(\mathscr{L}_{T})}\sup_{\alpha\in\mathbb{R}_{+}}\sup_{z\in(\Gamma_{\alpha}\times
T)\cup\\{\emptyset\\}}\mathbb{P}\left(\bar{R}((x,m_{x}),\alpha,\Xi_{x}^{{\lbag
z\rbag}}+\delta_{(x,m_{x})})\geq t\right)$

satisfies $\bar{\tau}(t)=0$ for some $t>0$ (resp. $\bar{\tau}(t)\leq
C_{1}e^{-C_{2}t}$ for all $t>0$, where $C_{1}$ and $C_{2}$ are some positive
constants independent of $t$).

A2.2 Translation Invariance

We write $d(x,A):=\inf\\{d(x,y);\leavevmode\nobreak\ y\in A\\}$, $A\pm
B:=\\{x\pm y;\ x\in A,\ y\in B\\}$ for $x\in\mathbb{R}^{d}$ and
$A,B\in\mathscr{B}\left(\mathbb{R}^{d}\right)$. Recall that the shift operator
is defined as $\Xi^{x}(\cdot\times D):=\Xi((\cdot+x)\times D)$ for all
$x\in\mathbb{R}^{d}$, $D\in\mathscr{T}$.

A2.2.1 Unrestricted Case:

######  Definition 2.5.

The score function $\eta$ is translation invariant if for all locally finite
configuration $\mathscr{X}$, $x\in\mathbb{R}^{d}$ and $m\in T$,
$\eta((\bm{0},m),\mathscr{X})=\eta((x,m),\mathscr{X}^{-x})=:g(\mathscr{X})$.

A2.2.2 Restricted Case:

As a translation may send a configuration to the outside of $\Gamma_{\alpha}$,
resulting in a completely different configuration inside $\Gamma_{\alpha}$, it
is necessary to focus on the part that affects the score function. Therefore,
we expect the score function to take the same value for two configurations if
the parts within their stabilising radii are completely inside
$\Gamma_{\alpha}$ and one is a translation of the other. More precisely, we
have the following definition.

######  Definition 2.6.

A stabilising score function $\eta$ with stabilisation radius $\bar{R}$ is
called translation invariant if for any $\alpha>0$, $x\in\Gamma_{\alpha}$ and
$\mathscr{X}\in\bm{C}_{\bm{S}}$ such that
$\bar{R}((x,m),\alpha,\mathscr{X})\leq d(x,\partial\Gamma_{\alpha})$, where
$\partial A$ stands for the boundary of $A$, then
$\eta\left((x,m),\mathscr{X},\Gamma_{\alpha}\right)=\eta\left((x^{\prime},m),\mathscr{X}^{\prime},\Gamma_{\alpha^{\prime}}\right)$
and
$\bar{R}((x^{\prime},m),\alpha^{\prime},\mathscr{X}^{\prime})=\bar{R}((x,m),\alpha,\mathscr{X})$
for all $\alpha^{\prime}>0$, $x^{\prime}\in\Gamma_{\alpha^{\prime}}$ and
$\mathscr{X}^{\prime}\in\bm{C}_{\bm{S}}$ such that
$\bar{R}((x^{\prime},m),\alpha^{\prime},\mathscr{X}^{\prime})\leq
d(x^{\prime},\partial\Gamma_{\alpha^{\prime}})$ and
$\left(\mathscr{X}^{\prime}_{B(x^{\prime},\bar{R}((x,m),\alpha,\mathscr{X}))}\right)^{x^{\prime}}=\left(\mathscr{X}_{B(x,\bar{R}((x,m),\alpha,\mathscr{X}))}\right)^{x}$.

Noting that there is a tacit assumption of consistency in Definition 2.6,
which implies that if $\eta$ is translation invariant in Definition 2.6, then
there exists a $\bar{g}:\bm{C}_{\bm{S}}\rightarrow\mathbb{R}$ such that

$\lim_{\alpha\rightarrow\infty}\eta\left((0,m),\mathscr{X},\Gamma_{\alpha}\right)=\bar{g}\left(\mathscr{X}\right)$

for $\mathscr{L}_{T}$ almost all $m\in T$ and almost all realisations
$\mathscr{X}$ of the marked point process $\Xi\sim\mathscr{P}$. The limit
$\bar{g}$ ensures that for each score function $\eta$ satisfying the
translation-invariance in Definition 2.6, there exists a score function for
the unrestricted case by setting
$\bar{\eta}((x,m),\mathscr{X}):=\bar{g}(\mathscr{X}^{x})\mathbf{1}_{(x,m)\in\mathscr{X}}$
with the radius of stabilisation
$R((x,m),\mathscr{X})=\lim_{\alpha\to\infty}\bar{R}((x,m),\alpha,\mathscr{X})$.
Consequently, $\bar{\eta}$ is range bound (resp. exponentially stabilising) in
the sense of Definition 2.3 if $\eta$ is range bound (resp. exponentially
stabilising) in the sense of Definition 2.4. Moreover, if
$B(x,R(x))\subset\Gamma_{\alpha}$, then $\bar{R}(x,\alpha)=R(x)$, and if
$B(x,R(x))\not\subset\Gamma_{\alpha}$, then
$\bar{R}(x,\alpha)>d(x,\partial\Gamma_{\alpha})$, but there is no definite
relationship between $\bar{R}$ and $R$.

A2.3 Moment condition

We need moment conditions of both the marked point process $\Xi$ and the score
function $\eta$. We say that the marked point process $\Xi\sim\mathscr{P}$
satisfies the $k$th moment condition if there exists a nonempty open set
$B\subset\mathbb{R}^{d}$ such that

$\mathbb{E}\left(\overline{\Xi}(B)^{k}\right)<\infty.$ (2.4)

For the score function $\eta$, there are two cases to consider.

Unrestricted Case: The score function $\eta$ is said to satisfy the $k$th
moment condition if

$\mathbb{E}\left[\left|\eta\left(({\bf 0},M_{\bf
0}),\Xi_{\bm{0}}+\delta_{(\bm{0},M_{\bf 0})}\right)\right|^{k}\right]<\infty.$
(2.5)

Restricted Case: The score function $\eta$ is said to satisfy the $k$th moment
condition if there exists a positive constant $C$ such that

$\sup_{\alpha\in\mathbb{R}_{+}}\sup_{x\in\Gamma_{\alpha}}\mathbb{E}\left[\left|\eta\left((x,M_{x}),(\Xi_{x})_{\Gamma_{\alpha}}+\delta_{(x,M_{x})}\right)\right|^{k}\right]\leq
C.$ (2.6)

One can verify that if $\eta$ is exponentially stabilising and satisfies
(2.6), then its induced $\bar{\eta}$ satisfies the moment condition of the
same order in the sense of (2.5).

A2.4 Variation Condition

The speed of convergence of the normal approximation is determined by the
order of ${\rm Var}(W_{\alpha})$ or ${\rm Var}(\bar{W}_{\alpha})$. If we have
the following condition, we can prove that the variances ${\rm
Var}(W_{\alpha})$ and ${\rm Var}(\bar{W}_{\alpha})$ have the same order as the
volume $\alpha$, cf. [Błaszczyszyn, Yogeshwaran and Yukich (2019), (1.22)].

Unrestricted Case: The score function is said to satisfy the variation
condition if

$\displaystyle\sigma^{2}=$
$\displaystyle\mathbb{E}\left(\eta((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(0,M_{\bm{0}})})^{2}\right)\lambda$
$\displaystyle+\frac{\mathbb{E}\left(\int_{\mathbb{R}^{d}\backslash\\{\bm{0}\\}}\left(\eta((x,M_{x}),\Xi)\overline{\Xi}(dx)-Pdx\right)\left(\eta((\bm{0},M_{\bm{0}}),\Xi)\overline{\Xi}(d\bm{0})-Pd\bm{0}\right)\right)}{d\bm{0}}>0,$
(2.7)

where
$P:=\lambda\mathbb{E}(\eta((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(0,M_{\bm{0}}})))$.

Restricted Case: We define the variation condition when the score function is
exponentially stabilising. The score function $\eta$ for restricted input
satisfies the variation condition if it is exponentially stabilising, and the
corresponding $\bar{\eta}$ satisfies

$\displaystyle\bar{\sigma}^{2}=$
$\displaystyle\mathbb{E}\left(\bar{\eta}((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(0,M_{\bm{0}})})^{2}\right)\lambda$
$\displaystyle+\frac{\mathbb{E}\left(\int_{\mathbb{R}^{d}\backslash\\{\bm{0}\\}}\left(\bar{\eta}((x,M_{x}),\Xi)\overline{\Xi}(dx)-\bar{P}dx\right)\left(\bar{\eta}((\bm{0},M_{\bm{0}}),\Xi)\overline{\Xi}(d\bm{0})-\bar{P}d\bm{0}\right)\right)}{d\bm{0}}>0,$
(2.8)

where
$\bar{P}:=\lambda\mathbb{E}(\bar{\eta}((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(0,M_{\bm{0}})}))$.

The above conditions are generally difficult to verify, we refer the
interested readers to the discussion at Remark (iii) of [Błaszczyszyn,
Yogeshwaran and Yukich (2019), Theorem 1.14]. For this reason, we formulate
the bounds of approximation errors in terms of the following variation
conditions. If $f_{1}$ and $f_{2}$ are two functions satisfying
$\liminf_{x\to\infty}f_{1}(x)/f_{2}(x)>0$, then we write
$f_{1}(x)=\Omega(f_{2}(x))$ as $x\to\infty$.

Unrestricted Case: ${\rm Var}(W_{\alpha})=\Omega(\alpha^{\nu})$ for some
$\nu\in(\frac{2}{3},1]$ as $\alpha\to\infty$.

Restricted Case: ${\rm Var}(\bar{W}_{\alpha})=\Omega(\alpha^{\nu})$ for some
$\nu\in(\frac{2}{3},1]$ as $\alpha\to\infty$.

In the proof of Theorem 2.11 and Remark 5.11, we can see that under the
conditions of Theorem 2.11, the variation condition (2.7) (resp. (2.8)) holds
if and only if the above condition holds for $\nu=1$.

Given the variation conditions, we can establish the convergence rate in terms
of the Wasserstein distance defined as

$d_{W}(X,Y):=\sup_{h\in\mathscr{F}_{\rm
Lip}}\mathbb{E}\left(h(X)-h(Y)\right),$ (2.9)

where $\mathscr{F}_{\rm Lip}$ is the set of all Lipschitz functions $h$ on
$\mathbb{R}$ such that $|h(x)-h(y)|\leq|x-y|$ for all $x,y\in\mathbb{R}$. Our
main result for $W_{\alpha}$ (unrestricted case) is summarised below.

######  Theorem 2.7.

Assume that the score function $\eta$ is translation invariant in Definition
2.5 and satisfies the sixth moment condition (2.5), $\Xi$ satisfies the EDD,
the fifth moment condition (2.4) and ${\rm
Var}(W_{\alpha})=\Omega(\alpha^{\nu})$ for some $\nu\in(\frac{2}{3},1]$ as
$\alpha\to\infty$.

(i) If $\eta$ is range-bound as in Definition 2.3, then

$d_{W}\left(\frac{W_{\alpha}-\mathbb{E}W_{\alpha}}{\sqrt{{\rm
Var}(W_{\alpha})}},Z\right)\leq O\left(\alpha^{-\frac{3}{2}\nu+1}\right).$

(ii) If $\eta$ is exponentially stabilising as in Definition 2.3, then

$d_{W}\left(\frac{W_{\alpha}-\mathbb{E}W_{\alpha}}{\sqrt{{\rm
Var}(W_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{3}{2}\nu+1}\ln(\alpha)^{5d}\right).$

######  Corollary 2.8.

Assume that the score function $\eta$ is translation invariant in Definition
2.5 and satisfies the sixth moment condition (2.5), $\Xi$ satisfies the EDD,
the fifth moment condition (2.4) and (2.7) holds.

(i) If $\eta$ is range-bound as in Definition 2.3, then

$d_{W}\left(\frac{W_{\alpha}-\mathbb{E}W_{\alpha}}{\sqrt{{\rm
Var}(W_{\alpha})}},Z\right)\leq O\left(\alpha^{-\frac{1}{2}}\right).$

(ii) If $\eta$ is exponentially stabilising as in Definition 2.3, then

$d_{W}\left(\frac{W_{\alpha}-\mathbb{E}W_{\alpha}}{\sqrt{{\rm
Var}(W_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{1}{2}}\ln(\alpha)^{5d}\right).$

The restricted case is of interest in many applications, by adapting the
conditions accordingly, we can show that the main result for
$\bar{W}_{\alpha}$ (restricted case) also holds.

######  Theorem 2.9.

Assume that the score function $\eta$ is translation invariant in Definition
2.6 and satisfies the sixth moment condition (2.6), $\Xi$ satisfies the EDD,
the fifth moment condition (2.4) and ${\rm
Var}(\bar{W}_{\alpha})=\Omega(\alpha^{\nu})$ for some $\nu\in(\frac{2}{3},1]$
as $\alpha\to\infty$.

(i) If $\eta$ is range-bound as in Definition 2.4, then

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{3}{2}\nu+1}\right).$

(ii) If $\eta$ is exponentially stabilising as in Definition 2.4, then

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{3}{2}\nu+1}\ln(\alpha)^{5d}\right).$

######  Corollary 2.10.

Assume that the score function $\eta$ is translation invariant in Definition
2.6 and satisfies the sixth moment condition (2.6), $\Xi$ satisfies the EDD,
the fifth moment condition (2.4) and (2.8) holds.

(i) If $\eta$ is range-bound as in Definition 2.4, then

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq O\left(\alpha^{-\frac{1}{2}}\right).$

(ii) If $\eta$ is exponentially stabilising as in Definition 2.4, then

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{1}{2}}\ln(\alpha)^{5d}\right).$

We write $f_{1}=\Theta(f_{2})$ if $f_{1}=\Omega(f_{2})$ and
$f_{2}=\Omega(f_{1})$. Then in terms of the order of ${\rm
Var}(\bar{W}_{\alpha})$ and ${\rm Var}(W_{\alpha})$, we have the following
results which can be regarded as the counterparts of [Lachièze-Rey, Schulte
and Yukich (2019), Theorem 1.15].

######  Theorem 2.11.

(a) (unrestricted case) Assume that $\Xi$ satisfies the EDD and the fifth
moment condition (2.4), and the score function $\eta$ satisfies the sixth
moment condition (2.5). If $\eta$ is exponentially stabilising in Definition
2.3, translation invariant in Definition 2.5 and $\sigma^{2}>0$, then ${\rm
Var}(W_{\alpha})=\Theta(\alpha)$.

(b) (restricted case) Assume that $\Xi$ satisfies the EDD and the fifth moment
condition (2.4), and the score function $\eta$ satisfies the sixth moment
condition (2.6). If $\eta$ is exponentially stabilising in Definition 2.4,
translation invariant in Definition 2.6 and $\bar{\sigma}^{2}>0$, then ${\rm
Var}(\bar{W}_{\alpha})=\Theta(\alpha)$.

The proofs of the main and auxiliary results are postponed to Section 5, and
we turn our attention to the EDD point processes and applications of the main
results first.

##  3 EDD point processes

The cornerstone model of point processes is the Poisson point process, where
points behave independently in different regions, and it is obvious that a
Poisson point process satisfies the EDD. There are a range of extensions of
Poisson point processes to capture dependent random structures and significant
development has been made in the determinantal point processes and the Gibbs
point processes [Stoyan, Kendall and Mecke (1995), Soshnikov (2000), Baddeley
et al. (2005), Daley and Vere-Jones (2003), Dereudre (2019)] with the
connections of the two classes investigated in [Georgii and Yoo (2005)]. Both
classes have been well assessed for statistical inferences [Møller and
Waagepetersen (2004), Møller and Waagepetersen (2007), Lavancier, Møller and
Rubak (2015)]. In this section, we show that the Gibbs point processes with
nearly finite range potentials, the determinantal point processes with fast
decay kernels, the $r$-dependent point processes and the Boolean models all
satisfy the EDD.

###  3.1 Rarified Gibbs point process

For ease of reading, we briefly introduce the idea of perfect simulation in
[Schreiber and Yukich (2013), Section 3] for the Gibbs point processes with
nearly finite range potentials $\Psi$. To this end, let $\Psi$ be a
$[0,\infty]$ valued functional on the finite configuration space
$\bm{C}_{\mathbb{R}^{d},b}:=\\{\xi\in\bm{C}_{\mathbb{R}^{d}}:\
\xi(\mathbb{R}^{d})<\infty\\}$ satisfying i) translation invariant:
$\Psi(\mathscr{X})=\Psi(x+\mathscr{X})$ for all $x\in\mathbb{R}^{d}$ and
$\mathscr{X}\in\bm{C}_{\mathbb{R}^{d},b}$; ii) rotation invariant:
$\Psi(\mathscr{X})=\Psi(\mathscr{X}^{\prime})$ for all
$\mathscr{X}\in\bm{C}_{\mathbb{R}^{d},b}$ and all rotations
$\mathscr{X}^{\prime}$ of $\mathscr{X}$; iii) non-decreasing:
$\Psi(\mathscr{X})\leq\Psi(\mathscr{X}^{\prime})$ for all
$\mathscr{X},\mathscr{X}^{\prime}\in\bm{C}_{\mathbb{R}^{d},b}$ such that
$\mathscr{X}\subset\mathscr{X}^{\prime}$; iv) non-degenerate:
$\Psi(\\{x\\})<\infty$ for all $x\in\mathbb{R}^{d}$. Let $D_{n}:=[-n,n]^{d}$
for $n\in\mathbb{N}:=\\{1,2,\dots\\}$,
$\Psi_{D}(\mathscr{X}):=\Psi(\mathscr{X}\cap D)$ for
$D\in\mathscr{B}(\mathbb{R}^{d})$, $\mathscr{P}^{\beta\Psi}$ and
$\mathscr{P}_{D}^{\beta\Psi}$ denote the Gibbs point process with inverse
temperature $\beta>0$ and potential $\Psi$ and $\Psi_{D}$ respectively. Write
$\Delta(\bm{0},\mathscr{X}):=\Delta^{\Psi}(\bm{0},\mathscr{X}):=\Psi(\mathscr{X}\cup\\{\bm{0}\\})-\Psi(\mathscr{X})$,
$\bm{0}\not\in\mathscr{X}$, with $\infty-\infty:=0$, and assume that
$\Delta(\bm{0},\mathscr{X})$ satisfies

$\Delta_{[r]}(\bm{0},\mathscr{X}\cap
B_{r}(\bm{0}))\leq\Delta(\bm{0},\mathscr{X})\leq\Delta^{[r]}(\bm{0},\mathscr{X}\cap
B_{r}(\bm{0}))$

for some non-negative, translation invariant functions $\Delta_{[r]}$ and
$\Delta^{[r]}$, and $\Delta_{[r]}$ and $\Delta^{[r]}$ are assumed to be
respectively increasing and decreasing in $r$. We say $\beta\Psi$ has nearly
finite range if there exists a decreasing continuous function
$\psi^{(\beta)}:\mathbb{R}^{+}\rightarrow[0,1]$ satisfying that
$\psi^{(\beta)}(0)=1$, $\psi^{(\beta)}(r)$ decays exponentially fast in $r$
and

$e^{-\beta\Delta_{[r]}(\bm{0},\mathscr{X}\cap
B_{r}(\bm{0}))}-e^{-\beta\Delta^{[r]}(\bm{0},\mathscr{X}\cap
B_{r}(\bm{0}))}\leq\psi^{\beta}(r)$

for all $r>0$ and $\mathscr{X}\in\mathbf{C}_{\mathbb{R}^{d}}$. [Schreiber and
Yukich (2013)] established that the class of Gibbs point processes having
nearly finite range $\beta\Psi$ includes

i) the point process with a pair potential function
$\Psi(\mathscr{X})=\sum_{x\neq y}\phi(\|x-y\|)$, where
$\phi:[0,\infty)\to[0,\infty)$ has a compact support or

$\phi(r)\left\\{\begin{array}[]{ll}\leq
K_{1}\exp(-K_{2}r),&r\in[r_{0},\infty),\\\
=\infty,&r\in(0,r_{0}),\end{array}\right.$

for constants $K_{1},\leavevmode\nobreak\ K_{2}\in\mathbb{R}_{+}$;

ii) the point process defined by the continuum Widom-Rowlinson model for
spheres of type $A$ having centres $\mathscr{X}$ and spheres of type $B$
having centres $\mathscr{Y}$:

$\Psi(\mathscr{X}\cup\mathscr{Y})=\left\\{\begin{array}[]{ll}\alpha_{1}\mbox{\rm
card}(\mathscr{X})+\alpha_{2}\mbox{\rm
card}(\mathscr{Y})+\alpha_{3},&d(\mathscr{X},\mathscr{Y})>2a,\\\ \infty,&{\rm
otherwise},\end{array}\right.$

where $\mbox{\rm card}(\mathscr{X})$ is the cardinality of $\mathscr{X}$, $a$
is the common radii of the spheres and $\alpha_{i}$’s are positive constants;

iii) the area interaction point process with

$\Psi(\mathscr{X})=\mathrm{Vol}\left(\cup_{x\in\mathscr{X}}(x+K)\right)+\alpha_{1}{\rm
card}(\mathscr{X})+\alpha_{2},$

where $\alpha_{i}$’s are positive constants and $K$ is a fixed compact convex
set;

iv) the hard-core process with

$\Psi(\mathscr{X})=\left\\{\begin{array}[]{ll}\alpha_{1}{\rm
card}(\mathscr{X})+\alpha_{2},&\inf_{x,y\in\mathscr{X},x\neq y}|x-y|\geq
r_{0},\\\ \infty,&\inf_{x,y\in\mathscr{X},x\neq
y}|x-y|<r_{0},\end{array}\right.$

where $r_{0}$ and $\alpha_{i}$’s are positive constants.

Moreover, extensions of the point process with a pair potential function have
been developed in [Geyer (1999), Rajala, Murrell and Olhede (2018)] to detect
multivariate interactions in spatial point patterns, and most of these
extensions have nearly finite range potentials. [Schreiber and Yukich (2013),
Section 3.3] states that the infinite volume Gibbs point process
$\mathscr{P}^{\beta\Psi}$ exists and is the thermodynamic limit of
$\mathscr{P}_{D_{n}}^{\beta\Psi}$. The next lemma says that
$\mathscr{P}^{\beta\Psi}$ also satisfies the EDD.

######  Lemma 3.1.

The Gibbs point process $\mathscr{P}^{\beta\Psi}$ with nearly finite range
potential satisfies the EDD.

Proof.  Using the idea of perfect simulation introduced in [Ferrari, Fernández
and Garcia (2002)] and [Schreiber and Yukich (2013), Sections 3.2 & 3.3], we
can construct a stationary homogeneous free birth and death process
$\\{\rho(t)\\}_{t\in\mathbb{R}}$ such that
$\rho(t)\overset{d}{=}\mathscr{P}^{\beta\Psi}$ for all $t$. For
$B\in\mathscr{B}(\mathbb{R}^{d})$, define the ancestor clan
$\bm{A}_{B}^{\beta\Psi}(0)=:\bm{A}_{B}^{\beta\Psi}$ with respect to the
process $\\{\rho(t)\\}_{t\in\mathbb{R}}$ as the accepted births in
$\rho(0)\cap B$, their ancestors, the ancestors of their ancestors and so
forth. From the construction, $\rho(0)\cap A$ and $\rho(0)\cap B$ are
conditionally independent given
$\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}=\emptyset$. [Schreiber and
Yukich (2013), (3.6)] states that the ancestor clan $\bm{A}_{B}^{\beta\Psi}$
satisfies that for all $r\in\mathbb{R}_{+}$,

$\mathbb{P}\left[\mbox{diam}(\bm{A}_{B}^{\beta\Psi})\geq
r+\mbox{diam}(B)\right]\leq C(\mathrm{Vol}(B)\vee 1)\exp(-r/C)$

for some positive constant $C$ depending on the distribution of the process
$\mathscr{P}^{\beta\Psi}$ only. Noting that the ancestor clan
$\bm{A}_{B}^{\beta\Psi}$ starts from the accepted births in $\rho(0)\cap B$,
if $\rho(0)\cap B\neq\emptyset$, then $\bm{A}_{B}^{\beta\Psi}\cap
B\neq\emptyset$, which ensures that if $B$ is a cube in $\mathbb{R}^{d}$ with
centre $x$ and diagonal length ${\rm diam}(B):=2r$, then by the rotation
invariance, there is a constant $C$ such that

$\mathbb{P}\left[\bm{A}_{B}^{\beta\Psi}\nsubseteq
B(x,3r+r^{\prime})\right]\leq C(r^{d}\vee 1)\exp(-r^{\prime}/C)$ (3.1)

for all $r^{\prime}\in\mathbb{R}_{+}$.

For two bounded sets $A,\leavevmode\nobreak\ B\in\mathscr{B}(\mathbb{R}^{d})$
with $d(A,B)=:r_{0}$, without loss, we assume that $r_{0}\geq 1$. Let
$\Xi\overset{d}{=}\mathscr{P}^{\beta\Psi}$ and
$\\{\tilde{\rho}(t)\\}_{t\in\mathbb{R}}$ be an independent copy of
$\\{\rho(t)\\}_{t\in\mathbb{R}}$. Since $\rho(0)\cap A$ and $\rho(0)\cap B$
are conditionally independent given
$\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}=\emptyset$, we have

$\displaystyle\beta_{A,B}$ $\displaystyle=$ $\displaystyle
d_{TV}(\rho(0)\cap(A\cup B),(\rho(0)\cap A)\cup(\tilde{\rho}(0)\cap B))$
$\displaystyle=$ $\displaystyle d_{TV}\left(\rho(0)\cap(A\cup B),(\rho(0)\cap
A)\cup(\tilde{\rho}(0)\cap
B)\middle|\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}=\emptyset\right)\mathbb{P}(\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}=\emptyset)$
$\displaystyle+d_{TV}\left(\rho(0)\cap(A\cup B),(\rho(0)\cap
A)\cup(\tilde{\rho}(0)\cap
B)\middle|\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}\neq\emptyset\right)\mathbb{P}(\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}\neq\emptyset)$
$\displaystyle\leq$
$\displaystyle\mathbb{P}(\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}\neq\emptyset).$
(3.2)

Since $A$ and $B$ are bounded, ${\rm diam}(A)$ and ${\rm diam}(B)$ are finite.
We can find a set of disjoint cubes
$\\{\mathbb{C}_{i,j}\\}_{i\in\\{1,2\\},0\leq j\leq n_{i}}$ with diagonal
length $\frac{r_{0}}{16}$ such that $A\subset\cup_{j\leq
n_{1}}\mathbb{C}_{1,j}$ and $B\subset\cup_{j\leq n_{2}}\mathbb{C}_{2,j}$ for
positive integers $n_{1}\leq C_{1}({\rm diam}(A)\vee 1)^{d}r_{0}^{-d}$ and
$n_{2}\leq C_{1}({\rm diam}(B)^{d}\vee 1)r_{0}^{-d}$. Write the centre of
$\mathbb{C}_{i,j}$ as $c_{i,j}$. Then for any $j_{1}\leq n_{1}$, $j_{2}\leq
n_{2}$, $d(\mathbb{C}_{1,j_{1}},\mathbb{C}_{2,j_{2}})\geq d(A,B)-({\rm
diam}(\mathbb{C}_{1,j_{1}})+{\rm
diam}(\mathbb{C}_{2,j_{2}}))=\frac{7r_{0}}{8}$, it follows from (3.1) with
$2r=r^{\prime}=\frac{r_{0}}{16}$ that

$\displaystyle\mathbb{P}(\bm{A}_{\mathbb{C}_{1,j_{1}}}^{\beta\Psi}\cap\bm{A}_{\mathbb{C}_{2,j_{2}}}^{\beta\Psi}\neq\emptyset)$
$\displaystyle\leq$
$\displaystyle\mathbb{P}\left(\bm{A}_{\mathbb{C}_{1,j_{1}}}^{\beta\Psi}\nsubseteq
B\left(c_{1,j_{1}},\frac{5r_{0}}{32}\right)\right)+\mathbb{P}\left(\bm{A}_{\mathbb{C}_{2,j_{2}}}^{\beta\Psi}\nsubseteq
B\left(c_{2,j_{2}},\frac{5r_{0}}{32}\right)\right)$ $\displaystyle\leq$
$\displaystyle C_{2}r_{0}^{d}\exp(-C_{3}r_{0})$ (3.3)

for some positive constants $C_{2}$ and $C_{3}$ independent of $j_{1}$ and
$j_{2}$. Also, from the definition of the ancestor clans, if a set $B$ is
covered by a class of sets $\\{B_{1},\dots,B_{n}\\}$, then
$\bm{A}_{B}^{\beta\Psi}\subset\cup_{i\leq n}\bm{A}_{B_{i}}^{\beta\Psi}$.
Together with (3.2) and (3.3), we have

$\displaystyle\beta_{A,B}\leq$
$\displaystyle\mathbb{P}(\bm{A}_{A}^{\beta\Psi}\cap\bm{A}_{B}^{\beta\Psi}\neq\emptyset)$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\left(\left(\cup_{j_{1}\leq
n_{1}}\bm{A}_{\mathbb{C}_{1,j_{1}}}^{\beta\Psi}\right)\cap\left(\cup_{j_{2}\leq
n_{2}}\bm{A}_{\mathbb{C}_{2,j_{2}}}^{\beta\Psi}\right)\neq\emptyset\right)$
$\displaystyle\leq$ $\displaystyle\sum_{j_{1}\leq n_{1}}\sum_{j_{2}\leq
n_{2}}\mathbb{P}\left(\bm{A}_{\mathbb{C}_{1,j_{1}}}^{\beta\Psi}\cap\bm{A}_{\mathbb{C}_{2,j_{2}}}^{\beta\Psi}\neq\emptyset\right)$
$\displaystyle\leq$ $\displaystyle
n_{1}n_{2}C_{2}r_{0}^{d}\exp(-C_{3}r_{0})\leq C_{1}^{2}C_{2}({\rm
diam}(A)^{d}\vee 1)({\rm diam}(B)^{d}\vee 1)\exp(-C_{3}r_{0}),$

completing the proof.

Lemma 3.1 ensures that Theorem 2.11 is applicable to all geometric statistics
arising from a Gibbs point process $\mathscr{P}^{\beta\Psi}$ with nearly
finite range potential. In Section 4, we demonstrate its use in two examples:
the total edge length of a $k$-nearest neighbour graph and the total log
volume in a given range of forest.

###  3.2 Determinantal point process

The determinantal point processes are a broad class of point processes such
that the distributions can be characterised by the determinants of given
functions. More precisely, we say that $\Xi$ is a determinantal point process
on space $\mathbb{R}^{d}$ with kernel $K$ if it is a simple point process on
$\mathbb{R}^{d}$ with the joint intensities given by

$\rho_{n}(x_{1},\dots,x_{n})=\det\left[K(x_{i},x_{j})\right]_{1\leq i,j\leq
n}$

for $n\in\mathbb{N}$ and $x_{1},\dots,x_{n}\in\mathbb{R}^{d}$ [Georgii and Yoo
(2005)]. These processes are widely used in random matrix theory and
mathematical physics. The next lemma says that the determinantal point process
satisfies the EDD if its kernel function decreases fast enough.

######  Lemma 3.2.

If the kernel $K$ of the determinantal point process $\Xi$ satisfies
$\|K\|_{\infty}:=\sup_{x,y\in\mathbb{R}^{d}}|K(x,y)|<\infty$ and there exist
constants $C_{i}\in\mathbb{R}_{+}$, $1\leq i\leq 4$, such that $|K(x,y)|\leq
C_{1}e^{-C_{2}e^{C_{3}|x-y|}}$ for all $x,\leavevmode\nobreak\
y\in\mathbb{R}^{d}$ such that $\|x-y\|\geq C_{4}$, then $\Xi$ is an EDD point
process.

Proof.  We use Lemma 2.1 to prove the claim. Let $p_{A}=\mathrm{Vol}(A)$ and
$p_{B}=\mathrm{Vol}(B)$, [Poinas (2019), Theorem 4.1] states that

$\displaystyle\beta_{A,B}$ $\displaystyle\leq
2p_{A}p_{B}(1+2p_{A}\|K\|_{\infty})(1+2p_{B}\|K\|_{\infty})e^{2\left(p_{A}+p_{B}\right)\|K\|_{\infty}}\omega\left(d(A,B)\right)^{2},$
(3.4)

where $\omega(r):=\sup_{|x-y|\geq r}|K(x,y)|$. It is easy to see that

$1+2p_{A}\|K\|_{\infty}\leq 3(p_{A}\vee 1)(\|K\|_{\infty}\vee 1)\mbox{ and
}1+2p_{B}\|K\|_{\infty}\leq 3(p_{B}\vee 1)(\|K\|_{\infty}\vee 1),$

hence

$2p_{A}p_{B}(1+2p_{A}\|K\|_{\infty})(1+2p_{B}\|K\|_{\infty})\leq
18(\|K\|_{\infty}^{2}\vee 1)(p_{A}^{2}\vee 1)(p_{B}^{2}\vee 1).$ (3.5)

For the remaining part, we take
$C_{5}=\left(1\vee\frac{2}{C_{3}}\right)\left(1\vee\ln\left(\frac{4\|K\|_{\infty}}{C_{2}}\right)\right)$,
then there exists an $r_{0}\in\mathbb{R}_{+}$ such that $s:=d(A,B)\geq
C_{5}\ln s$ for all $s\geq r_{0}$. Set $\theta_{3}^{\prime}=C_{5}\vee r_{0}$,
$\theta_{4}^{\prime}=e$ and $\Theta=p_{A}\vee p_{B}\vee e$, if
$s\geq\theta_{3}^{\prime}\ln\Theta$, then

$\displaystyle
e^{2\left(p_{A}+p_{B}\right)\|K\|_{\infty}}\omega\left(d(A,B)\right)^{2}$
$\displaystyle\leq
C_{1}^{2}e^{4\Theta\|K\|_{\infty}}\exp\left\\{-2C_{2}e^{C_{3}s}\right\\}$
$\displaystyle\leq
C_{1}^{2}e^{4\Theta\|K\|_{\infty}}\exp\left\\{-C_{2}e^{C_{3}C_{5}\ln\Theta}-C_{2}e^{C_{3}C_{5}\ln
s}\right\\}$
$\displaystyle=C_{1}^{2}e^{4\Theta\|K\|_{\infty}}\exp\left\\{-C_{2}\Theta^{C_{3}C_{5}}-C_{2}s^{C_{3}C_{5}}\right\\}$
$\displaystyle\leq
C_{1}^{2}e^{4\Theta\|K\|_{\infty}}\exp\left\\{-C_{2}\Theta^{1+0.5C_{3}C_{5}}-C_{2}s\right\\}$
$\displaystyle\leq C_{1}^{2}e^{4\Theta\|K\|_{\infty}}\exp\left\\{-C_{2}\Theta
e^{\ln\left(\frac{4\|K\|_{\infty}}{C_{2}}\right)}-C_{2}s\right\\}$
$\displaystyle=C_{1}^{2}e^{-C_{2}s},$ (3.6)

hence the EDD follows from combining (3.4), (3.5), (3.6) and applying Lemma
2.1 with $\theta_{0}^{\prime}=2$,
$\theta_{1}^{\prime}=18C_{1}^{2}(\|K\|_{\infty}^{2}\vee 1)$ and
$\theta_{2}^{\prime}=C_{2}.$

Lemma 3.2 ensures that Theorem 2.11 is also applicable to all geometric
statistics and the log volume arising from a determinantal point process with
its kernel satisfying the conditions in Lemma 3.2.

Now, we consider two more models that satisfy the EDD so that Theorem 2.11 can
be applied to geometric statistics driven by these point processes.

###  3.3 $r$-dependent point process

A point process $\Xi$ on $\mathbb{R}^{d}$ is said to be $r$-dependent if for
any Borel sets $A$, $B\in\mathscr{B}(\mathbb{R}^{d})$ with distance
$d(A,B)\geq r$, $\Xi_{A}$ and $\Xi_{B}$ are independent. The definition
implies that $\beta_{A,B}=0$ for $d(A,B)\geq r$, hence the following lemma is
trivial.

######  Lemma 3.3.

An $r$-dependent point process satisfies the EDD.

One example of the $r$-dependent point processes is the Matérn hard-core
process [Daley and Vere-Jones (2003), p. 298]. Let
$\Xi^{\prime}:=\sum_{i\in\mathbb{N}}\delta_{X_{i}}$ be a homogeneous Poisson
point process on $\mathbb{R}^{d}$. Then we can construct a hard-core Poisson
process by setting
$\Xi:=\sum_{i\in\mathbb{N}}\delta_{X_{i}}\mathbf{1}_{B(X_{i},r/2)\cap\Xi^{\prime}=\\{X_{i}\\}}$,
then for any Borel sets $A$, $B\in\mathscr{B}(\mathbb{R}^{d})$ with distance
$d(A,B)\geq r$, $\sigma(\Xi_{A})\subset\sigma(\Xi^{\prime}_{B(A,r/2)})$ is
independent of $\sigma(\Xi_{B})\subset\sigma(\Xi^{\prime}_{B(B,r/2)})$, where
$B(A,r^{\prime}):=\\{x:\ d(x,A)<r^{\prime}\\}$ for all
$A\in\mathscr{B}(\mathbb{R}^{d})$ and $r^{\prime}\in\mathbb{R}_{+}$, so $\Xi$
is $r$-dependent.

###  3.4 Boolean model

The Boolean model is a special class of the germ-grain model (see, for
example, [Heinrich and Molchanov (1999)]). We call a point process
$\Xi=\cup_{n\in\mathbb{N}}(X_{n}+\Xi(n))$ a germ-grain model, where the grains
$\\{\Xi(n)\\}_{n\in\mathbb{N}}$ are ${i.i.d.}$ point processes and germs
$\\{X_{n}\\}$ are independent of $\\{\Xi(n)\\}_{n\in\mathbb{N}}$ and form a
stationary point process $\Xi^{\prime}:=\sum_{n\in\mathbb{N}}\delta_{X_{n}}$.
A germ-grain model is called the Boolean model if $\Xi^{\prime}$ is Poisson
[Daley and Vere-Jones (2003), p. 206].

For the Boolean model, assume that $\Xi(n)$’s are bounded, that is, there
exists some $r\in\mathbb{R}_{+}$ such that $\mathbb{P}(\Xi(n)\cap B({\bf
0},r)^{c}=\emptyset)=1$, then for any Borel sets $A$,
$B\in\mathscr{B}(\mathbb{R}^{d})$ with distance $d(A,B)\geq 4r$,
$\sigma(\Xi_{A})\subset\sigma\left(\left\\{\Xi^{\prime}_{B(A,r)},\Xi(n)\leavevmode\nobreak\
\mbox{such that }X_{n}\in B(A,r)\right\\}\right)$ is independent of
$\sigma(\Xi_{B})$, so $\Xi$ is a $(4r)$-dependent process and, consequently,
it satisfies the EDD. More generally, we have the following conclusion.

######  Lemma 3.4.

For the Boolean model defined above, if there exist positive constants
$r_{0}$, $C_{1}$ and $C_{2}$ such that $\mathbb{P}(\Xi(n)\cap B({\bf
0},r)^{c}\neq\emptyset)\leq C_{1}\exp(-C_{2}r)$ for all $r\geq r_{0}$, then
$\Xi$ satisfies the EDD.

Proof.  Since $\beta_{A_{1},A_{2}}$ is a non-decreasing function in the sense
of inclusion (see Remark 2.2), without loss, we take $A_{1}$ and $A_{2}$ as
two balls with centres $O_{1}$ and $O_{2}$, radii $r_{1}$ and $r_{2}$, set
$R=d(A_{1},A_{2})$, and we assume $R\geq 4r_{0}$. For the points
$\mathscr{X}_{i}$ in $A_{i}$ contributed by $\Xi(n)$ with its germ $X_{n}\in
B(O_{i},r_{i}+R/4)$ and $\Xi(n)\cap B({\bf 0},R/4)^{c}=\emptyset$, these
points $\mathscr{X}_{i}$ and their dependents are in $B(O_{i},r_{i}+R/2)$, so
$\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ are independent. In other words, the
contribution of dependent points is from those violating these conditions. By
abuse of notation, we write $\Xi(x)$ as the grain of the germ at
$x\in\Xi^{\prime}$, $V(r)=\frac{\pi^{d/2}}{\Gamma(1+d/2)}r^{d}$ as the volume
of the ball with radius $r$, and add up the probabilities of all possible
cases leading to dependent points in $A_{1}$ and $A_{2}$ to get

$\displaystyle\beta_{A_{1},A_{2}}\leq$
$\displaystyle\sum_{i=1}^{2}\int_{B(O_{i},r_{i}+R/4)}\mathbb{P}\left(\Xi(x)\cap
B({\bf 0},R/4)^{c}\neq\emptyset\right)\mathbb{E}\Xi^{\prime}(dx)$
$\displaystyle+\sum_{i=1}^{2}\int_{B(O_{i},r_{i}+R/4)^{c}}\mathbb{P}\left(\Xi(x)\cap
B({\bf
0},\|x-O_{i}\|-r_{i})^{c}\neq\emptyset\right)\mathbb{E}\Xi^{\prime}(dx)$
$\displaystyle\leq$ $\displaystyle\sum_{i=1}^{2}C_{1}\lambda
e^{-C_{2}R/4}V(r_{i}+R/4)+\sum_{i=1}^{2}\int_{r_{i}+R/4}^{\infty}C_{1}\lambda
e^{-C_{2}(r-r_{i})}dV(r)$ $\displaystyle\leq$ $\displaystyle
C_{3}e^{-C_{2}R/4}(R\vee r_{1}\vee r_{2}\vee 1)^{d},$

for $C_{3}\in\mathbb{R}_{+},$ where $\lambda=\mathbb{E}\Xi^{\prime}(d{\bf
0})/d{\bf 0}$. Choose $R_{0}\geq 1\vee(4r_{0})$ such that $\frac{R}{\ln
R}\geq\frac{8d}{C_{2}}$ for all $R\geq R_{0}$, set $\theta_{4}=e^{R_{0}}$,
$\theta_{3}=1$, then for $R\geq\ln((2r_{1})\vee(2r_{2})\vee\theta_{4})$, we
have

$\beta_{A_{1},A_{2}}\leq C_{3}((2r_{1})^{d}\vee 1)((2r_{2})^{d}\vee
1)e^{-C_{2}R/8},$

which implies (2.1) with $\theta_{0}=d$, $\theta_{1}=C_{3}$,
$\theta_{2}=C_{2}/8$.

##  4 Applications

The asymptotic behaviour of geometric functionals has been of considerable
interest in the last three decades, and our main normal approximation results
can be applied to a large class of geometric functionals, including
$k$-nearest neighbour graph, Voronoi graph, sphere of influence graph,
Delaunay triangulation, Gabriel graph and relative neighbourhood graph
[Devroye (1988), Toussaint (1982)] with vertices driven by a point process
satisfying the EDD.

The limit theory of geometric functionals with determinantal point process
input or Gibbs point process input is investigated in [Błaszczyszyn,
Yogeshwaran and Yukich (2019)]. Error bounds of a normal approximation in
terms of the Kolmogorov distance for the geometric functions with Gibbsian
input were derived in [Xia and Yukich (2015)]. To illustrate the use of the
main results in Section 2, we bound the errors of a normal approximation to
the total edge length in the $k$-nearest neighbour graph with vertices forming
a rarified Gibbs point process or a determinantal point process with fast
decay kernel. We also bound the error of a normal approximation to the total
log volume in a given range of forest with trees following a marked Gibbs
point process.

###  4.1 The total edge length of $k$-nearest neighbour graphs

The $k$-nearest neighbour graph $NG(\mathscr{X})$ with respect to a
configuration $\mathscr{X}\in\bm{C}_{\mathbb{R}^{d}}$ is a graph with vertices
$\mathscr{X}$ and edges $\\{x,y\\}$ such that $y$ is one of the $k$ points
nearest to $x$ or $x$ is one of the $k$ points nearest to $y$ in
$\mathscr{X}$. A variant $NG^{\prime}\left(\mathscr{X}\right)$ of the
$NG(\mathscr{X})$ can be constructed by inserting directed edges $x\rightarrow
y$ if $y$ is one of the $k$ nearest neighbours of $x$ instead of the
undirected edges in $NG(\mathscr{X})$. As in [Schreiber and Yukich (2013)], we
take the score function $\eta(x,\mathscr{X},\Gamma_{\alpha})$ (resp.
$\eta^{\prime}(x,\mathscr{X},\Gamma_{\alpha})$) as one half the sum of the
edge lengths of edges in
$NG\left(\Gamma_{\alpha}\cap\left(\mathscr{X}\cup\\{x\\}\right)\right)$ (resp.
$NG^{\prime}\left(\Gamma_{\alpha}\cap\left(\mathscr{X}\cup\\{x\\}\right)\right)$)
which are incident to $x$, and set

$\bar{W}_{\alpha}=\sum_{{x\in\Xi_{\Gamma_{\alpha}}}}\eta(x,\Xi,\Gamma_{\alpha})\mbox{
and
}\bar{W}_{\alpha}^{\prime}=\sum_{{x\in\Xi_{\Gamma_{\alpha}}}}\eta^{\prime}(x,\Xi,\Gamma_{\alpha}).$
(4.1)

We now state the error bounds for a normal approximation of the total edge
lengths $\bar{W}_{\alpha}$ and $\bar{W}_{\alpha}^{\prime}$ of
$NG\left(\Xi_{\Gamma_{\alpha}}\right)$ if $\Xi$ follows
$\mathscr{P}^{\beta\Psi}$ or a determinantal point process with fast decay
dependence.

######  Theorem 4.1.

(a) If $\Xi$ is an infinite range Gibbs point process with nearly finite range
potential, then

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{1}{2}}\ln(\alpha)^{5d}\right).$

The statement holds if $\bar{W}_{\alpha}$ is replaced by
$\bar{W}_{\alpha}^{\prime}$ in (4.1).

(b) If $\Xi$ is a determinantal point process with continuous kernel $K$
satisfying the conditions in Lemma 3.2, and the total edge length
$\bar{W}_{\alpha}$ satisfies ${\rm
Var}(\bar{W}_{\alpha})=\Omega(\alpha^{\nu})$ for some $\nu>\frac{2}{3}$, then

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{3}{2}\nu+1}\ln(\alpha)^{5d}\right).$

The statement holds true if $\bar{W}_{\alpha}$ is replaced by
$\bar{W}_{\alpha}^{\prime}$ in (4.1).

######  Remark 4.2.

[Xia and Yukich (2015)] proved that a normal approximation error of
$W_{\alpha}:=\sum_{{x\in\Xi_{\Gamma_{\alpha}}}}\eta(x,\Xi,\Gamma_{\infty})$ in
terms of the Kolmogorov distance can be bounded above by
$O\left(\alpha^{-\frac{1}{2}}\ln(\alpha)^{2d}\right)$, which is slightly
better than the error bound for $\bar{W}_{\alpha}$ in terms of the Wasserstein
distance in Theorem 4.1 (a).

Proof of Theorem 4.1.  We only show the claims for the undirected case, and
the directed case can be handled using the same idea. In this case, the total
edge length $\bar{W}_{\alpha}$ can be represented as

$\eta\left(x,\mathscr{X},\Gamma_{\alpha}\right):=\frac{1}{2}\sum_{y\in\mathscr{X}_{\Gamma_{\alpha}}}\|y-x\|\mathbf{1}_{\\{(x,y)\in
NG(\mathscr{X}_{\Gamma_{\alpha}})\\}}.$

The score function $\eta$ is translation invariant according to the
construction and the EDD is ensured by Lemma 3.1 for (a) and Lemma 3.2 for
(b). To apply Theorem 2.9, we need to check the stabilising condition as in
Definition 2.4, the moment conditions (2.4), (2.6) and the order of
$\mathrm{Var}(\bar{W}_{\alpha}).$

(a) According to Lemma $3.3$ in [Schreiber and Yukich (2013)], the Gibbs point
process $\Xi$ is Poisson-like, which means that $\Xi$ is stochastically
dominated by a Poisson point process on $\mathbb{R}^{d}$ with intensity
$\lambda^{\prime}>0$, and there exist strictly positive constants
$C:=C(\lambda^{\prime})$ and $r_{1}$ such that for all $r\geq r_{1}$,
$x\in\mathbb{R}^{d}$ and $\mathscr{X}\in\bm{C}_{\mathbb{R}^{d}\backslash
B(x,r)}$, the conditional probability that $B(x,r)$ is not hitted by $\Xi$
given $\Xi_{B(x,r)^{c}}=\mathscr{X}$ satisfies that

$\mathbb{P}\left[\Xi_{B(x,r)}=\emptyset\middle|\Xi_{B(x,r)^{c}}=\mathscr{X}\right]\leq
e^{-Cr^{d}}.$ (4.2)

Consequently, the fifth moment condition (2.4) of $\Xi$ is ensured by the
Poisson-like property and the moment property of the Poisson point process.

To examine the remaining conditions, for simplicity, we take $d=2$ and follow
the proof of Theorem 3.1 in [Cong and Xia (2020)] using the idea initiated in
[Penrose and Yukich (2001)] to achieve the purpose. For completeness, we recap
the main steps in [Cong and Xia (2020)]. For $x\in\Gamma_{\alpha}$ and $t>0$,
we carve the disk with centre $x$ and radius $t$ into six disjoint circular
sectors $T_{j}(t)$, $1\leq j\leq 6$, of the same size with $x$ as the centre
and $\frac{\pi}{3}$ as their central angle. The sectors are rotated around $x$
such that all straight edges of the sectors have at least the minimal angle
$\pi/12$ with respect to the edges of $\Gamma_{\alpha}$. Let
$T_{j}(\infty)=\cup_{t>0}T_{j}(t)$ for $1\leq j\leq 6$ and define

$t_{x,\alpha}(\Xi_{x})=\inf\\{t:\ \mbox{\rm
card}(T_{j}(t)\cap\Gamma_{\alpha}\cap\Xi_{x})\geq k+1\mbox{ or
}T_{j}(t)\cap\Gamma_{\alpha}=T_{j}(\infty)\cap\Gamma_{\alpha},\ 1\leq j\leq
6\\}$

and $\bar{R}\left(x,\alpha\right)=3t_{x,\alpha}(\Xi_{x}+\delta_{x})$. It was
demonstrated in [Cong and Xia (2020)] that $\bar{R}$ is a radius of
stabilisation. For the tail distribution of $\bar{R}$, let $A_{t}$ be an
obtuse triangle with the longest side length $t$ and two angles $\pi/12$ and
$\pi/3$, define $\tau:=\inf\\{t:\ \mbox{\rm card}(\Xi_{x}\cap A_{t})\geq
k+1\\}$, then [Cong and Xia (2020)] established that
$\mathbb{P}\left(\bar{R}\left(x,\alpha\right)>t\right)\leq
6\mathbb{P}\left(\tau>t/3\right)$. We can find a constant
$C_{1}\in\mathbb{R}_{+}$ such that there are $k+1$ disjoint disks
$\\{B_{1},\dots,B_{k+1}\\}$ of radius $C_{1}t$ such that
$\cup_{i=1}^{k+1}B_{i}\subset A_{t/3}$. Using Poisson-like property (4.2), we
obtain

$\displaystyle\mathbb{P}\left(\bar{R}\left(x,\alpha\right)>t\right)\leq
6\mathbb{P}\left(\tau>t/3\right)$ $\displaystyle\leq
6\mathbb{P}\left(\mbox{\rm card}(\Xi_{x}\cap A_{t/3})\leq k\right)\leq
6\mathbb{P}\left(\cup_{i=1}^{k+1}\\{\Xi_{x}\cap B_{i}=\emptyset\\}\right)$
$\displaystyle\leq 6(k+1)e^{-C\left(C_{1}t\right)^{2}},$ (4.3)

which ensures the exponential stabilisation in Definition 2.4. For the moment
condition (2.6), we again make use of the proof of Theorem 3.1 in [Cong and
Xia (2020)] that

$\eta\left(x,(\Xi_{x})_{\Gamma_{\alpha}}+\delta_{x}\right)\leq
3.5kt_{x,\alpha}(\Xi_{x}+\delta_{x}).$ (4.4)

Since $\bar{R}\left(x,\alpha\right)=3t_{x,\alpha}(\Xi_{x})$, (4.3) implies

$\displaystyle\sup_{\alpha\in\mathbb{R}_{+}}\sup_{x\in\Gamma_{\alpha}}\mathbb{P}\left(t_{x,\alpha}(\Xi_{x}+\delta_{x})>t\right)\leq
6(k+1)e^{-C\left(3C_{1}t\right)^{2}}$

for all $t>0$. This ensures
$\sup_{\alpha\in\mathbb{R}_{+}}\sup_{x\in\Gamma_{\alpha}}\mathbb{E}\left(t_{x,\alpha}(\Xi_{x}+\delta_{x})\right)^{6}<\infty$
and the moment condition (2.6) is an immediate consequence of (4.4). Finally,
we establish (2.8). To this end, define

$\bar{\eta}\left(x,\mathscr{X}\right):=\frac{1}{2}\sum_{y\in\mathscr{X}}\|y-x\|\mathbf{1}_{\\{(x,y)\in
NG(\mathscr{X})\\}},$

and
$\bar{W}_{\infty,\alpha}=\sum_{{x\in\Xi_{\Gamma_{\alpha}}}}\bar{\eta}(x,\Xi)$,
then we can apply [Xia and Yukich (2015), Theorem 1.1] to obtain (2.8). The
proof of (a) is completed by applying Corollary 2.10 (ii).

(b) Since the kernel $K$ is continuous and fast-decreasing, it follows from
Section 2.2.2 and Section 2.1, Remark (i) of [Błaszczyszyn, Yogeshwaran and
Yukich (2019)] that $\mathbb{E}\left(\Xi(B)^{k}\right)$ is finite for all
bounded $B\in\mathscr{B}(\mathbb{R}^{d})$ and all $k\in\mathbb{N}$, which
ensures the fifth moment condition (2.4) of $\Xi$.

To apply Theorem 2.9, as the order of $\mathrm{Var}(\bar{W}_{\alpha})$ is
assumed, it remains to check the stabilising condition as in Definition 2.4
and the moment condition (2.6). For simplicity, we again take $d=2$. Following
the same argument as that for (4.3) and applying Lemma 5.6 of [Błaszczyszyn,
Yogeshwaran and Yukich (2019), Supplement], we obtain

$\displaystyle\mathbb{P}\left(\bar{R}\left(x,\alpha\right)>t\right)\leq
6\mathbb{P}\left(\cup_{i=1}^{k+1}\\{\Xi_{x}\cap B_{i}=\emptyset\\}\right)\leq
6(k+1)e^{1/8-K({\bf 0},{\bf 0})\pi\left(C_{1}t\right)^{2}/8},$ (4.5)

which implies the exponential stabilisation as in Definition 2.4. For the
moment condition (2.6), we again use the relationship
$\bar{R}\left(x,\alpha\right)=3t_{x,\alpha}(\Xi_{x})$ and (4.5) to get

$\sup_{\alpha\in\mathbb{R}_{+}}\sup_{x\in\Gamma_{\alpha}}\mathbb{P}\left(t_{x,\alpha}(\Xi_{x}+\delta_{x})>t\right)\leq
6(k+1)e^{1/8-K({\bf 0},{\bf 0})\pi\left(3C_{1}t\right)^{2}/8}$

for all $t>0$. The tail behaviour of $t_{x,\alpha}(\Xi_{x}+\delta_{x})$ and
(4.4) ensure that

$\sup_{\alpha\in\mathbb{R}_{+}}\sup_{x\in\Gamma_{\alpha}}\mathbb{E}\left(\eta\left(x,(\Xi_{x})_{\Gamma_{\alpha}}+\delta_{x}\right)^{6}\right)\leq(3.5k)^{6}\sup_{\alpha\in\mathbb{R}_{+}}\sup_{x\in\Gamma_{\alpha}}\mathbb{E}\left(t_{x,\alpha}(\Xi_{x}+\delta_{x})^{6}\right)<\infty.$

The proof of (b) is completed by applying Theorem 2.9 (ii).

###  4.2 The log volume of a forest with Gibbs point process tree locations

Marks play an important role when it is necessary to classify the points. For
example, in insurance, marks may be introduced to represent the types of
claims [Zeller and Scherer (2021)]; in thinning [Daley and Vere-Jones (2003),
p. 32], marks may be used to stand for the points retained and discarded. In
this subsection, we consider the total log volume in a random forest [Cong and
Xia (2020), Section $3.3$], where marks are used to label the species of the
trees. The estimation of the total log volume in a given range is of great
interest in forest science and forest management [Cai (1980), Li et al.
(2015)]. When modelling the natural forest, it is reasonable to assume the
locations of trees form a Gibbs point process $\overline{\Xi}$, such as a
Poisson point process or a hard-core process. As the contribution of the log
volume from different species of trees varies, we use marks to classify the
species. That is, for $x\in\overline{\Xi}$, let $M_{x}\in{T}:=\\{1,\dots,n\\}$
to be the species of the tree at position $x$. We can assume that the marks
are independent of other marks and the locations $\overline{\Xi}$. Then
$\Xi:=\sum_{x\in\overline{\Xi}}\delta_{(x,M_{x})}$ forms a marked Gibbs point
process recording the locations and species of trees in a forest. We can model
the timber volume of the tree at location $x$ by a function of the location,
the species of the tree and the configuration of trees in a finite range
around $x$, adjusted by a quantity ${\epsilon}_{x}$ due to other unspecified
factors. Formally speaking, the timber volume of a tree at location $x$ can be
denoted by
$\left(\eta(\left(x,m\right),\Xi_{\Gamma_{\alpha}},\Gamma_{\alpha})+{\epsilon}_{x}\right)\vee
0$, where $\eta$ is a non-negative bounded score function such that

$\eta(\left(x,m\right),\Xi_{\Gamma_{\alpha}},\Gamma_{\alpha})=\eta(\left(x,m\right),\Xi_{\Gamma_{\alpha}\cap
B(x,r)},\Gamma_{\alpha})$

for some positive constant $r$ and

$\eta(\left(x,m\right),\Xi_{B(x,r)},\Gamma_{\alpha_{1}})=\eta(\left(x,m\right),\Xi_{B(x,r)},\Gamma_{\alpha_{2}})$

for all $\alpha_{1}$ and $\alpha_{2}$ with
$B(x,r)\subset\Gamma_{\alpha_{1}\wedge\alpha_{2}}$. Then we have the following
result analogous to [Cong and Xia (2020), Theorem $3.3$].

######  Theorem 4.3.

If $\Xi$ is an infinite range Gibbs point process with nearly finite range
potential, ${\epsilon}_{x}$’s are ${i.i.d.}$ random variables with finite
sixth moment and the positive part ${\epsilon}_{x}^{+}:={\epsilon}_{x}\vee 0$
is non-degenerate (i.e., ${\rm Var}({\epsilon}_{x}^{+})>0$), and
${\epsilon}_{x}$’s are independent of $\Xi$, then the log volume in the range
$\Gamma_{\alpha}$ is

${\bar{W}}_{\alpha}:=\sum_{x\in\overline{\Xi}_{\Gamma_{\alpha}}}\left(\eta(\left(x,m\right),\Xi_{\Gamma_{\alpha}},\Gamma_{\alpha})+{\epsilon}_{x}\right)\vee
0$

and it satisfies

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq O\left(\alpha^{-\frac{1}{2}}\right).$

Proof.  The proof is adapted from [Cong and Xia (2020), Theorem $3.3$]. We can
construct a new marked Gibbs point process
$\Xi^{\prime}:=\sum_{x\in\overline{\Xi}}\delta_{(x,(M_{x},{\epsilon}_{x}))}$
by replacing the marks $\\{M_{x}\\}_{x\in\bar{\Xi}}$ of $\Xi$ by ${i.i.d.}$
marks $\\{(M_{x},{\epsilon}_{x})\\}_{x\in\bar{\Xi}}$ on the space
$(T\times\mathbb{R},\mathscr{T}\times\mathscr{B}(\mathbb{R}))$ independent of
the ground process $\overline{\Xi}^{\prime}=\overline{\Xi}$. The fifth moment
condition (2.4) of $\Xi^{\prime}$ follows from the Poisson-like property, as
shown in the proof of Theorem 4.1 (a), and the EDD is ensured by Lemma 3.1. As
$\bar{W}_{\alpha}$ can be represented as the sum of score function

$\displaystyle\eta^{\prime}((x,{(m,{\epsilon}_{x})}),\Xi^{\prime},\Gamma_{\alpha}):=$
$\displaystyle\eta^{\prime}((x,(m,{\epsilon}_{x})),\Xi^{\prime}_{\Gamma_{\alpha}},\Gamma_{\alpha})$
$\displaystyle:=$
$\displaystyle\left[\left(\eta(\left(x,m\right),\Xi_{\Gamma_{\alpha}},\Gamma_{\alpha})+{\epsilon}_{x}\right)\vee
0\right]\mathbf{1}_{(x,(m,{\epsilon}_{x}))\in\Xi^{\prime}_{\Gamma_{\alpha}}},$

the translation invariant property and the range-bound property are direct
results of the construction, the sixth moment condition (2.6) is guaranteed by
the boundedness of $\eta$, the moment condition of ${\epsilon}_{x}$’s and the
Minkowski inequality. Now, we can apply [Xia and Yukich (2015), Theorem 1.1]
again to get $\mathrm{Var}(\bar{W}_{\alpha})=\Omega(\alpha)$ and the proof is
then completed by applying Theorem 2.9 (i).

######  Remark 4.4.

As discussed in [Cong and Xia (2020), Remark $3.2$], if the timber volume is
determined by its $k$-nearest neighbouring trees, we can adjust the above
proof to show the distribution of the log volume $\bar{W}_{\alpha}$ satisfies
$d_{W}\left(\frac{\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha}}{\sqrt{{\rm
Var}(\bar{W}_{\alpha})}},Z\right)\leq
O\left(\alpha^{-\frac{1}{2}}\ln(\alpha)^{5d}\right).$

##  5 The proofs of the auxiliary and main results

Recalling the shift operator defined in Section 2, we can write
$g(\mathscr{X}^{x}):=\eta(\left({\bf 0},m\right),\mathscr{X}^{x})$
$=\eta(\left(x,m\right),\mathscr{X})$ (resp.
$g_{\alpha}(x,\mathscr{X}):=\eta\left(\left(x,m\right),\mathscr{X},\Gamma_{\alpha}\right)$)
for all configurations $\mathscr{X}$ with $(x,m)\in\mathscr{X}$ and $\alpha>0$
so that notations can be simplified, e.g.,

$\displaystyle W_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{(x,m)\in\Xi_{\Gamma_{\alpha}}}\eta(\left(x,m\right),\Xi)=\int_{\Gamma_{\alpha}}g(\Xi^{x})\overline{\Xi}(dx)=\sum_{x\in\overline{\Xi}_{\Gamma_{\alpha}}}g(\Xi^{x}),$
$\displaystyle\bar{W}_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{(x,m)\in\Xi_{\Gamma_{\alpha}}}\eta(\left(x,m\right),\Xi,\Gamma_{\alpha})=\int_{\Gamma_{\alpha}}{g_{\alpha}(x,{\Xi})\overline{\Xi}(dx)}=\sum_{x\in\overline{\Xi}_{\Gamma_{\alpha}}}g_{\alpha}(x,{\Xi}),$

where $\overline{\Xi}$ is the projection of $\Xi$ on $\mathbb{R}^{d}$.

We now proceed to establish a few lemmas needed in the proofs. The following
lemma bounds the difference between a normal distribution and the standard
normal distribution under the Wasserstein distance, and it can be verified
directly (see also [Chafaï and Malrieu (2010), Lemma 2.4]).

######  Lemma 5.1.

Let $F_{\mu,\sigma}$ be the distribution of $N(\mu,\sigma^{2})$, the normal
distribution with mean $\mu$ and variance $\sigma^{2}$, and $\Phi=F_{0,1}$,
then

$d_{W}(F_{\mu,\sigma},\Phi)\leq|\mu|+\frac{2}{\sqrt{2\pi}}|\sigma-1|.$

The following lemma says that the cost of throwing away the terms with large
radii of stabilisation is negligible under stabilising conditions. For
convenience, we define
$W_{\alpha,r}:=\sum_{(x,m){\in\Xi_{\Gamma_{\alpha}}}}\eta(\left(x,m\right),\Xi)\mathbf{1}_{R(x)\leq
r}$,
$\bar{W}_{\alpha,r}:={\sum_{(x,m)\in\Xi_{\Gamma_{\alpha}}}\eta(\left(x,m\right),\Xi,\Gamma_{\alpha})\mathbf{1}_{\bar{R}(x,\alpha)\leq
r}}$, which means that we through away the terms with stabilisation radii
greater than $r$ from $W_{\alpha}$ and $\bar{W}_{\alpha}$.

######  Lemma 5.2.

(a) (unrestricted case) If the score function is exponentially stabilising in
Definition 2.3, then we have

$d_{TV}(W_{\alpha},W_{\alpha,r})\leq C_{1}\alpha e^{-C_{2}r}$

for some positive constants $C_{1}$, $C_{2}$.

(b) (restricted case) If the score function is exponentially stabilising in
Definition 2.4, then we have

$d_{TV}(\bar{W}_{\alpha},\bar{W}_{\alpha,r})\leq C_{1}\alpha e^{-C_{2}r}$

for some positive constants $C_{1}$, $C_{2}$.

Proof. We first prove (b). Recall that $M_{x}\sim\mathscr{L}_{T}$ is the mark
of the point $x\in\overline{\Xi}$, and it is independent of $\Xi_{x}$. From
the construction of $\bar{W}_{\alpha}$ and $\bar{W}_{\alpha,r}$, we can see
that the event $\\{\bar{W}_{\alpha}\neq\bar{W}_{\alpha,r}\\}\subset\\{\mbox{at
least one }x\in\overline{\Xi}\cap\Gamma_{\alpha}\leavevmode\nobreak\
\mbox{with }\bar{R}(x,\alpha)>r\\}$, so from (2.3), we have

$\displaystyle d_{TV}(\bar{W}_{\alpha},\bar{W}_{\alpha,r})\leq$
$\displaystyle\mathbb{P}\left(\\{\bar{W}_{\alpha}\neq\bar{W}_{\alpha,r}\\}\right)$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\left(\\{\mbox{at least one
}x\in\overline{\Xi}\cap\Gamma_{\alpha}\leavevmode\nobreak\ {\mbox{such that
}}\bar{R}(x,\alpha)>r\\}\right)$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{\Gamma_{\alpha}}\mathbf{1}_{\bar{R}(x,\alpha)>r}\overline{\Xi}(dx)$
$\displaystyle=$
$\displaystyle\int_{\Gamma_{\alpha}}\mathbb{P}\left(\bar{R}((x,M_{x}),\alpha,\Xi_{x}+\delta_{(x,M_{x})})>r\right)\lambda
dx$ $\displaystyle\leq$ $\displaystyle\alpha\lambda\bar{\tau}(r).$ (5.1)

This, together with the stabilisation condition in Definition 2.4, gives the
claim in (b).

The claim (a) can be proved by replacing corresponding counterparts
$\bar{W}_{\alpha}$ by $W_{\alpha}$; $\bar{W}_{\alpha,r}$ by $W_{\alpha,r}$;
$\bar{R}(x,\alpha)$ by $R(x)$;
$\bar{R}((x,M_{x}),\alpha,\Xi_{x}+\delta_{(x,M_{x})})$ by
$R((x,M_{x}),\Xi_{x}+\delta_{(x,M_{x})})$ and $\bar{\tau}$ by $\tau.$

The moments of $W_{\alpha,r}$ and $W_{\alpha}$ (resp. $\bar{W}_{\alpha,r}$ and
$\bar{W}_{\alpha}$) can be established using the moment conditions required.
To begin with, we first show a statement about the moments of
$\overline{\Xi}(\Gamma_{\alpha})$. Let
$\|X\|_{p}:=\mathbb{E}\left(|X|^{p}\right)^{\frac{1}{p}}$ be the $L_{p}$ norm
of $X$ provided it is finite.

######  Lemma 5.3.

For $k\in\mathbb{N}$, if the marked point process $\Xi\sim\mathscr{P}$
satisfies the $k$th moment condition (2.4), then
$\mathbb{E}\left(\overline{\Xi}(\Gamma_{\alpha})^{k}\right)\leq O(\alpha^{k})$
for $\alpha>0$.

Proof. Since $\overline{\Xi}(B)$ is non-decreasing in $B$ in the sense of
inclusion and it is also stationary, the condition (2.4) is equivalent to that
there exists an $\alpha_{0}>0$ such that
$\mathbb{E}\left(\overline{\Xi}(\Gamma_{\alpha_{0}})^{k}\right)=\mathbb{E}\left(\overline{\Xi}(\Gamma_{\alpha_{0}}+x)^{k}\right)=:C<\infty$
for all $x\in\mathbb{R}^{d}$.

We can find a cover of $\Gamma_{\alpha}$ of the form
$\\{\Gamma_{\alpha_{0}}+x_{i}\\}_{i\leq n_{\alpha}}$ with
$n_{\alpha}=\left\lceil\left(\frac{\alpha}{\alpha_{0}}\right)^{\frac{1}{d}}\right\rceil^{d}=O(\alpha)$,
it follows from Minkowski’s inequality that

$\displaystyle\mathbb{E}\left(\overline{\Xi}(\Gamma_{\alpha})^{k}\right)=\|\overline{\Xi}(\Gamma_{\alpha})\|_{k}^{k}\leq\left(\sum_{i=1}^{n_{\alpha}}\|\overline{\Xi}(\Gamma_{\alpha_{0}}+x_{i})\|_{k}\right)^{k}=n_{\alpha}^{k}C^{k}=O(\alpha^{k}),$

as claimed.

######  Remark 5.4.

From the proof above, we can see that for arbitrary
$A\in\mathscr{B}\left(\mathbb{R}^{d}\right)$, if $A$ can be covered by
$\\{\Gamma_{\alpha_{0}}+x_{i}\\}_{i\leq n_{A}}$, then
$\mathbb{E}\left(\overline{\Xi}(A)^{k}\right)\leq n_{A}^{k}C$ for some
constant $C$.

######  Lemma 5.5.

(a) (unrestricted case) If $\Xi$ satisfies the $(2n-1)$th moment condition
(2.4) and the score function $\eta$ satisfies the $(2n)$th moment condition
(2.5), then

$\mathbb{E}\left(|W_{\alpha}|^{k}\right)\vee\mathbb{E}\left(|W_{\alpha,r}|^{k}\right)\leq
C\alpha^{2k-1}$

for some positive constant $C$ for all integers $1\leq k\leq n$.

(b) (restricted case) If $\Xi$ satisfies the $(2n-1)$th moment condition (2.4)
and the score function $\eta$ satisfies the $(2n)$th moment condition (2.6),
then

$\mathbb{E}\left(|\bar{W}_{\alpha}|^{k}\right)\vee\mathbb{E}\left(|\bar{W}_{\alpha,r}|^{k}\right)\leq
C\alpha^{2k-1}$

for some positive constant $C$ for all integers $1\leq k\leq n$.

Proof. We start with the restricted case, and the unrestricted case follows in
the same way. For $k\in\mathbb{N}$,

$\displaystyle\mathbb{E}\left(|\bar{W}_{\alpha}|^{k}\right)=$
$\displaystyle\mathbb{E}\left(\left|\int_{\Gamma_{\alpha}}g_{\alpha}(x,\Xi)\overline{\Xi}(dx)\right|^{k}\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\left(\int_{\Gamma_{\alpha}}\left|g_{\alpha}(x,\Xi)\right|\overline{\Xi}(dx)\right)^{k}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\underbrace{\int_{\Gamma_{\alpha}}\dots\int_{\Gamma_{\alpha}}}_{k\leavevmode\nobreak\
\mbox{\scriptsize{of them}}}\left|g_{\alpha}(x_{1},\Xi)\dots
g_{\alpha}(x_{k},\Xi)\right|\overline{\Xi}(dx_{1})\dots\overline{\Xi}(dx_{k})$
$\displaystyle\leq$
$\displaystyle\frac{1}{k}\mathbb{E}\underbrace{\int_{\Gamma_{\alpha}}\dots\int_{\Gamma_{\alpha}}}_{k\leavevmode\nobreak\
\mbox{\scriptsize{of
them}}}\left(\left|g_{\alpha}(x_{1},\Xi)\right|^{k}+\dots+\left|g_{\alpha}(x_{k},\Xi)\right|^{k}\right)d\overline{\Xi}(dx_{1})\dots\overline{\Xi}(dx_{k})$
$\displaystyle=$
$\displaystyle\mathbb{E}\int_{\Gamma_{\alpha}}\overline{\Xi}(\Gamma_{\alpha})^{k-1}\left|g_{\alpha}(x,\Xi)\right|^{k}\overline{\Xi}(dx)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\mathbb{E}\int_{\Gamma_{\alpha}}\left(\overline{\Xi}(\Gamma_{\alpha})^{2k-2}+\left|g_{\alpha}(x,\Xi)\right|^{2k}\right)\overline{\Xi}(dx)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\mathbb{E}\left(\overline{\Xi}(\Gamma_{\alpha})^{2k-1}\right)+\mathbb{E}\left(\int_{\Gamma_{\alpha}}|g_{\alpha}(x,\Xi_{x}+\delta_{(x,M_{x})})|^{2k}\lambda
dx\right)\right]$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}C_{1}\alpha^{2k-1}+\frac{1}{2}C_{2}\alpha,$ (5.2)

for some positive constants $C_{1}$ and $C_{2}$, the second and third
inequalities follow from the fact that
$\Pi_{i=1}^{j}y_{i}\leq\frac{1}{j}\sum_{i=1}^{j}y_{i}^{j}$ for all $y_{i}\geq
0$, $1\leq i\leq j$, the last equality follows from (2.3) and the last
inequality follows from Lemma 5.3. Then we can find a common positive constant
$C$ such that $\mathbb{E}\left(|\bar{W}_{\alpha}|^{k}\right)\leq
C\alpha^{2k-1}$ for all integers $k\leq n$. The claim
$\mathbb{E}\left(|\bar{W}_{\alpha,r}|^{k}\right)\leq C\alpha^{2k-1}$ can be
proved by following exactly the same steps but replacing $g_{\alpha}(x,\Xi)$
with $g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x,\alpha)\leq r}$.

The statement for the unrestricted case is also true, which can be proved by
replacing the corresponding counterparts $\bar{W}_{\alpha}$ by $W_{\alpha}$,
$\bar{W}_{\alpha,r}$ by $W_{\alpha,r}$, and $g_{\alpha}$ by $g$.

######  Remark 5.6.

The proof of Lemma 5.5 does not depend on the shape of $\Gamma_{\alpha}$, so
the claims still hold if we replace $\Gamma_{\alpha}$ with a set
$A\in\mathscr{B}(\mathbb{R})$ such that $A$ satisfies the assumption in Remark
5.4 with $n_{A}\leq O(\mathrm{Vol}(A))$.

######  Remark 5.7.

Using the same idea as in the proof of (5.2), for each $1\leq i\leq k$, by
taking the range of $x_{i}$ in $A_{i}$ satisfying the condition in Remark 5.6
instead of $\Gamma_{\alpha}$, we have, for all integers $1\leq k\leq n$,

$\displaystyle\mathbb{E}\int_{A_{1}}\dots\int_{A_{k}}\left|g_{\alpha}(x_{1},\Xi)\dots
g_{\alpha}(x_{k},\Xi)\right|\overline{\Xi}(dx_{1})\dots\overline{\Xi}(dx_{k})\leq
C\max_{1\leq j\leq k}\mathrm{Vol}(A_{j})^{2k-1}$

for some constant $C$.

With these preparations, we are ready to bound the differences $\left|{\rm
Var}\left(W_{\alpha}\right)-{\rm Var}\left(W_{\alpha,r}\right)\right|$ and
$\left|{\rm Var}\left(\bar{W}_{\alpha}\right)-{\rm
Var}\left(\bar{W}_{\alpha,r}\right)\right|$.

######  Lemma 5.8.

(a) (unrestricted case) Assume that $\Xi$ satisfies the fifth moment condition
(2.4) and the score function $\eta$ satisfies the sixth moment condition
(2.5). If $\eta$ is exponentially stabilising in Definition 2.3, then there
exist positive constants $\alpha_{0}$ and $C$ such that

$\left|{\rm Var}\left(W_{\alpha}\right)-{\rm
Var}\left(W_{\alpha,r}\right)\right|\leq\frac{1}{\alpha}$ (5.3)

for all $\alpha\geq\alpha_{0}$ and $r\geq C\ln(\alpha)$.

(b) (restricted case) Assume that $\Xi$ satisfies the fifth moment condition
(2.4) and the score function $\eta$ satisfies the sixth moment condition
(2.6). If $\eta$ is exponentially stabilising in Definition 2.4, then there
exist positive constants $\alpha_{0}$ and $C$ such that

$\left|{\rm Var}\left(\bar{W}_{\alpha}\right)-{\rm
Var}\left(\bar{W}_{\alpha,r}\right)\right|\leq\frac{1}{\alpha}$ (5.4)

for all $\alpha\geq\alpha_{0}$ and $r\geq C\ln(\alpha)$.

Proof. We start with (5.4). From Lemma 5.5 (b), taking $n=3$ and $k=1$ or
$k=3$, we have

$\max\left\\{\|\bar{W}_{\alpha}\|_{1},\|\bar{W}_{\alpha,r}\|_{1}\right\\}\leq
C_{0}\alpha,\leavevmode\nobreak\
\max\left\\{\|\bar{W}_{\alpha}\|_{3},\|\bar{W}_{\alpha,r}\|_{3}\right\\}\leq
C_{0}\alpha^{\frac{5}{3}}$ (5.5)

for some positive constant $C_{0}\geq 1$. Without loss, we assume $\alpha>1$.
Since

$\left|{\rm Var}\left(\bar{W}_{\alpha}\right)-{\rm
Var}\left(\bar{W}_{\alpha,r}\right)\right|\leq\left|\mathbb{E}\left(\bar{W}_{\alpha}^{2}-\bar{W}_{\alpha,r}^{2}\right)\right|+\left|\left(\mathbb{E}\bar{W}_{\alpha}\right)^{2}-\left(\mathbb{E}\bar{W}_{\alpha,r}\right)^{2}\right|,$
(5.6)

using the assumption of stabilisation, we show that each of the terms at the
right hand side of (5.6) is bounded by $\frac{1}{2\alpha}$ for $\alpha$ and
$r$ sufficiently large. Clearly, the definition of $\bar{W}_{\alpha,r}$
implies that $\bar{W}_{\alpha}^{2}-\bar{W}_{\alpha,r}^{2}=0$ if
$\bar{R}(x,\alpha)\leq r$ for all $x\in\overline{\Xi}_{\Gamma_{\alpha}}$,
hence it remains to tackle $E_{r,\alpha}:=\\{\bar{R}(x,\alpha)\leq
r\leavevmode\nobreak\ \mbox{for all
}x\in\overline{\Xi}_{\Gamma_{\alpha}}\\}^{c}$. As shown in the proof of (5.1),
$\mathbb{P}\left(E_{r,\alpha}\right)\leq\alpha C_{1}e^{-C_{2}r}$, which,
together with (5.5), Hölder’s inequality and Minkowski’s inequality, ensures

$\displaystyle\left|\mathbb{E}\left(\bar{W}_{\alpha}^{2}-\bar{W}_{\alpha,r}^{2}\right)\right|$
$\displaystyle=\left|\mathbb{E}\left[\left(\bar{W}_{\alpha}^{2}-\bar{W}_{\alpha,r}^{2}\right)\mathbf{1}_{E_{r,\alpha}}\right]\right|$
$\displaystyle\leq\|\bar{W}_{\alpha}^{2}-\bar{W}_{\alpha,r}^{2}\|_{\frac{3}{2}}\|\mathbf{1}_{E_{r,\alpha}}\|_{3}$
$\displaystyle\leq\left(\|\bar{W}_{\alpha}^{2}\|_{\frac{3}{2}}+\|\bar{W}_{\alpha,r}^{2}\|_{\frac{3}{2}}\right)\mathbb{P}(E_{r,\alpha})^{\frac{1}{3}}$
$\displaystyle=\left(\|\bar{W}_{\alpha}\|_{3}^{2}+\|\bar{W}_{\alpha,r}\|_{3}^{2}\right)\mathbb{P}(E_{r,\alpha})^{\frac{1}{3}}\leq
2C_{0}^{2}\alpha^{\frac{10}{3}}\left(\alpha
C_{1}e^{-C_{2}r}\right)^{\frac{1}{3}}.$ (5.7)

For the remaining term of (5.6), we have

$\left|\left(\mathbb{E}\bar{W}_{\alpha}\right)^{2}-\left(\mathbb{E}\bar{W}_{\alpha,r}\right)^{2}\right|=\left|\mathbb{E}\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha,r}\right|\left|\mathbb{E}\bar{W}_{\alpha}+\mathbb{E}\bar{W}_{\alpha,r}\right|.$

The bound (5.5) implies
$\left|\mathbb{E}\bar{W}_{\alpha}+\mathbb{E}\bar{W}_{\alpha,r}\right|\leq
2C_{0}\alpha$. However, using Hölder’s inequality, Minkowski’s inequality and
(5.5) again, we have

$\displaystyle\left|\mathbb{E}\bar{W}_{\alpha}-\mathbb{E}\bar{W}_{\alpha,r}\right|$
$\displaystyle=\left|\mathbb{E}\left[\left(\bar{W}_{\alpha}-\bar{W}_{\alpha,r}\right)\mathbf{1}_{E_{r,\alpha}}\right]\right|$
$\displaystyle\leq\|\bar{W}_{\alpha}-\bar{W}_{\alpha,r}\|_{3}\|\mathbf{1}_{E_{r,\alpha}}\|_{\frac{3}{2}}$
$\displaystyle\leq\left(\|\bar{W}_{\alpha}\|_{3}+\|\bar{W}_{\alpha,r}\|_{3}\right)\mathbb{P}(E_{r,\alpha})^{\frac{2}{3}}$
$\displaystyle\leq 2C_{0}\alpha^{\frac{5}{3}}\left(\alpha
C_{1}e^{-C_{2}r}\right)^{\frac{2}{3}},$ (5.8)

giving

$\left|\left(\mathbb{E}\bar{W}_{\alpha}\right)^{2}-\left(\mathbb{E}\bar{W}_{\alpha,r}\right)^{2}\right|\leq
4C_{0}^{2}\alpha^{\frac{8}{3}}\left(\alpha
C_{1}e^{-C_{2}r}\right)^{\frac{2}{3}}.$ (5.9)

We set $r=C\ln(\alpha)$ in the upper bounds of (5.7) and (5.9) and find $C$
such that both bounds are bounded by $1/(2\alpha)$, completing the proof of
(5.4).

A line-by-line repetition of the above proof with $\bar{W}_{\alpha}$ and
$\bar{W}_{\alpha,r}$ replaced by $W_{\alpha}$ and $W_{\alpha,r}$ and
$\bar{R}(x,\alpha)$ replaced by $R(x)$ gives (5.3).

We can now establish the lower bounds for ${\rm Var}\left(W_{\alpha}\right)$
and ${\rm Var}\left(\bar{W}_{\alpha}\right)$ using the variation conditions
(2.7) and (2.8). To this end, we start with a lemma. Recall that
$P:=\lambda\mathbb{E}(\eta((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(0M_{\bm{0}})})$
for the unrestricted case and
$\bar{P}:=\lambda\mathbb{E}(\bar{\eta}((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(0,M_{\bm{0}})})$
for the restricted case.

######  Lemma 5.9.

(a) (unrestricted case) Assume that $\Xi$ satisfies the EDD and the fifth
moment condition (2.4), and the score function $\eta$ is translation invariant
and satisfies the sixth moment condition (2.5). If $\eta$ is exponentially
stabilising in Definition 2.3, then

$\int_{\Gamma_{\alpha}}\int_{\mathbb{R}^{d}}\mathbb{E}\left[(g(x,\Xi)\overline{\Xi}(dx)-Pdx)(g(y,\Xi)\overline{\Xi}(dy)-Pdy)\right]=\alpha\sigma^{2}<\infty.$

Furthermore, for any fixed $\alpha_{1}>0$,

$\int_{\Gamma_{\alpha_{1}}}\int_{\mathbb{R}^{d}\backslash\Gamma_{\alpha_{2}}}\mathbb{E}\left[(g(y,\Xi)\overline{\Xi}(dy)-Pdy)(g(x,\Xi)\overline{\Xi}(dx)-Pdx)\right]$

converges to $0$ exponentially fast as $\alpha_{2}\rightarrow\infty$.

(b) (restricted case) Assume that $\Xi$ satisfies the EDD and the fifth moment
condition (2.4), and the score function $\eta$ is translation invariant and
satisfies the sixth moment condition (2.6). If $\eta$ is exponentially
stabilising in Definition 2.4, then

$\int_{\Gamma_{\alpha}}\int_{\mathbb{R}^{d}}\mathbb{E}\left[(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)\right]=\alpha\bar{\sigma}^{2}<\infty.$

Furthermore, for any fixed $\alpha_{1}>0$,

$\int_{\Gamma_{\alpha_{1}}}\int_{\mathbb{R}^{d}\backslash\Gamma_{\alpha_{2}}}\mathbb{E}\left[(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\right]$

converges to $0$ exponentially fast as $\alpha_{2}\rightarrow\infty$.

Proof. We start with the restricted case first. It is sufficient to show that,
for any fixed $\alpha_{1}>0$,

$\int_{\Gamma_{\alpha_{1}}}\int_{\mathbb{R}^{d}\backslash\Gamma_{\alpha_{2}}}\mathbb{E}\left[(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\right]$

converges to $0$ exponentially fast as $\alpha_{2}\rightarrow\infty$. Bearing
in mind Remark 5.7, at the cost of no more than $C_{0}(\alpha_{1}\vee 1)^{3}$,
without loss, we may assume that
$\alpha^{1/d}_{2}>\left(6\alpha^{1/d}_{1}\right)\vee 2$ and
$\frac{\alpha_{2}^{1/d}}{\ln\left(\alpha_{2}^{1/d}(2\sqrt{d}+1/3)\right)}\geq
12\theta_{3}$ with $\theta_{3}$ in the definition of the EDD. The space
$\mathbb{R}^{d}\backslash\Gamma_{\alpha_{2}}$ can be divided into sets of the
form
$\left\\{\Gamma_{\alpha_{2}(1+l)^{d}}\backslash\Gamma_{\alpha_{2}l^{d}}\right\\}_{l\in\mathbb{N}}=:\\{A_{l}\\}_{l\in\mathbb{N}}$.
Then

$\displaystyle\int_{\Gamma_{\alpha_{1}}}\int_{\mathbb{R}^{d}\backslash\Gamma_{\alpha_{2}}}\mathbb{E}\left[(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\right]$
$\displaystyle=$
$\displaystyle\sum_{l\in\mathbb{N}}\int_{\Gamma_{\alpha_{1}}}\int_{A_{l}}\mathbb{E}\left[(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\right]$
$\displaystyle=$
$\displaystyle\sum_{l\in\mathbb{N}}\mathbb{E}\left[\int_{\Gamma_{\alpha_{1}}}\int_{A_{l}}(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\right]$

if the sum in the last line is absolutely convergent.

For $l\in\mathbb{N}$, ${\rm
diam}\left(B\left(A_{l},l\alpha_{2}^{1/d}/6\right)\right)=\alpha_{2}^{1/d}(\sqrt{d}(1+l)+l/3)\geq{\rm
diam}\left(B\left(\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6\right)\right)=\sqrt{d}\alpha_{1}^{1/d}+l\alpha_{2}^{1/d}/3$,
and
$d\left(B\left(\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6\right),B\left(A_{l},l\alpha_{2}^{1/d}/6\right)\right)=\frac{l\alpha_{2}^{1/d}}{6}-\frac{\alpha_{1}^{1/d}}{2}\geq\frac{l\alpha_{2}^{1/d}}{12}$.
Since $x/\ln(ax)$ for $a>0$ is an increasing function of $x\geq e/a$, we have

$\frac{l\alpha_{2}^{1/d}}{\ln\left(\alpha_{2}^{1/d}(\sqrt{d}(1+l)+l/3)\right)}\geq\frac{l\alpha_{2}^{1/d}}{\ln\left(l\alpha_{2}^{1/d}(2\sqrt{d}+1/3)\right)}\geq\frac{\alpha_{2}^{1/d}}{\ln\left(\alpha_{2}^{1/d}(2\sqrt{d}+1/3)\right)}\geq
12\theta_{3},$

which ensures

$\displaystyle
d\left(B\left(\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6\right),B\left(A_{l},l\alpha_{2}^{1/d}/6\right)\right)$
$\displaystyle\geq\theta_{3}\ln\left({\rm
diam}\left(B\left(\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6\right)\right)\vee{\rm
diam}\left(B\left(A_{l},l\alpha_{2}^{1/d}/6\right)\right)\vee 1\right).$

With (2.3) we can show that
$\mathbb{E}\int_{A}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)=0$ for all
bounded measurable set $A\subset\mathbb{R}^{d}$. Let $\tilde{\Xi}$ denote an
independent copy of $\Xi$, then
$\mathbb{E}(\int_{\Gamma_{\alpha_{1}}}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\int_{A_{l}}(\bar{g}(y,\tilde{\Xi})\overline{\tilde{\Xi}}(dy)-\bar{P}dy))=0$.
For simplicity, we write $S_{A}:=\int_{A}\bar{g}(x,\Xi)\overline{\Xi}(dx)$,
$S_{A,r}:=\int_{A}\bar{g}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)$ and the corresponding counterparts with $\tilde{\Xi}$
instead of $\Xi$ as $\tilde{S}_{A}$ and $\tilde{S}_{A,r}$ for all bounded
measurable sets $A\subset\mathbb{R}^{d}$. Using the stabilising condition in
Definition 2.4 and the EDD, we can get an upper bound for
$d_{TV}((S_{\Gamma_{\alpha_{1}}},\tilde{S}_{A_{l}}),(S_{\Gamma_{\alpha_{1}}},S_{A_{l}}))$
as follows:

$\displaystyle
d_{TV}((S_{\Gamma_{\alpha_{1}}}-\alpha_{1}\bar{P},\tilde{S}_{A_{l}}-\mathrm{Vol}(A_{l})\bar{P}),(S_{\Gamma_{\alpha_{1}}}-\alpha_{1}\bar{P},S_{A_{l}}-\mathrm{Vol}(A_{l})\bar{P}))$
$\displaystyle=$ $\displaystyle
d_{TV}((S_{\Gamma_{\alpha_{1}}},\tilde{S}_{A_{l}}),(S_{\Gamma_{\alpha_{1}}},S_{A_{l}}))$
$\displaystyle\leq$ $\displaystyle\mathbb{P}(S_{\Gamma_{\alpha_{1}}}\neq
S_{\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6})+2\mathbb{P}(\tilde{S}_{A_{l}}\neq\tilde{S}_{A_{l},l\alpha_{2}^{1/d}/6})$
$\displaystyle+d_{TV}((S_{\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6},\tilde{S}_{A_{l},l\alpha_{2}^{1/d}/6}),(S_{\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6},S_{A_{l},l\alpha_{2}^{1/d}/6}))$
$\displaystyle\leq$
$\displaystyle\lambda(\alpha_{1}+2\mathrm{Vol}(A_{l}))\bar{\tau}(l\alpha_{2}^{1/d}/6)+\beta_{B(\Gamma_{\alpha_{1}},l\alpha_{2}^{1/d}/6),B(A_{l},l\alpha_{2}^{1/d}/6)}$
$\displaystyle\leq$ $\displaystyle C_{3}e^{-C_{4}l\alpha_{2}^{1/d}},$

for some positive constants $C_{3}$ and $C_{4}$ which are independent of $l$
and $\alpha_{2}$, where the second inequality follows from the same argument
as that for (5.1), and the last inequality follows from the stabilising
condition in Definition 2.4 and the EDD.

Using [Barbour, Holst and Janson (1992), p. 254], we can find a suitable
coupling $((X_{1},X_{2}),(Y_{1},Y_{2}))$ of
$\left(S_{\Gamma_{\alpha_{1}}}-\alpha_{1}\bar{P},\tilde{S}_{A_{l}}-\mathrm{Vol}(A_{l})\bar{P}\right)$
and
$\left(S_{\Gamma_{\alpha_{1}}}-\alpha_{1}\bar{P},S_{A_{l}}-\mathrm{Vol}(A_{l})\bar{P}\right)$
such that
$(X_{1},X_{2})\overset{d}{=}\left(S_{\Gamma_{\alpha_{1}}}-\alpha_{1}\bar{P},\tilde{S}_{A_{l}}-\mathrm{Vol}(A_{l})\bar{P}\right)$,
$(Y_{1},Y_{2})\overset{d}{=}\left(S_{\Gamma_{\alpha_{1}}}-\alpha_{1}\bar{P},S_{A_{l}}-\mathrm{Vol}(A_{l})\bar{P}\right)$,
$\mathbb{P}(E):=\mathbb{P}((X_{1},X_{2})\neq(Y_{1},Y_{2}))\leq
C_{3}e^{-C_{4}l\alpha_{2}^{1/d}}$. With Remark 5.6, we can see that
$\|S_{A}-\mathrm{Vol}(A)\bar{P}\|_{3}\leq C_{5}\mathrm{Vol}(A)^{\frac{5}{3}}$
for $A=\Gamma_{\alpha}$ or $A_{l}$ for $l\geq 1$, $\alpha>0$ for some constant
$C_{5}$.

From Hölder’s inequality and Remark 5.6, we can see that

$\displaystyle\left|\mathbb{E}\left(\int_{\Gamma_{\alpha_{1}}}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\int_{A_{l}}(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)\right)\right|$
$\displaystyle=$
$\displaystyle\left|\mathbb{E}\left(\int_{\Gamma_{\alpha_{1}}}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\int_{A_{l}}(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)\right)\right.$
$\displaystyle\left.-\mathbb{E}\left(\int_{\Gamma_{\alpha_{1}}}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\int_{A_{l}}(\bar{g}(y,\tilde{\Xi})\overline{\tilde{\Xi}}(dy)-\bar{P}dy)\right)\right|$
$\displaystyle=$
$\displaystyle\left|\mathbb{E}\left(\int_{\Gamma_{\alpha_{1}}}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\int_{A_{l}}(\bar{g}(y,\Xi)\overline{\Xi}(dy)-\bar{P}dy)\mathbf{1}_{E}\right)\right.$
$\displaystyle\left.-\mathbb{E}\left(\int_{\Gamma_{\alpha_{1}}}(\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx)\int_{A_{l}}(\bar{g}(y,\tilde{\Xi})\overline{\tilde{\Xi}}(dy)-\bar{P}dy)\mathbf{1}_{E}\right)\right|$
$\displaystyle\leq$ $\displaystyle C_{6}e^{-C_{7}l\alpha_{2}^{1/d}}$ (5.10)

for some positive constants $C_{6}$ and $C_{7}$. This, together with the
translation invariant property and the definition of $\bar{\sigma}^{2}$,
completes the proof for the restricted case.

The statement for the unrestricted case can be proved by replacing
corresponding counterparts $g$ by $\bar{g}$; $\bar{R}$ by $R$; $\bar{P}$ by
$P$; $\bar{\tau}$ by $\tau$ and $\bar{\sigma}^{2}$ by $\sigma^{2}$.

To show the volume of the variance, we need a lemma saying that we can
approximate $\alpha\sigma^{2}$ (resp. $\alpha\bar{\sigma}^{2}$) using the
score function $\eta$ restricted to $R\leq r$ (resp. $\bar{R}\leq r$). For
convenience, let
$P_{r}:=\lambda\mathbb{E}(\eta((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(\bm{0},M_{\bm{0}})})\mathbf{1}_{R((\bm{0},M_{\bm{0}}),\Xi_{\bm{0}}+\delta_{(\bm{0},M_{\bm{0}})})\leq
r})$,
$\bar{P}_{\alpha,x,r}:=\lambda\mathbb{E}(\eta((x,M_{x}),\Xi_{x}+\delta_{(x,M_{x})},\Gamma_{\alpha})\mathbf{1}_{\bar{R}((x,M_{x}),\alpha,\Xi_{x}+\delta_{(x,M_{x})})\leq
r})$.

######  Lemma 5.10.

(a) (unrestricted case) If the conditions in Lemma 5.9 (a) hold and
$\sigma^{2}>0$, then for a fixed sufficiently large $\alpha_{0}>0$, there
exist positive constants $\alpha_{1}$ and $C$ such that

$\displaystyle\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}(g(x,\Xi)\mathbf{1}_{R(x)\leq
r}\overline{\Xi}(dx)-P_{r}dx)\int_{\Gamma_{\alpha}}(g(y,\Xi)\mathbf{1}_{R(y)\leq
r}\overline{\Xi}(dy)-P_{r}dy)\right]$ $\displaystyle\in$
$\displaystyle[\frac{1}{2}\alpha_{0}\sigma^{2},\frac{3}{2}\alpha_{0}\sigma^{2}]$

for all $\alpha\geq\alpha_{1}$ and $r\geq C\ln(\alpha)$, $z\in\Gamma_{\alpha}$
such that $d(z,\partial\Gamma_{\alpha})\geq 5r$.

(b) (restricted case) If the conditions in Lemma 5.9 (b) hold and
$\bar{\sigma}^{2}>0$, then for a fixed sufficiently large $\alpha_{0}>0$,
there exist positive constants $\alpha_{1}$ and $C$ such that

$\displaystyle\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}(g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx)\int_{\Gamma_{\alpha}}(g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)-\bar{P}_{\alpha,y,r}dy)\right]$ $\displaystyle\in$
$\displaystyle[\frac{1}{2}\alpha_{0}\bar{\sigma}^{2},\frac{3}{2}\alpha_{0}\bar{\sigma}^{2}]$

for all $\alpha\geq\alpha_{1}$ and $r\geq C\ln(\alpha)$, $z\in\Gamma_{\alpha}$
such that $d(z,\partial\Gamma_{\alpha})\geq 5r$.

Proof. We prove the restricted case only, and the unrestricted case can be
proved similarly.

Let
$\bar{P}_{r}:=\lambda\mathbb{E}(\bar{g}(\bm{0},\Xi_{\bm{0}}+\delta_{(\bm{0},M_{\bm{0}})})\mathbf{1}_{\bar{R}((\bm{0},M_{\bm{0}}),\alpha,\Xi_{\bm{0}}+\delta_{(\bm{0},M_{\bm{0}})})\leq
r})$, then from the moment condition (2.6), we can see that
$\max_{\alpha\in\mathbb{R}_{+},r\in\mathbb{R}_{+},x\in\Gamma_{\alpha}}\\{|\bar{P}_{r}|,|\bar{P}_{\alpha,x,r}|\\}\leq
C_{1}$ for some constant $C_{1}$. By the translation-invariance, if $x$ and
$r$ satisfy $B(x,r)\subset\Gamma_{\alpha}$, then
$g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}=\bar{g}(x,\Xi)\mathbf{1}_{R(x)\leq r}$ and
$\bar{P}_{r}=\bar{P}_{\alpha,x,r}$, where
$R(x)=\lim_{\alpha\to\infty}\bar{R}(x)$, see the discussion after Definition
2.6. The stabilising condition ensures that
$\mathbb{P}(\bar{R}((\bm{0},M_{\bm{0}}),\alpha,\Xi_{\bm{0}}+\delta_{(\bm{0},M_{\bm{0}})})\geq
r)\leq\bar{\tau}(r)$ decreases exponentially fast. Arguing in the same way as
that for (5.7), it follows from the moment condition (2.6) and Hölder’s
inequality that $|\bar{P}-\bar{P}_{r}|\leq\alpha^{-2}$ for all $r\geq
C_{2}\ln(\alpha)$ for some positive constant $C_{2}$. Writing for brevity
$T_{\alpha,r}(x,\Xi,dx):=g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx$ and
$T_{r}(x,\Xi,dx):=\bar{g}(x,\Xi)\mathbf{1}_{R(x)\leq
r}\overline{\Xi}(dx)-\bar{P}dx$, we have from Lemma 5.5 (b) that there exists
a positive constant $\alpha_{1}^{\prime}\geq e$ such that

$\displaystyle\left|\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}T_{\alpha,r}(x,\Xi,dx)\int_{B(z+\Gamma_{\alpha_{0}},3r)}T_{\alpha,r}(y,\Xi,dy)\right]\right.$
$\displaystyle\ \ \
-\left.\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}T_{r}(x,\Xi,dx)\int_{{B(z+\Gamma_{\alpha_{0}}},3r)}T_{r}(y,\Xi,dy)\right]\right|$
$\displaystyle\leq\frac{1}{8}\alpha_{0}\bar{\sigma}^{2}$ (5.11)

for all $\alpha\geq\alpha_{1}^{\prime},r\geq C_{2}\ln(\alpha)$.

Following the proof of (5.1), we have
$\mathbb{P}(E^{c}):=\mathbb{P}\left(\\{\bar{R}(x)\leq r\leavevmode\nobreak\
\mbox{for all}\leavevmode\nobreak\
x\in\overline{\Xi}_{\Gamma_{\alpha}}\\}^{c}\right)\leq\alpha\bar{\tau}(r)$.
Also, we can see that on the event $E$,

$\int_{z+\Gamma_{\alpha_{0}}}T_{r}(x,\Xi,dx)\int_{{B(z+\Gamma_{\alpha_{0}}},3r)}T_{r}(y,\Xi,dy)$

is the same as

$\int_{z+\Gamma_{\alpha_{0}}}T_{\infty}(x,\Xi,dx)\int_{{B(z+\Gamma_{\alpha_{0}}},3r)}T_{\infty}(y,\Xi,dy),$

where $T_{\infty}(x,\Xi,dx):=\bar{g}(x,\Xi)\overline{\Xi}(dx)-\bar{P}dx$.
Using Hölder’s inequality as for (5.7), we can find a $C_{3}\in\mathbb{R}_{+}$
such that

$\displaystyle\left|\int_{z+\Gamma_{\alpha_{0}}}T_{r}(x,\Xi,dx)\int_{{B(z+\Gamma_{\alpha_{0}}},3r)}T_{r}(y,\Xi,dy)\right.$
$\displaystyle\ \ \
\left.-\int_{z+\Gamma_{\alpha_{0}}}T_{\infty}(x,\Xi,dx)\int_{{B(z+\Gamma_{\alpha_{0}}},3r)}T_{\infty}(y,\Xi,dy)\right|$
$\displaystyle\leq\frac{1}{8}\alpha_{0}\bar{\sigma}^{2}$ (5.12)

for all $r\geq C_{3}\ln(\alpha)$.

Using the translation invariant property, and replacing
$\mathbb{R}^{d}\backslash\Gamma_{\alpha_{2}}$ by $\mathbb{R}^{d}\backslash
B(\Gamma_{\alpha_{0}},3r)$ in the proof of Lemma 5.9, with necessary minor
adjustment, we obtain

$\displaystyle\left|\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}T_{\infty}(x,\Xi,dx)\int_{{B(z+\Gamma_{\alpha_{0}}},3r)}T_{\infty}(y,\Xi,dy)\right]-\alpha_{0}\bar{\sigma}^{2}\right|$
$\displaystyle=\left|\mathbb{E}\left[\int_{\Gamma_{\alpha_{0}}}T_{\infty}(x,\Xi,dx)\int_{B(\Gamma_{\alpha_{0}},3r)}T_{\infty}(y,\Xi,dy)\right]-\alpha_{0}\bar{\sigma}^{2}\right|$
$\displaystyle\leq\frac{1}{8}\alpha_{0}\bar{\sigma}^{2}$ (5.13)

for $r\geq C_{4}\ln(\alpha)$. On the other hand, the EDD ensures that we can
find an independent copy $\tilde{\Xi}$ of $\Xi$ such that

$E_{1}:=\\{\Xi_{B(z+\Gamma_{\alpha_{0}},r)\cup(\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},2r))}\neq\Xi_{B(z+\Gamma_{\alpha_{0}},r)}\cup\tilde{\Xi}_{\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},2r)}\\}$

satisfies

$\mathbb{P}(E_{1})=\beta_{B(z+\Gamma_{\alpha_{0}},r),\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},2r)}\leq\theta_{1}\left(\left(\sqrt{d}\alpha^{2\theta_{0}/d}\right)\vee
1\right)e^{-\theta_{2}r}.$ (5.14)

Since $\mathbb{E}T_{\alpha,r}(x,\Xi,dx)=0$, following the same argument as
that for (5.10) and applying Hölder’s inequality in the first inequality, the
moment condition (2.6) and (5.14) in the last inequality below, we get

$\displaystyle\left|\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}T_{\alpha,r}(x,\Xi,dx)\int_{\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},3r)}T_{\alpha,r}(y,\Xi,dy)\right]\right|$
$\displaystyle=\left|\mathbb{E}\left[{\bf
1}_{E_{1}}\int_{z+\Gamma_{\alpha_{0}}}T_{\alpha,r}(x,\Xi,dx)\int_{\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},3r)}T_{\alpha,r}(y,\Xi,dy)\right]\right.$
$\displaystyle\ \ \ -{\bf
1}_{E_{1}}\left.\mathbb{E}\left[\int_{z+\Gamma_{\alpha_{0}}}T_{\alpha,r}(x,\Xi,dx)\int_{\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},3r)}T_{\alpha,r}(y,\tilde{\Xi},dy)\right]\right|$
$\displaystyle\leq\left\|\int_{z+\Gamma_{\alpha_{0}}}T_{\alpha,r}(x,\Xi,dx)\int_{\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},3r)}T_{\alpha,r}(y,\Xi,dy)\right\|_{3/2}\mathbb{P}(E_{1})^{1/3}$
$\displaystyle\ \ \
+\left\|\int_{z+\Gamma_{\alpha_{0}}}T_{\alpha,r}(x,\Xi,dx)\int_{\Gamma_{\alpha}\backslash
B(z+\Gamma_{\alpha_{0}},3r)}T_{\alpha,r}(y,\tilde{\Xi},dy)\right\|_{3/2}\mathbb{P}(E_{1})^{1/3}$
$\displaystyle\leq\frac{1}{8}\alpha_{0}\bar{\sigma}^{2}$ (5.15)

for all $z$ satisfying $d(z,\partial\Gamma_{\alpha})\geq 5r$, $r\geq
C_{5}\ln(\alpha)$ and $\alpha\geq\alpha_{2}^{\prime}$. Collecting (5.11),
(5.12), (5.13) and (5.15), we obtain claim (b) with
$C=\max\\{C_{2},C_{3},C_{4},C_{5}\\}$ and
$\alpha_{1}=\max\\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\\}$.

The statement for the unrestricted case can be proved by replacing
$\bar{P}_{\alpha,x,r}$ and $\bar{P}_{r}$ by $P_{\alpha,x,r}$ and $P_{r}$;
$\bar{g}$ by $g$; $\bar{R}$ by $R$; $\bar{\tau}$ by $\tau$; $\bar{P}$ by $P$.

Together with the variation conditions (2.7) and (2.8), we can show that the
variances of $W_{\alpha}$ and $\bar{W}_{\alpha}$ are of the order $\alpha$ as
claimed in Theorem 2.11.

Proof of Theorem 2.11. We show the statement for the restricted case first,
and the statement for the unrestricted case can be shown in the same way.

To begin with, we choose $\alpha_{0}$ and $C$ such that Lemma 5.8 holds, i.e.,
$|{\rm Var}(\bar{W}_{\alpha})-{\rm
Var}(\bar{W}_{\alpha,r})|\leq\frac{1}{\alpha}$ for $r\geq C\ln(\alpha)$ and
$\alpha\geq\alpha_{0}$. We then choose $C_{1}\geq C$ and
$\alpha_{1}\geq\alpha_{0}\vee e$ such that Lemma 5.10 (b) holds for all $r\geq
C_{1}\ln(\alpha)$ and $\alpha\geq\alpha_{1}$. In the rest of the proof, we fix
$r=C_{1}\ln(\alpha)$. Replacing $\theta_{3}$ in the definition of the EDD with
$2C_{1}$ if necessary, we assume $\theta_{3}\geq 2C_{1}$. Cover
$\Gamma_{\alpha}\backslash B(\partial\Gamma_{\alpha},5\theta_{3}\ln(\alpha))$
with disjoint cubes $\mathbb{C}_{1},\dots,\mathbb{C}_{n_{\alpha}}$ each with a
volume between $\alpha_{0}$ and $2\alpha_{0}$ and intersects
$\Gamma_{\alpha}\backslash B(\partial\Gamma_{\alpha},5\theta_{3}\ln(\alpha))$,
then the number of cubes $n_{\alpha}$ has the same order as $\alpha$. For
convenience, let $\mathbb{C}_{\alpha}:=\cup_{1\leq i\leq
n_{\alpha}}\mathbb{C}_{i}$, then from Lemma 5.10 (b), for
$\alpha\geq\alpha_{1}$,

$\displaystyle\mathbb{E}\left[\int_{\mathbb{C}_{\alpha}}(g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx)\int_{\Gamma_{\alpha}}(g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)-\bar{P}_{\alpha,y,r}dy)\right]$
$\displaystyle\in[\frac{1}{2}n_{\alpha}\alpha_{0}\bar{\sigma}^{2},3n_{\alpha}\alpha_{0}\bar{\sigma}^{2}].$
(5.16)

Next, we show that

$\displaystyle\left|\mathbb{E}\left[\int_{\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha}}(g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx)\int_{\Gamma_{\alpha}}(g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)-\bar{P}_{\alpha,y,r}dy)\right]\right|$ $\displaystyle\leq
O\left((\ln\alpha)^{2d+1}\alpha^{\frac{d-1}{d}}\right).$ (5.17)

To this end, we choose $C_{2}>1$ such that $\theta_{2}\theta_{3}C_{2}\geq
13.5+\theta_{0}/d$ and $C_{2}\theta_{3}>2C_{1}$, where $\theta_{0},\theta_{2}$
are as in the EDD, divide $\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha}$ into
at most $n_{\mathbb{D}}=O\left(\alpha^{\frac{d-1}{d}}\ln(\alpha)\right)$ cubes
$\\{\mathbb{D}_{i}\\}_{1\leq i\leq n_{\mathbb{D}}}$ having diameters between
$1$ and $\theta_{3}\ln(\alpha)$. For each $x\in\mathbb{D}_{i}$,
$T(x,\Xi,dx):=g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx$ is completely determined by
$\Xi_{B(\mathbb{D}_{i},r)}$ and $T(y,\Xi,dy)$ is completely determined by
$\Xi_{\Gamma_{\alpha}\cap B(\mathbb{D}_{i},2\theta_{3}C_{2}\ln(\alpha))^{c}}$
if $x\in B(\mathbb{D}_{i},r)$ and $\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)$. By
the EDD, we can find an independent copy $\tilde{\Xi}$ of $\Xi$ such that

$\displaystyle\mathbb{P}(E_{i}):=$
$\displaystyle\mathbb{P}\left(\Xi_{B(\mathbb{D}_{i},r)}\cup\Xi_{\Gamma_{\alpha}\cap
B(\mathbb{D}_{i},2\theta_{3}C_{2}\ln(\alpha))^{c}}\neq\Xi_{B(\mathbb{D}_{i},r)}\cup\tilde{\Xi}_{\Gamma_{\alpha}\cap
B(\mathbb{D}_{i},2\theta_{3}C_{2}\ln(\alpha))^{c}}\right)$ $\displaystyle=$
$\displaystyle\beta_{B(\mathbb{D}_{i},r),\Gamma_{\alpha}\cap
B(\mathbb{D}_{i},2\theta_{3}C_{2}\ln(\alpha))^{c}}\leq
C_{3}\ln(\alpha)^{\theta_{0}}\alpha^{\theta_{0}/d}e^{-\theta_{2}\theta_{3}C_{2}\ln(\alpha)}.$
(5.18)

Hence, following the same steps as those for (5.10) and using Hölder’s
inequality in the first inequality, (5.18) and Remark 5.7 with $k=3$ in the
second inequality below, we have

$\displaystyle\left|\mathbb{E}\left[\int_{x\in\mathbb{D}_{i}}T(x,\Xi,dx)\int_{y\in\Gamma_{\alpha},\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)}T(y,\Xi,dy)\right]\right|$
$\displaystyle=\left|\mathbb{E}\left[{\bf
1}_{E_{i}}\int_{x\in\mathbb{D}_{i}}T(x,\Xi,dx)\int_{y\in\Gamma_{\alpha},\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)}T(y,\Xi,dy)\right]\right.$
$\displaystyle\ \ \ \ \ \ \ -\left.\mathbb{E}\left[{\bf
1}_{E_{i}}\int_{x\in\mathbb{D}_{i}}T(x,\Xi,dx)\int_{y\in\Gamma_{\alpha},\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)}T(y,\tilde{\Xi},dy)\right]\right|$
$\displaystyle\leq\left\|\int_{x\in\mathbb{D}_{i}}T(x,\Xi,dx)\int_{y\in\Gamma_{\alpha},\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)}T(y,\Xi,dy)\right\|_{3/2}\mathbb{P}(E_{i})^{1/3}$
$\displaystyle\ \ \ \ \ \ \
+\left\|\int_{x\in\mathbb{D}_{i}}T(x,\Xi,dx)\int_{y\in\Gamma_{\alpha},\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)}T(y,\tilde{\Xi},dy)\right\|_{3/2}\mathbb{P}(E_{i})^{1/3}$
$\displaystyle\leq
O\left(\alpha^{\frac{5}{2}+\frac{\theta_{0}}{3d}-\frac{1}{3}\theta_{2}\theta_{3}C_{2}}\ln(\alpha)^{\theta_{0}/3}\right)\leq
O\left(\ln(\alpha)^{\theta_{0}/3}\alpha^{-2}\right).$ (5.19)

Adding the estimates of (5.19) for $1\leq i\leq n_{\mathbb{D}}$ and using the
fact $n_{\mathbb{D}}=O\left(\alpha^{\frac{d-1}{d}}\ln\alpha\right)$, we obtain

$\displaystyle\left|\mathbb{E}\left[\int_{x\in\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha}}T(x,\Xi,dx)\int_{y\in\Gamma_{\alpha},\|y-x\|>4\theta_{3}C_{2}\ln(\alpha)}T(y,\Xi,dy)\right]\right|\leq
O\left(\alpha^{-1}\right).$ (5.20)

For the remaining part, we have

$\displaystyle\left|\mathbb{E}\left[\iint_{x\in\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha},y\in\Gamma_{\alpha},\|y-x\|\leq
4\theta_{3}C_{2}\ln(\alpha)}g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)\right]\right|$
$\displaystyle\leq\mathbb{E}\left[\iint_{x\in\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha},y\in\Gamma_{\alpha},\|y-x\|\leq
4\theta_{3}C_{2}\ln(\alpha)}\frac{1}{2}\left(g_{\alpha}(x,\Xi)^{2}\mathbf{1}_{\bar{R}(x)\leq
r}+g_{\alpha}(y,\Xi)^{2}\mathbf{1}_{\bar{R}(y)\leq
r}\right)\overline{\Xi}(dx)\overline{\Xi}(dy)\right]$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\left[\int_{x\in\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha}}\overline{\Xi}(B(x,4\theta_{3}C_{2}\ln(\alpha)))g_{\alpha}(x,\Xi)^{2}\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)\right]$ $\displaystyle\ \ \
+\frac{1}{2}\mathbb{E}\left[\int_{y\in\Gamma_{\alpha}\cap
B(\partial\Gamma_{\alpha},9\theta_{3}C_{2}\ln(\alpha))}\overline{\Xi}(B(y,4\theta_{3}C_{2}\ln(\alpha)))g_{\alpha}(y,\Xi)^{2}\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{y\in\Gamma_{\alpha}\cap
B(\partial\Gamma_{\alpha},9\theta_{3}C_{2}\ln(\alpha))}\overline{\Xi}(B(y,4\theta_{3}C_{2}\ln(\alpha)))g_{\alpha}(y,\Xi)^{2}\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{y\in\Gamma_{\alpha}\cap
B(\partial\Gamma_{\alpha},9\theta_{3}C_{2}\ln(\alpha))}\frac{1}{2}\left(\overline{\Xi}(B(y,4\theta_{3}C_{2}\ln(\alpha)))^{2}+g_{\alpha}(y,\Xi)^{4}\mathbf{1}_{\bar{R}(y)\leq
r}\right)\overline{\Xi}(dy)\right]$
$\displaystyle\leq\int_{y\in\Gamma_{\alpha}\cap
B(\partial\Gamma_{\alpha},9\theta_{3}C_{2}\ln(\alpha))}\left(\left(\overline{\Xi}_{\bf
0}(B({\bf 0},4\theta_{3}C_{2}\ln(\alpha)))+1\right)^{2}\lambda
dy+g_{\alpha}(y,\Xi)^{4}\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)\right).$ (5.21)

However, by Remark 5.4,

$\displaystyle O\left(\ln(\alpha)^{3d}\right)$ $\displaystyle=$
$\displaystyle\mathbb{E}\overline{\Xi}(B({\bf
0},8\theta_{3}C_{2}\ln(\alpha)))^{3}$ $\displaystyle\geq$
$\displaystyle\mathbb{E}\int_{B({\bf
0},4\theta_{3}C_{2}\ln(\alpha))}\overline{\Xi}(B(x,4\theta_{3}C_{2}\ln(\alpha)))^{2}\overline{\Xi}(dx)$
$\displaystyle=$ $\displaystyle\lambda\int_{B({\bf
0},4\theta_{3}C_{2}\ln(\alpha))}\mathbb{E}\left(\overline{\Xi}_{x}(B(x,4\theta_{3}C_{2}\ln(\alpha)))+1\right)^{2}dx$
$\displaystyle=$
$\displaystyle\frac{\lambda(4\theta_{3}C_{2}\ln(\alpha))^{d}\pi^{d/2}}{\Gamma(1+d/2)}\mathbb{E}\left(\overline{\Xi}_{\bf
0}(B({\bf 0},4\theta_{3}C_{2}\ln(\alpha)))+1\right)^{2},$

which implies

$\mathbb{E}\left(\overline{\Xi}_{\bf 0}(B({\bf
0},4\theta_{3}C_{2}\ln(\alpha)))+1\right)^{2}\leq
O\left(\ln(\alpha)^{2d}\right).$ (5.22)

Combining (2.6), (5.21) and (5.22) gives

$\displaystyle\left|\mathbb{E}\left[\iint_{x\in\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha},y\in\Gamma_{\alpha},\|y-x\|\leq
4\theta_{3}C_{2}\ln(\alpha)}g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)\right]\right|$ $\displaystyle\leq
O\left((\ln\alpha)^{2d+1}\alpha^{\frac{d-1}{d}}\right).$ (5.23)

Direct verification using (2.6) again gives

$\displaystyle\left|\mathbb{E}\left[\iint_{x\in\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha},y\in\Gamma_{\alpha},\|y-x\|\leq
4\theta_{3}C_{2}\ln(\alpha)}\bar{P}_{\alpha,x,r}dx\bar{P}_{\alpha,y,r}dy\right]\right|\leq
O\left(\alpha^{\frac{d-1}{d}}\ln(\alpha)^{d+1}\right).$ (5.24)

Collecting (5.20), (5.23) and (5.24), we have (5.17).

Now, since $\bar{W}_{\alpha,r}$ can be decomposed as

$\displaystyle{\rm Var}(\bar{W}_{\alpha,r})$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{\mathbb{C}_{\alpha}}(g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx)\int_{\Gamma_{\alpha}}(g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)-\bar{P}_{\alpha,y,r}dy)\right]$
$\displaystyle+\mathbb{E}\left[\int_{\Gamma_{\alpha}\backslash\mathbb{C}_{\alpha}}(g_{\alpha}(x,\Xi)\mathbf{1}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-\bar{P}_{\alpha,x,r}dx)\int_{\Gamma_{\alpha}}(g_{\alpha}(y,\Xi)\mathbf{1}_{\bar{R}(y)\leq
r}\overline{\Xi}(dy)-\bar{P}_{\alpha,y,r}dy)\right],$

it follows from (5.16) and (5.17) that, for
$\alpha\geq\alpha_{1}\vee\alpha_{0}\vee e$, ${\rm
Var}(\bar{W}_{\alpha,r})\in[C_{4}\alpha,C_{5}\alpha]$ for some positive
constants $C_{4}$, $C_{5}$. This, together with $|{\rm
Var}(\bar{W}_{\alpha})-{\rm Var}(\bar{W}_{\alpha,r})|\leq\frac{1}{\alpha}$,
ensures that ${\rm Var}(\bar{W}_{\alpha})=\Theta({\rm
Var}(\bar{W}_{\alpha,r}))=\Theta(\alpha)$.

The statement for the unrestricted case can be proved by replacing
$g_{\alpha}$ by $g$; $\bar{P}_{\alpha,x,r}$ by $P_{r}$; $\bar{W}_{\alpha}$ by
$W_{\alpha}$ and $\bar{W}_{\alpha,r}$ by $W_{\alpha,r}$.

######  Remark 5.11.

Since the variance is always non-negative, the proof of this also shows that
$\sigma^{2}$ (resp. $\bar{\sigma}^{2}$) defined in (2.7) (resp. (2.8)) is non-
negative.

######  Remark 5.12.

Using the same idea as in the proof of Theorem 2.11, we can see that ${\rm
Var}(W_{\alpha})$ and ${\rm Var}(\bar{W}_{\alpha})$ cannot have an order
greater than $\alpha$.

Proof of Theorem 2.9. Let
$\mu_{\alpha}:=\mathbb{E}\left(\bar{W}_{\alpha}\right)$,
$\mu_{\alpha,r}:=\mathbb{E}\left(\bar{W}_{\alpha,r}\right)$,
$\sigma_{\alpha}^{2}:={\rm Var}\left(\bar{W}_{\alpha}\right)$,
$\sigma_{\alpha,r}^{2}:={\rm Var}\left(\bar{W}_{\alpha,r}\right)$ and
$\bar{Z}_{\alpha,r}\sim
N\left(\frac{\mu_{\alpha,r}-\mu_{\alpha}}{\sigma_{\alpha}},\frac{\sigma_{\alpha,r}^{2}}{\sigma_{\alpha}^{2}}\right)$,
then it follows from the triangle inequality that

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mu_{\alpha}}{\sigma_{\alpha}},Z\right)\leq
d_{W}\left(\frac{\bar{W}_{\alpha}-\mu_{\alpha}}{\sigma_{\alpha}},\frac{\bar{W}_{\alpha,r}-\mu_{\alpha}}{\sigma_{\alpha}}\right)+d_{W}\left(Z,\bar{Z}_{\alpha,r}\right)+d_{W}\left(\frac{\bar{W}_{\alpha,r}-\mu_{\alpha}}{\sigma_{\alpha}},\bar{Z}_{\alpha,r}\right).$
(5.25)

Next, we analyse the three terms of (5.25) separately. We start with the
exponentially stabilising case (ii).

The first term of (5.25) can be bounded using Lemma 5.2 (b), the variation
condition and the property of the Wasserstein distance. Let
$U_{\alpha}:=\left(\bar{W}_{\alpha}-\mu_{\alpha}\right)/\sigma_{\alpha}$ and
$U_{\alpha,r}:=\left(\bar{W}_{\alpha,r}-\mu_{\alpha,r}\right)/\sigma_{\alpha}$.
According to the property of the total variation distance and [Barbour, Holst
and Janson (1992), p. 254], we can find a coupling
$(\bar{U}_{\alpha},\bar{U}_{\alpha,r})$ of $U_{\alpha}$ and $U_{\alpha,r}$
such that $\bar{U}_{\alpha}\overset{d}{=}U_{\alpha}$,
$\bar{U}_{\alpha,r}\overset{d}{=}U_{\alpha,r}$ and

$\mathbb{P}(\bar{U}_{\alpha}\neq\bar{U}_{\alpha,r})=:\mathbb{P}(E_{\alpha,r})=d_{TV}(U_{\alpha},U_{\alpha,r})=d_{TV}(\bar{W}_{\alpha},\bar{W}_{\alpha,r})\leq
C_{1}\alpha e^{-C_{2}r}.$

Then from Hölder’s inequality, the variation condition and Lemma 5.5,

$\displaystyle
d_{W}\left(\frac{\bar{W}_{\alpha}-\mu_{\alpha}}{\sigma_{\alpha}},\frac{\bar{W}_{\alpha,r}-\mu_{\alpha}}{\sigma_{\alpha}}\right)$
$\displaystyle=$
$\displaystyle\inf_{X\overset{d}{=}U_{\alpha},Y\overset{d}{=}U_{\alpha,r}}\mathbb{E}(|X-Y|)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}(|\bar{U}_{\alpha}-\bar{U}_{\alpha,r}|)\leq\mathbb{E}\left(|\bar{U}_{\alpha}|\mathbf{1}_{E_{\alpha,r}}\right)+\mathbb{E}\left(|\bar{U}_{\alpha,r}|\mathbf{1}_{E_{\alpha,r}}\right)\leq\frac{1}{\alpha},$
(5.26)

for $r>C_{3}\ln(\alpha).$

For the second term of (5.25), we can establish an upper bound using Lemma
5.1. To this end, (5.4) gives

$\left|\sigma_{\alpha}^{2}-\sigma_{\alpha,r}^{2}\right|\leq\frac{1}{\alpha},$
(5.27)

which, together with the condition given in the theorem, implies

$\sigma_{\alpha,r}^{2}=\Omega(\alpha^{\nu}),\ \ \ \ \
\sigma_{\alpha}^{2}=\Omega(\alpha^{\nu}),$ (5.28)

for $r>C_{4}\ln(\alpha).$ We combine (5.8) and (5.28) to obtain

$\frac{\left|\mu_{\alpha}-\mu_{\alpha,r}\right|}{\sigma_{\alpha}}\leq
O\left(\alpha^{-1}\right),$ (5.29)

for $r>C_{5}\ln(\alpha).$ Therefore, it follows from (5.27), (5.28), (5.29)
and Lemma 5.1 that

$\displaystyle
d_{W}(Z,\bar{Z}_{\alpha,r})\leq\frac{\left|\mu_{\alpha}-\mu_{\alpha,r}\right|}{\sigma_{\alpha}}+\frac{\left|\sigma_{\alpha,r}-\sigma_{\alpha}\right|}{\sigma_{\alpha}}\leq
O(\alpha^{-1})$ (5.30)

for $r>C_{6}\ln(\alpha).$

It remains to tackle the last term of (5.25). From the definition of the
Wasserstein distance, we have

$\displaystyle
d_{W}\left(\frac{\bar{W}_{\alpha,r}-\mu_{\alpha}}{\sigma_{\alpha}},\bar{Z}_{\alpha,r}\right)=d_{W}\left(\frac{\bar{W}_{\alpha,r}-\mu_{\alpha,r}}{\sigma_{\alpha}},\bar{Z}_{\alpha,r}+\frac{\mu_{\alpha}-\mu_{\alpha,r}}{\sigma_{\alpha}}\right)$
$\displaystyle\leq\frac{\sigma_{\alpha,r}}{\sigma_{\alpha}}d_{W}\left(\frac{\bar{W}_{\alpha,r}-\mu_{\alpha,r}}{\sigma_{\alpha,r}},Z\right)\leq
2d_{W}\left(V_{\alpha,r},Z\right)$ (5.31)

for $r>C_{4}\ln(\alpha)$ when $\alpha$ large, where
$V_{\alpha,r}:=\left(\bar{W}_{\alpha,r}-\mu_{\alpha,r}\right)/\sigma_{\alpha,r}$.
We now use Stein’s method to bound the Wasserstein distance between
$V_{\alpha,r}$ and $Z$. Stein’s method for the normal approximation hinges on
a Stein equation (see [Chen, Goldstein and Shao (2011), pp. 15–16])

$f^{\prime}(w)-wf(w)=h(w)-Nh,$ (5.32)

where $Nh:=\mathbb{E}h(Z)$. The solution of (5.32) is given by

$f_{h}(w)=e^{w^{2}/2}\int^{w}_{-\infty}e^{-t^{2}/2}(h(t)-Nh)dt=-e^{w^{2}/2}\int_{w}^{\infty}e^{-t^{2}/2}(h(t)-Nh)dt.$

Recall the definition of the Wasserstein distance (2.9), we have, for any
random variable $X$,

$d_{W}(X,Z)=\sup_{h\in\mathscr{F}_{\rm
Lip}}|\mathbb{E}(h(X)-h(Z))|\leq\sup_{f\in\mathscr{F}}\left|\mathbb{E}\left(f^{\prime}(X)-Xf(X)\right)\right|,$
(5.33)

where $\mathscr{F}:=\left\\{f;\leavevmode\nobreak\
\mathbb{R}\rightarrow\mathbb{R},\|f\|\leq 2,\leavevmode\nobreak\
\|f^{\prime}\|\leq\sqrt{\frac{2}{\pi}},\leavevmode\nobreak\
\|f^{\prime\prime}\|\leq 2\right\\}$. From the definition of $V_{\alpha,r}$,
we can represent it as
$V_{\alpha,r}=\frac{1}{\sigma_{\alpha,r}}\int_{\Gamma_{\alpha}}\left(g_{\alpha}(x,\Xi){\mathbf{1}}_{\bar{R}(x)\leq
r}\overline{\Xi}(dx)-P_{\alpha,x,r}dx\right)=:\int_{\Gamma_{\alpha}}V(dx)$ if
this does not cause confusion. Then, from the definition of $V_{\alpha,r}$, we
have

$\displaystyle 1={\rm
Var}(V_{\alpha,r})=\mathbb{E}\left(\int_{\Gamma_{\alpha}}V(dx)\right)^{2}.$

To bound $d_{W}(V_{\alpha,r},Z)$, by (5.33), it is sufficient to bound

$|\mathbb{E}\left(f^{\prime}(V_{\alpha,r})-V_{\alpha,r}f(V_{\alpha,r})\right)|$
(5.34)

for all $f\in\mathscr{F}$. To do this, let’s consider the two terms
separately.

Before analysing (5.34), for large $\alpha$, we can divide $\Gamma_{\alpha}$
into disjoint cubes with volumes at least $\alpha_{0}\leq\alpha$ for some
positive constant $\alpha_{0}$. To this end, we can find a partition of
$\Gamma_{\alpha}$,
$\mathscr{C}:=\\{\mathbb{C}_{1},\dots,\mathbb{C}_{n_{\alpha}}\\}$, where
$\mathbb{C}_{i}$ are cubes with edge length
$\frac{\alpha^{1/d}}{\left\lfloor(\alpha/\alpha_{0})^{1/d}\right\rfloor}$ for
all $1\leq i\leq n_{\alpha}$. Then each cube in $\mathscr{C}$ has a volume no
more than $2^{d}\alpha_{0}$ and $n_{\alpha}$ has the same order as $\alpha$.
Let $N_{i,\alpha,r}^{\prime}=B(\mathbb{C}_{i},3r)\cap\Gamma_{\alpha}$ and
$N_{i,\alpha,r}^{\prime\prime}=B(\mathbb{C}_{i},6r)\cap\Gamma_{\alpha}$, we
have $N_{i,\alpha,r}^{\prime}\subset B(\mathbb{C}_{i},3r)$ and
$N_{i,\alpha,r}^{\prime\prime}\subset B(\mathbb{C}_{i},6r)$, so the volumes of
$N_{i,\alpha,r}^{\prime}$ and $N_{i,\alpha,r}^{\prime\prime}$ are bounded by
$O(r^{d})$ for all $1\leq i\leq n_{\alpha}$. Define
$S_{i,\alpha,r}=\int_{\mathbb{C}_{i}}V(dy)$,
$S_{i,\alpha,r}^{\prime}=\int_{N_{i,\alpha,r}^{\prime}}V(dy)$ and
$S_{i,\alpha,r}^{\prime\prime}=\int_{N_{i,\alpha,r}^{\prime\prime}}V(dy)$.
Clearly, $V(dx)$ is a function of $\Xi_{B(x,r)}\cap\Gamma_{\alpha}$,
$S_{i,\alpha,r}^{\prime}$, $S_{i,\alpha,r}^{\prime\prime}$
$V_{\alpha,r}-S_{i,\alpha,r}^{\prime}$ and
$V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime}$ are functions of
$\Xi_{B(\mathbb{C}_{i},4r)}$, $\Xi_{B(\mathbb{C}_{i},7r)}$,
$\Xi_{\Gamma_{\alpha}\backslash B(\mathbb{C}_{i},2r)}$ and
$\Xi_{\Gamma_{\alpha}\backslash B(\mathbb{C}_{i},5r)}$ respectively. For
convenience, we write $\tilde{\Xi}$ as an independent copy of $\Xi$ and
$\tilde{V}_{\alpha,r}$, $\tilde{V}(dx)$, $\tilde{S}^{\prime}_{i,\alpha,r}$,
$\tilde{S}_{i,\alpha,r}^{\prime\prime}$ as the corresponding counterparts of
$V_{\alpha,r}$, $V(dx)$, $S^{\prime}_{i,\alpha,r}$,
$S_{i,\alpha,r}^{\prime\prime}$.

For the first term in (5.34),

$\displaystyle\mathbb{E}f^{\prime}(V_{\alpha,r})$ $\displaystyle=$
$\displaystyle\mathbb{E}\left(\int_{\Gamma_{\alpha}}V(dx)\right)^{2}\mathbb{E}f^{\prime}(V_{\alpha,r})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right)\mathbb{E}f^{\prime}(V_{\alpha,r})+\mathbb{E}\left(\left(\int_{\Gamma_{\alpha}}V(dx)\right)^{2}-\sum_{i=1}^{n_{\alpha}}S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right)\mathbb{E}f^{\prime}(V_{\alpha,r})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right)(\mathbb{E}f^{\prime}(V_{\alpha,r})-\mathbb{E}f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})+\mathbb{E}f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime}))+\epsilon_{1}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left(\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right)\mathbb{E}\left(\int_{0}^{S_{i,\alpha,r}^{\prime\prime}}f^{\prime\prime}(V_{\alpha,r}-x)dx\right)\right)+\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}f^{\prime}(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime\prime})\right)$
$\displaystyle+\epsilon_{1}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\left(f^{\prime}(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime\prime})-f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})+f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})\right)\right)+\epsilon_{1}+\epsilon_{2}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})\right)+\epsilon_{1}+\epsilon_{2}+\epsilon_{3},$
(5.35)

where

$\displaystyle\epsilon_{1}=$
$\displaystyle\mathbb{E}\left(\left(\int_{\Gamma_{\alpha}}V(dx)\right)^{2}-\sum_{i=1}^{n_{\alpha}}S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right)\mathbb{E}f^{\prime}(V_{\alpha,r});$
$\displaystyle\epsilon_{2}=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left(\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right)\mathbb{E}\left(\int_{0}^{S_{i,\alpha,r}^{\prime\prime}}f^{\prime\prime}(V_{\alpha,r}-x)dx\right)\right);$
$\displaystyle\epsilon_{3}=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\left(f^{\prime}(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime\prime})-f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})\right)\right).$

Using the same idea as in the proof of Lemma 5.10 and the fact that
$\|f^{\prime}\|\leq\sqrt{\frac{2}{\pi}}$, we have

$|\epsilon_{1}|\leq\sqrt{\frac{2}{\pi}}\left|\mathbb{E}\sum_{i=1}^{n_{\alpha}}S_{i,\alpha,r}\left(\int_{\Gamma_{\alpha}}V(dx)-S_{i,\alpha,r}^{\prime}\right)\right|\leq\alpha^{-1}$
(5.36)

for $r>C_{7}\ln(\alpha)$ for some positive constant $C_{7}$.

For $\epsilon_{2}$, using the fact that $\|f^{\prime\prime}\|\leq 2$ and
Remark 5.7, we have

$\displaystyle|\epsilon_{2}|\leq$ $\displaystyle
2\sum_{i=1}^{n_{\alpha}}\left(\mathbb{E}\left(\left|S_{i,\alpha,r}S^{\prime}_{i,\alpha,r}S^{\prime\prime}_{i,\alpha,r}\right|\right)\right)$
$\displaystyle\leq$ $\displaystyle
O(\sigma_{\alpha,r}^{-3}n_{\alpha}\max_{i}\\{\mathrm{Vol}(N_{i,\alpha,r}^{\prime}),\mathrm{Vol}(N_{i,\alpha,r}^{\prime\prime})\\}^{5})=O(\alpha^{-\frac{3}{2}\nu+1}r^{5d}).$
(5.37)

To bound $\epsilon_{3}$, we can use the coupling method. Since
$(S_{i,\alpha,r},S_{i,\alpha,r}^{\prime})$ is a function of
$\Xi_{B(\mathbb{C}_{i},4r)}$ and $V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime}$
is a function of $\Xi_{\Gamma_{\alpha}\backslash B(\mathbb{C}_{i},5r)}$, from
the EDD and [Barbour, Holst and Janson (1992), p. 254], there is a coupling
$(X_{i,1},X_{i,2},X_{i,3})$ and $(Y_{i,1},Y_{i,2},Y_{i,3})$ of
$(S_{i,\alpha,r},S_{i,\alpha,r}^{\prime},f^{\prime}(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime\prime}))$
and
$(S_{i,\alpha,r},S_{i,\alpha,r}^{\prime},f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime}))$
such that
$\mathbb{P}(E_{i}):=\mathbb{P}((X_{i,1},X_{i,2},X_{i,3})\neq(Y_{i,1},Y_{i,2},Y_{i,3}))\leq\alpha^{\theta_{0}/d}C_{8}r^{\theta_{0}}e^{-C_{9}r}$,
for all $1\leq i\leq n_{\alpha}$. Then for $r>C_{10}\ln(\alpha)$ where
$C_{10}$ is large enough, $\mathbb{P}(E_{i})\leq C_{11}\alpha^{-6}$. Now,

$\displaystyle|\epsilon_{3}|\leq$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left|\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\left(f^{\prime}(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime\prime})-f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})\right)\right)\right|$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left|\mathbb{E}\left(X_{i,1}X_{i,2}X_{i,3}-Y_{i,1}Y_{i,2}Y_{i,3}\right)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left(\left|\mathbb{E}\left(X_{i,1}X_{i,2}X_{i,3}\mathbf{1}_{E_{i}}\right)\right|+\left|\mathbb{E}\left(Y_{i,1}Y_{i,2}Y_{i,3}\mathbf{1}_{E_{i}}\right)\right|\right).$
(5.38)

However, applying Hölder’s inequality, the fact that
$\|f^{\prime}\|\leq\sqrt{\frac{2}{\pi}}\leq 1$ and Remark 5.7, we have

$\displaystyle\left|\mathbb{E}\left(X_{i,1}X_{i,2}X_{i,3}\mathbf{1}_{E_{i}}\right)\right|\leq\|X_{i,1}X_{i,2}\|_{3/2}\mathbb{P}(E_{i})^{1/3}\leq
O\left(r^{10d/3}\right)\sigma_{\alpha,r}^{-2}\mathbb{P}(E_{i})^{1/3},$
$\displaystyle\left|\mathbb{E}\left(Y_{i,1}Y_{i,2}Y_{i,3}\mathbf{1}_{E_{i}}\right)\right|\leq\|Y_{i,1}Y_{i,2}\|_{3/2}\mathbb{P}(E_{i})^{1/3}\leq
O\left(r^{10d/3}\right)\sigma_{\alpha,r}^{-2}\mathbb{P}(E_{i})^{1/3}.$

Combining with (5.38), we get

$|\epsilon_{3}|\leq n_{\alpha}\sigma_{\alpha,r}^{-2}O(r^{10d/3})\max_{1\leq
i\leq n_{\alpha}}\mathbb{P}(E_{i})^{1/3}\leq O(\alpha^{-1-\nu}r^{10d/3}).$
(5.39)

For the second term in (5.34), we have

$\displaystyle\mathbb{E}V_{\alpha,r}f(V_{\alpha,r})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}f(V_{\alpha,r})\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}\left(f(V_{\alpha,r}-S_{i,\alpha,r}^{\prime})+f(V_{\alpha,r})-f(V_{\alpha,r}-S_{i,\alpha,r}^{\prime})\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\int_{0}^{1}f^{\prime}(V_{\alpha,r}-uS_{i,\alpha,r}^{\prime})du\right)+\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime})\right)$
$\displaystyle+\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}\left(f(V_{\alpha,r}-S_{i,\alpha,r}^{\prime})-f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime})\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\int_{0}^{1}f^{\prime}(V_{\alpha,r}-uS_{i,\alpha,r}^{\prime})du\right)+\epsilon_{4}+\epsilon_{5},$
(5.40)

where

$\displaystyle\epsilon_{4}=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime})\right);$
$\displaystyle\epsilon_{5}=$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}\left(f(V_{\alpha,r}-S_{i,\alpha,r}^{\prime})-f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime})\right)\right).$

Since $f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime})$ is independent
of $S_{i,\alpha,r}$ for all $1\leq i\leq n_{\alpha}$,

$\displaystyle\epsilon_{4}=\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}\right)\mathbb{E}\left(f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime})\right)=0$
(5.41)

from the definition of $S_{i,\alpha,r}$ and $\|f\|\leq 2$.

To bound $\epsilon_{5}$, we can use the same idea as that for bounding
$\epsilon_{3}$, i.e., we can construct a coupling of
$(S_{i,\alpha,r},f(V_{\alpha,r}-S_{i,\alpha,r}^{\prime}))$ and
$(S_{i,\alpha,r},f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime}))$.
Since $S_{i,\alpha,r}$ is a function of $\Xi_{B(\mathbb{C}_{i},r)}$ and
$V_{\alpha,r}-S_{i,\alpha,r}^{\prime}$ is a function of
$\Xi_{\Gamma_{\alpha}\backslash B(\mathbb{C}_{i},2r)}$, from the EDD and
[Barbour, Holst and Janson (1992), p. 254], there is a coupling
$((X_{i,1},X_{i,2}),(Y_{i,1},Y_{i,2}))$ of
$(S_{i,\alpha,r},f(V_{\alpha,r}-S_{i,\alpha,r}^{\prime}))$ and
$(S_{i,\alpha,r},f(\tilde{V}_{\alpha,r}-\tilde{S}_{i,\alpha,r}^{\prime}))$
such that
$\mathbb{P}\left((X_{i,1},X_{i,2})\neq(Y_{i,1},Y_{i,2})\right)=:\mathbb{P}(E^{\prime})\leq\alpha^{\theta_{0}/d}C_{12}r^{\theta_{0}}e^{-C_{13}r}$
for all $1\leq i\leq n_{\alpha},$ where $C_{12},\ C_{13}\in\mathbb{R}_{+}$ are
independent of $i$’s. Then we follow the steps as for (5.39) to get

$\displaystyle|\epsilon_{5}|\leq O(\alpha^{-1-\nu}r^{10d/3}),$ (5.42)

for $r>C_{14}\ln(\alpha)$ with sufficiently large $C_{14}$.

Also, from the fact that $\|f^{\prime\prime}\|\leq 2$ and Remark 5.7,

$\displaystyle|\epsilon_{6}|:=$
$\displaystyle\left|\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})\right)-\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\int_{0}^{1}f^{\prime}(V_{\alpha,r}-uS_{i,\alpha,r}^{\prime})du\right)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left|\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\int_{0}^{1}\left(f^{\prime}(V_{\alpha,r}-uS_{i,\alpha,r}^{\prime})-f^{\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime})\right)du\right)\right|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n_{\alpha}}\left|\mathbb{E}\left(S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\int_{0}^{1}\int_{0}^{S_{i,\alpha,r}^{\prime\prime}-uS_{i,\alpha,r}^{\prime}}f^{\prime\prime}(V_{\alpha,r}-S_{i,\alpha,r}^{\prime\prime}+v)dvdu\right)\right|$
$\displaystyle\leq$ $\displaystyle
2\sum_{i=1}^{n_{\alpha}}\mathbb{E}\left(\left|S_{i,\alpha,r}S_{i,\alpha,r}^{\prime}\right|\left(\left|S_{i,\alpha,r}^{\prime\prime}\right|+\left|S_{i,\alpha,r}^{\prime}\right|\right)\right)\leq
O(\alpha^{-\frac{3}{2}\nu+1}r^{5d}).$ (5.43)

Combining (5.31), (5.34), (5.35), (5.36), (5.37), (5.39), (5.40), (5.41),
(5.42) and (5.43), we obtain the bound

$d_{W}\left(\frac{\bar{W}_{\alpha,r}-\mu_{\alpha}}{\sigma_{\alpha}},\bar{Z}_{\alpha,r}\right)\leq
O(\alpha^{-\frac{3}{2}\nu+1}r^{5d})$ (5.44)

for $r>C_{15}\ln(\alpha)$, where $C_{15}=\max\\{C_{7},C_{10},C_{14}\\}$.

From (5.25), taking $r=\max\\{C_{3},C_{4},C_{6},C_{15}\\}\ln(\alpha)$ for
large $\alpha$, together with the bounds in (5.26), (5.30) and (5.44), we have

$d_{W}\left(\frac{\bar{W}_{\alpha}-\mu_{\alpha}}{\sigma_{\alpha}},Z\right)\leq\alpha^{-\frac{3}{2}\nu+1}\ln(\alpha)^{5d}.$

(i) If $\eta$ is range-bound, then there exists an $r_{1}>0$ such that
$\bar{W}_{\alpha,r_{1}}=\bar{W}_{\alpha}$ $a.s.$ for all $\alpha$, so
$d_{W}\left(\frac{\bar{W}_{\alpha}-\mu_{\alpha}}{\sigma_{\alpha}},Z\right)=d_{W}\left(\frac{\bar{W}_{\alpha,r_{1}}-\mu_{\alpha}}{\sigma_{\alpha}},\bar{Z}_{\alpha,r_{1}}\right)$.
Since $\eta$ is range-bound, it is also exponentially stabilising, (5.28) and
(5.44) still hold, then
$d_{W}\left(\frac{\bar{W}_{\alpha}-\mu_{\alpha}}{\sigma_{\alpha}},Z\right)=O(\alpha^{-\frac{3}{2}\nu+1}r_{1}^{5d})=O(\alpha^{-\frac{3}{2}\nu+1})$,
completing the proof.

Proof of Theorem 2.7. The statement can be shown by replacing
$\bar{W}_{\alpha}$, $\bar{W}_{\alpha,r}$, $\bar{Z}_{\alpha}$,
$\bar{Z}_{\alpha,r}$, $g_{\alpha}(x,\Xi)$ and $\bar{R}(x,\alpha)$ by their
counterparts $W_{\alpha}$, $W_{\alpha,r}$, $Z_{\alpha}$, $Z_{\alpha,r}$,
$g(\Xi^{x})$ and $R(x)$ in the proof of Theorem 2.9.

According to Theorem 2.11, Corollary 2.8 and Corollary 2.10 are special cases
of Theorem 2.7 and Theorem 2.9 respectively.

## References

  * [Avram and Bertsimas (1993)] Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. _Ann. Appl. Probab._ 3, 1033–1046.
  * [Baddeley et al. (2005)] Baddeley, A., Gregori, P., Mateu, J., Stoica, R., and Stoyan, D. (2005). Case studies in spatial point process models. Lecture Notes in Statistics 185, Springer-Verlag, New-York.
  * [Barbour, Holst and Janson (1992)] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson approximation. Oxford University Press.