# Limit theorems for critical faces above the vanishing threshold

Zifu Wei  Department of Statistics
Purdue University
West Lafayette, 47907, USA<EMAIL_ADDRESS>,  Takashi Owada  Department of
Statistics
Purdue University
West Lafayette, 47907, USA<EMAIL_ADDRESS>and  D.Yogeshwaran  Theoretical
Statistics and Mathematics Unit
Indian Statistical Institute
Bangalore, India<EMAIL_ADDRESS>
###### Abstract.

We investigate convergence of point processes associated with critical faces
for a Čech filtration built over a homogeneous Poisson point process in the
$d$-dimensional flat torus. The convergence of our point process is
established in terms of the $\mathcal{M}_{0}$-topology, when the connecting
radius of a Čech complex decays to $0$, so slowly that critical faces are even
less likely to occur than those in the regime of threshold for homological
connectivity. We also obtain a series of limit theorems for positive and
negative critical faces, all of which are considerably analogous to those for
critical faces.

###### Key words and phrases:

$\mathcal{M}_{0}$-convergence, point process, Morse critical points,
stochastic geometry.

###### 2020 Mathematics Subject Classification:

Primary 60D05. Secondary 60G55, 55U10.

TO’s research was partially supported by the AFOSR grant FA9550-22-1-0238. DY
was funded through CPDA of the Indian Statistical Institute and SERB-MATRICS:
MTR-2020-000470.

##  1\. Introduction

The main objective of this paper is to study geometric and topological
configuration of point processes associated to the spatial distribution of
critical faces. We will introduce necessary notions informally and outline our
results. More rigorous definitions and statements are postponed to Section 2.

Consider the $d$-dimensional flat torus
$\mathbb{T}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ (with a periodic boundary),
that is equipped with the toroidal metric

$\mathsf{dist}(x,y)=\min_{z\in{\mathbb{Z}}^{d}}\|x-y+z\|,\ \ \
x,y\in\mathbb{T}^{d},$

where $\|\cdot\|$ denotes the Euclidean norm. Given a finite point set
${\mathcal{X}}\subset{\mathbb{R}}^{d}$, the distance function
$d_{\mathcal{X}}\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{R}}^{d}\to[0,\infty)$, that will be examined in this paper,
is defined by

(1.1) $d_{\mathcal{X}}(x)=\min_{y\in{\mathcal{X}}}\mathsf{dist}(x,y),\ \
x\in{\mathbb{R}}^{d}.$

Morse theory informs us that the homology of the sub-level set
$C({\mathcal{X}},r)=\\{x\in{\mathbb{R}}^{d}\mathrel{\mathop{\mathchar
58\relax}}d_{\mathcal{X}}(x)\leq r\\}$ changes at critical levels of the
distance function. The distance function is not smooth, but it is of min-type
and so critical levels and critical points can be defined; see [10]. In
particular, a critical point of index $k\in\\{1,\ldots,d\\}$ can be generated
by a $(k+1)$-tuple ${\mathcal{Y}}\subset{\mathcal{X}}$. Such a critical point
shall be denoted by $c({\mathcal{Y}})$ and a critical value or level by
$\rho({\mathcal{Y}})$; see Definition 2.2.

This connection with Morse theory has been exploited to study thresholds for
vanishing of homology of $C(\mathcal{P}_{n},r)$ (see [4, 5, 8, 6, 3]), where
$\mathcal{P}_{n}$ is a homogeneous Poisson point process with intensity $n$ in
$\mathbb{T}^{d}$. To better understand the behaviour of homology above the
vanishing threshold, we investigate point processes of the corresponding
critical levels and points.

We now introduce some notation to outline our results. Since we are working
with the torus, we do not consider sub-level sets for large levels and hence
we focus only on $(C(\mathcal{P}_{n},r))_{r\in[0,R_{n}]}$, where $R_{n}$ is a
sequence of positive numbers tending to $0$ slowly enough. Given such $R_{n}$,
we are interested in the stochastic features of the following point process
induced by critical points and values:

(1.2) $\eta_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}s_{k,n}({\mathcal{Y}},\mathcal{P}_{n})\,\delta_{(c({\mathcal{Y}}),\,n\omega_{d}\rho({\mathcal{Y}})^{d}-a_{n})},$

where $\delta_{(x_{1},x_{2})}$ is the Dirac measure at
$(x_{1},x_{2})\in\mathbb{T}^{d}\times(-\infty,\infty]$, and
$s_{k,n}({\mathcal{Y}},\mathcal{P}_{n})$ is an indicator function, requiring
that ${\mathcal{Y}}$ forms a critical $k$-face, such that
$\rho({\mathcal{Y}})\leq R_{n}$. Furthermore, $\omega_{d}$ denotes volume of
the $d$-dimensional unit ball, and $a_{n}$ is a properly defined centering
sequence. Closely related to the above point process is the statistics of
total number of critical $k$-faces with large critical values. Given
$u\in{\mathbb{R}}$, let $(r_{n}(u))_{n\geq 1}$ be a sequence of positive
numbers defined by

(1.3) $r_{n}(u)=\Big{(}\frac{a_{n}+u}{n\omega_{d}}\Big{)}^{1/d},\ \ \ n\geq
1.$

Fixing $u=u_{0}$ and assuming $r_{n}(u_{0})\to 0$, $r_{n}(u_{0})/R_{n}\to 0$
as $n\to\infty$, we also study the behavior of the statistics

(1.4) $G_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}s_{k,n}({\mathcal{Y}},\mathcal{P}_{n})\,{\mathbbm{1}}\big{\\{}\rho({\mathcal{Y}})\geq
r_{n}(u_{0})\big{\\}},$

which counts the number of critical $k$-faces whose critical values are
between $r_{n}(u_{0})$ and $R_{n}$. Note that
$G_{k,n}=\eta_{k,n}(\mathbb{T}^{d}\times[u_{0},\infty))$.

In this paper, we put our focus on the dense regime i.e.,
$nr_{n}(u)^{d}\to\infty$. Poisson process approximation for the process
$\eta_{k,n}$ in the regime

(1.5) $a_{n}=\log n+(k-1)\log\log n+\mathsf{const},$

as well as Poisson convergence for $G_{k,n}$ in (1.4), were established in
Theorem 6.1 of [7] and Theorem 8.1 of [3], respectively. These results were
crucial to describing the behaviour of homology at the vanishing threshold.

We place ourselves above the Poisson regime (1.5), or equivalently, above the
vanishing threshold. More accurately, we are interested in the asymptotics of
$G_{k,n}$, in the case that

$r_{n}(u)\to 0,\ \ \ r_{n}(u)\gg\Big{(}\frac{\log n+(k-1)\log\log
n}{n\omega_{d}}\Big{)}^{1/d}.$

Equivalently, we consider the following centering term $(a_{n})$ in (1.2):

(1.6) $a_{n}-\log n-(k-1)\log\log n\to\infty,\ \ \ a_{n}=o(n),\ \ \
n\to\infty.$

Under this assumption, the process $\eta_{k,n}$ converges to the null measure
(i.e., the measure that assigns zeros to all Borel measurable sets in
$\mathbb{T}^{d}\times(-\infty,\infty]$), and $G_{k,n}$ converges to a
degenerate zero random variable ([8]). Our main results - Theorems 2.3 and 2.5
below - quantify the rate of convergence to the null measure and a zero random
variable using appropriate notions. More specifically, we identify a sequence
$b_{k,n}\mathrel{\mathop{\mathchar 58\relax}}=na_{n}^{k-1}e^{-a_{n}}\to 0$ to
show that

(1.7) $\big{(}b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\in\cdot),\,n\geq 1\big{)}$

and

(1.8) $\big{(}b_{k,n}^{-1}\mathbb{P}(G_{k,n}\in\cdot),\,n\geq 1\big{)}$

tend to non-degenerate limits as $n\to\infty$, while identifying the limits
themselves explicitly. Up to a constant (dependent on $k$), the limit in (1.7)
is supported on singletons with a product density independent of $k$. The
topology underlying the convergence of (1.7) is called _$\mathcal{M}_{0}$
-topology_, a useful notion for dealing with convergence of probability
measures defined on a complete and separable metric space. In recent times,
the notion of $\mathcal{M}_{0}$-topology has been used for the study of
geometrically and/or topologically rare events ([17, 12, 18]), as well as
regular variation of point processes and stochastic processes ([14, 15, 9,
21]). For further analyses, we recall that critical faces can be divided into
positive and negative critical faces (see [3]). Loosely speaking, positive
critical $k$-faces will create a (nontrivial) $k$-dimensional cycle in the
$k$th homology group of $C({\mathcal{X}},r)$, while negative critical
$k$-faces terminate a $(k-1)$-dimensional cycle. According to Propositions 4.2
and 5.1 in [3], for each $1\leq k\leq d-2$, the vanishing thresholds for
critical $k$-faces, positive critical $k$-faces, and negative critical
$(k+1)$-faces all coincide with one another (with high probability). Extending
this further, we can prove that even when the radius $(r_{n}(u))_{n\geq 1}$
satisfies condition (1.6), the functionals of positive critical $k$-faces and
negative critical $(k+1)$-faces exhibit asymptotic results similar to those
for critical $k$-faces; see Corollary 2.6 for more details. The Poisson
convergence results for positive and negative critical faces in the regime
(1.5) were used to understand the vanishing threshold for homology in [3].
Similarly, with the equivalence between positive (resp. negative) critical
faces and birth (resp. death) times in persistence diagrams, our results can
be rephrased in terms of the birth and death times within the interval
$[r_{n}(u),R_{n}]$. In other words, we can quantify the distribution of ’noisy
barcodes’ in the persistence diagram above the homological connectivity
regime.

As a final remark, we point out that in addition to these studies on the
vanishing homology, together with the related Poisson convergence, there have
also been a number of attempts at deducing other types of limit theorems,
including the central limit theorems ([4, 5, 22]) and the large deviation
principle in [13], when the radius $(r_{n}(u))_{n\geq 1}$ belongs to the
sparse regime (i.e., $nr_{n}(u)^{d}\to 0$) or critical regime (i.e.,
$nr_{n}(u)^{d}\to c\in(0,\infty)$).

The remainder of the paper is structured as follows. Section 2 gives a precise
setup for the processes $\eta_{k,n}$ and $G_{k,n}$. After that, the
convergence results (1.7) and (1.8) will be formally described in Theorems 2.3
and 2.5. Corollary 2.6 states the convergence results for positive and
negative critical faces. All the proofs are postponed to Section 3. The main
machinery in our proof is given by Theorems 4.1 and 6.1 of [7]; this will help
us to show that under the $\mathcal{M}_{0}$-topology, the process $\eta_{k,n}$
can be approximated by some Poisson point process whose intensity measure
tends to the null measure as $n\to\infty$. We then rely upon the estimates in
[3] to approximate the spatial distributions of positive and negative critical
faces by those of critical faces.

##  2\. Setup and main results

To better understand critical faces, it is important to define a _Čech
complex_ , which is one of the most studied geometric complexes (see [11]).

######  Definition 2.1.

Given a point set ${\mathcal{X}}=\\{x_{1},\dots,x_{n}\\}\subset\mathbb{T}^{d}$
and $r>0$, the Čech complex $\check{C}({\mathcal{X}},r)$ is defined as
follows:

  * •

The $0$-simplices are the points in ${\mathcal{X}}$.

  * •

For each $m\geq 1$, $\\{x_{i_{0}},\dots,x_{i_{m}}\\}\subset{\mathcal{X}}$
forms an $m$-simplex if $\bigcap_{j=0}^{m}B_{r}(x_{i_{j}})\neq\emptyset$,
where $B_{r}(x)\mathrel{\mathop{\mathchar
58\relax}}=\\{y\in\mathbb{T}^{d}\mathrel{\mathop{\mathchar
58\relax}}\mathsf{dist}(x,y)\leq r\\}$ is the closed ball of radius $r>0$
centered at $x$.

Intrinsically, the Čech complex possesses inclusion property
$\check{C}({\mathcal{X}},r)\subset\check{C}({\mathcal{X}},r^{\prime})$ for all
$r\leq r^{\prime}$ and thus induces a _Čech filtration_
$\big{(}\check{C}({\mathcal{X}},r)\big{)}_{r\geq 0}$. To analyze the homology
of a Čech filtration, the authors in [4] employed an approach based on an
extension of classical Morse theory (see [10]) to the min-type distance
function $d_{\mathcal{X}}$ in (1.1). Then, for each $k\in\\{1,\dots,d\\}$, the
change in the $k$th homology group of a Čech filtration can be characterized
by the _Morse critical point_ with index $k$ of $d_{\mathcal{X}}$.

######  Definition 2.2.

A point $c\in{\mathbb{R}}^{d}$ is said to be a (Morse) critical point of index
$k$ if there exists a subset ${\mathcal{Y}}\subset{\mathcal{X}}$ of $k+1$
points such that

  * •

the points in ${\mathcal{Y}}$ are in general position, i.e., ${\mathcal{Y}}$
spans a $k$-dimensional simplex in ${\mathbb{R}}^{d}$, so that there is a
unique $(k-1)$-dimensional sphere containing ${\mathcal{Y}}$.

  * •

$d_{\mathcal{X}}(c)=\mathsf{dist}(c,y)$ for any $y\in{\mathcal{Y}}$ and
$d_{\mathcal{X}}(c)<\min_{z\in{\mathcal{X}}\setminus{\mathcal{Y}}}\mathsf{dist}(c,z)$.

  * •

$c\in\sigma({\mathcal{X}})$, where $\sigma({\mathcal{X}})$ denotes an open
geometric $k$-simplex in ${\mathbb{R}}^{d}$ spanned by ${\mathcal{X}}$.

Whenever such ${\mathcal{Y}}$ exists, we say that ${\mathcal{Y}}$ forms a
_critical $k$-face_ (or simplex) for which the critical point is given by
$c=c({\mathcal{Y}})$. Moreover, denote by $\rho({\mathcal{Y}})$ its critical
value, i.e., the radius of a ball spanned by ${\mathcal{Y}}$ centered at
$c({\mathcal{Y}})$. The $0$-dimensional critical points are ${\mathcal{X}}$
itself and hence are not of interest to us.

Recall that $\mathcal{P}_{n}$ is a homogeneous Poisson point process with
intensity $n$ in the $d$-dimensional flat torus $\mathbb{T}^{d}$. Let
$(a_{n})_{n\geq 1}$ be a sequence satisfying (1.6), and $(r_{n}(u))_{n\geq 1}$
be defined as in (1.3). As mentioned in the Introduction, an extra caution is
needed when the Čech filtration is defined on the torus $\mathbb{T}^{d}$. For
example, if the radius $r$ is large enough, the intersection of balls on the
torus is not contractible; hence, the Čech complex
$\check{C}(\mathcal{P}_{n},r)$ will not be homotopy equivalent to the union of
balls $\bigcup_{p\in\mathcal{P}_{n}}B_{r}(p)$ (see the Nerve Lemma in Theorem
10.7 of [1]). Moreover, when $r$ is large, the notion of a critical point, as
well as its critical value, is not always well-defined on the torus. To
overcome this issue, we follow the convention of the previous studies in [3,
6], and focus only on a “bounded” filtration
$(\check{C}(\mathcal{P}_{n},r))_{r\in[0,R_{n}]}$, where $R_{n}$ satisfies

(2.1) $R_{n}\to 0,\ \ \ \frac{r_{n}(u)}{R_{n}}\to 0,\ \ \ n\to\infty,$

for all $u\in{\mathbb{R}}$. Note that (2.1) implies $a_{n}/(nR_{n}^{d})\to 0$
as $n\to\infty$.

Let $k\in\\{1,\dots,d-1\\}$ be a positive integer. For a $(k+1)$-point subset
${\mathcal{Y}}\subset\mathcal{P}_{n}$, which is in general position, we define

(2.2) $s_{k,n}({\mathcal{Y}},\mathcal{P}_{n})\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\\{{\mathcal{Y}}\text{ forms a critical
}k\text{-face}\\}\times{\mathbbm{1}}\\{\rho({\mathcal{Y}})\leq R_{n}\\},$

where $\rho({\mathcal{Y}})$ is the critical value defined after Definition
2.2. Next, for $u\in{\mathbb{R}}$, we define

(2.3)
$\displaystyle\begin{split}g_{k,n}({\mathcal{Y}},\mathcal{P}_{n};u)&\mathrel{\mathop{\mathchar
58\relax}}=s_{k,n}({\mathcal{Y}},\mathcal{P}_{n}){\mathbbm{1}}\\{\rho({\mathcal{Y}})\geq
r_{n}(u)\\}\\\ &={\mathbbm{1}}\\{{\mathcal{Y}}\text{ forms a critical
}k\text{-face}\\}\times{\mathbbm{1}}\big{\\{}\rho({\mathcal{Y}})\in[r_{n}(u),R_{n}]\big{\\}}.\end{split}$

Using (2.2), the point process of our interest can be formally defined as

(2.4) $\eta_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}s_{k,n}({\mathcal{Y}},\mathcal{P}_{n})\,\delta_{(c({\mathcal{Y}}),\,n\omega_{d}\rho({\mathcal{Y}})^{d}-a_{n})}.$

Notice that $\omega_{d}\rho({\mathcal{Y}})^{d}$ represents the volume of an
open ball in ${\mathbb{R}}^{d}$ with radius $\rho({\mathcal{Y}})$ centered at
$c({\mathcal{Y}})$. The process (2.4) is viewed as a random element in the
space $M_{p}(\mathbb{Y})$ of point measures on
$\mathbb{Y}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{T}^{d}\times(-\infty,\infty]$. Defining

$b_{k,n}\mathrel{\mathop{\mathchar 58\relax}}=na_{n}^{k-1}e^{-a_{n}},\ \ \
n\geq 1,$

we consider the sequence of probability measures

(2.5) $\big{(}b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\in\cdot),\,n\geq 1\big{)}.$

As described in the Introduction, the convergence of (2.5) has to be treated
under the $\mathcal{M}_{0}$-topology.

The formal definition of $\mathcal{M}_{0}$-topology is given as follows.
First, let $B_{\emptyset,r}$ denote an open ball of radius $r>0$ centered at
the null measure $\emptyset$ (in terms of the vague metric). Denote by
$\mathcal{M}_{0}=\mathcal{M}_{0}(M_{p}(\mathbb{Y}))$ the space of Borel
measures on $M_{p}(\mathbb{Y})$, the restriction of which to
$M_{p}(\mathbb{Y})\setminus B_{\emptyset,r}$ is finite for all $r>0$.
Moreover, define $\mathcal{C}_{0}=\mathcal{C}_{0}(M_{p}(\mathbb{Y}))$ to be
the space of continuous and bounded real-valued functions on
$M_{p}(\mathbb{Y})$ that vanish in the neighborhood of $\emptyset$. Given
$\xi_{n},\xi\in\mathcal{M}_{0}$, we say that $\xi_{n}$ converges to $\xi$ in
the $\mathcal{M}_{0}$-topology, denoted as $\xi_{n}\to\xi$ in
$\mathcal{M}_{0}$, if it holds that
$\int_{\mathcal{M}_{p}(\mathbb{Y})}g(\mu)\xi_{n}(\operatorname{d\\!}\mu)\to\int_{\mathcal{M}_{p}(\mathbb{Y})}g(\mu)\xi(\operatorname{d\\!}\mu)$
for all $g\in\mathcal{C}_{0}$. One may refer to [14] for more detailed
discussion on $\mathcal{M}_{0}$-topology.

######  Theorem 2.3.

Let $1\leq k\leq d-1$. We have, as $n\to\infty$,

(2.6) $b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\in\cdot)\to\lambda_{k},\ \ \text{ in
}\mathcal{M}_{0},$

where

(2.7)
$\lambda_{k}(\cdot)=D_{k}\int_{\mathbb{Y}}{\mathbbm{1}}\\{\delta_{(c,u)}\in\cdot\\}\,e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u,$

with $D_{k}$ being a positive constant defined specifically at (3.1).

Observe that $D_{k}^{-1}\lambda_{k}$ is a measure independent of $k$ and
concentrated on singletons. For technical reasons, we are forced to skip the
case $k=d$.

######  Example 2.4.

We consider the centering term

(2.8) $a_{n}=\log n+(k-1)\log\log n+\log\log\log n.$

It is then easy to calculate that $b_{k,n}\sim(\log\log n)^{-1}$ as
$n\to\infty$, and by Theorem 2.3,

(2.9) $(\log\log n)\mathbb{P}(\eta_{k,n}\in\cdot)\to\lambda_{k},\ \ \text{ in
}\mathcal{M}_{0}.$

Note that $(a_{n})_{n\geq 1}$ in (2.8) satisfies condition (1.6), but its
growth rate is close to the sequence (1.5) of the Poisson regime (the
difference is at most of order $\mathcal{O}(\log\log\log n)$). As a
consequence, the probability distribution of $\eta_{k,n}$ decays only
logarithmically.

In contrast, if one takes $a_{n}=2\log n$, then much fewer number of critical
$k$-faces will be counted by the process $(\eta_{k,n})_{n\geq 1}$. It then
follows that $b_{k,n}\sim(2\log n)^{k-1}n^{-1}$, and

$(2\log n)^{-(k-1)}n\mathbb{P}(\eta_{k,n}\in\cdot)\to\lambda_{k},\ \ \text{ in
}\mathcal{M}_{0}.$

In this case, the probability distribution of $\eta_{k,n}$ decays much faster
than (2.9).

In parallel, we also study the asymptotics of the sequence
$(b_{k,n}^{-1}\mathbb{P}\circ G_{k,n}^{-1})_{n\geq 1}$, where $G_{k,n}$ is
formally defined as follows. For a fixed $u_{0}\in{\mathbb{R}}$,

(2.10) $G_{k,n}\mathrel{\mathop{\mathchar
58\relax}}=G_{k,n}(u_{0})=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}g_{k,n}({\mathcal{Y}},\mathcal{P}_{n};u_{0}),$

which counts the number of critical points $c({\mathcal{Y}})\in\mathbb{T}^{d}$
with index $k$, such that $\rho({\mathcal{Y}})\in[r_{n}(u_{0}),R_{n}]$.

Let $E\mathrel{\mathop{\mathchar 58\relax}}=(0,\infty]$ and $M_{+}(E)$ be the
space of Radon measures on $E$. Define $C_{K}^{+}(E)$ to be the collection of
non-negative and continuous functions on $E$ with compact support. Given
$\xi_{n},\xi\in M_{+}(E)$, we say that $\xi_{n}$ converges vaguely to $\xi$,
denoted as $\xi_{n}\stackrel{{\scriptstyle v}}{{\to}}\xi$ in $M_{+}(E)$, if it
holds that
$\int_{E}g(x)\xi_{n}(\operatorname{d\\!}x)\to\int_{E}g(x)\xi(\operatorname{d\\!}x)$
for all $g\in C_{K}^{+}(E)$.

######  Theorem 2.5.

Under the assumptions of Theorem 2.3, as $n\to\infty$,

$b_{k,n}^{-1}\mathbb{P}(G_{k,n}\in\cdot)\stackrel{{\scriptstyle
v}}{{\to}}D_{k}e^{-u_{0}}\delta_{1},\ \ \text{ in }M_{+}(E),$

where $u_{0}$ is a fixed real number as at (2.10) and $D_{k}$ is a positive
constant defined at (3.1).
This implies that

$b_{k,n}^{-1}\mathbb{P}(G_{k,n}\geq 1)\to D_{k}e^{-u_{0}}.$

From the Morse-theoretic analyses on critical faces in [3], a critical
$k$-face either generates a (nontrivial) $k$-dimensional cycle in the $k$th
homology group of a Čech filtration or terminates a $(k-1)$-dimensional cycle
of the same complex. In the former case, we call such a critical $k$-face a
_positive critical $k$-face_, while the latter one is called a _negative
critical $k$-face_. For example, a negative critical $1$-face is nothing but
an edge in the minimal spanning tree on $\mathcal{P}_{n}$ with weights being
the Euclidean distance. Now, we introduce a series of indicator functions
analogous to (2.2) and (2.3): for a $(k+1)$-point subset
${\mathcal{Y}}\subset\mathcal{P}_{n}$, which is in general position,

$\displaystyle s_{k,n}^{+}({\mathcal{Y}},\mathcal{P}_{n})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\\{{\mathcal{Y}}\text{ forms a positive critical
}k\text{-face}\\}\times{\mathbbm{1}}\\{\rho({\mathcal{Y}})\leq R_{n}\\},$
$\displaystyle s_{k,n}^{-}({\mathcal{Y}},\mathcal{P}_{n})$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\\{{\mathcal{Y}}\text{ forms a negative critical
}k\text{-face}\\}\times{\mathbbm{1}}\\{\rho({\mathcal{Y}})\leq R_{n}\\},$

and, for $u\in{\mathbb{R}}$,

$g_{k,n}^{\pm}({\mathcal{Y}},\mathcal{P}_{n};u)\mathrel{\mathop{\mathchar
58\relax}}=s_{k,n}^{\pm}({\mathcal{Y}},\mathcal{P}_{n}){\mathbbm{1}}\\{\rho({\mathcal{Y}})\geq
r_{n}(u)\\}.$

Analogously to (2.4) and (2.10), we define the point processes,

(2.11) $\eta_{k,n}^{\pm}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}s_{k,n}^{\pm}({\mathcal{Y}},\mathcal{P}_{n})\,\delta_{(c({\mathcal{Y}}),\,n\omega_{d}\rho({\mathcal{Y}})^{d}-a_{n})}\in
M_{p}(\mathbb{Y}),$

and for a fixed $u_{0}\in{\mathbb{R}}$,

$G_{k,n}^{\pm}\mathrel{\mathop{\mathchar
58\relax}}=G_{k,n}^{\pm}(u_{0})=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}g_{k,n}^{\pm}({\mathcal{Y}},\mathcal{P}_{n};u_{0}).$

It was shown in Proposition 4.2 and 5.1 of [3] that if $(a_{n})$ satisfies
condition (1.5), then $(G_{k,n})_{n\geq 1}$, $(G_{k,n}^{+})_{n\geq 1}$, and
$(G_{k+1,n}^{-})_{n\geq 1}$ exhibit the same phase transition in terms of the
$k$th homological connectivity. From this point of view, the results below
claim that even under the assumption (1.6), $(G_{k,n}^{+})_{n\geq 1}$ and
$(G_{k+1,n}^{-})_{n\geq 1}$ still satisfy the same limit theorem as Theorem
2.5. Moreover, the processes $(\eta_{k,n}^{+})_{n\geq 1}$ and
$(\eta_{k+1,n}^{-})_{n\geq 1}$ also satisfy the same limit theorem as Theorem
2.3. For the latter process, however, we can deduce the corresponding limit
result only for the process with a restricted state space, i.e.,

(2.12) $\eta_{k+1,n}^{(-,\mathsf{r})}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}s_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n})\,\delta_{n\omega_{d}\rho({\mathcal{Y}})^{d}-a_{n}}\in
M_{p}\big{(}(-\infty,\infty]\big{)},$

for which the locational coordinate $c({\mathcal{Y}})$ has been removed from
(2.11). We conjecture that the same limit theorem holds even for the process
$(\eta_{k+1,n}^{-})_{n\geq 1}$ as well. However, due to a technical
complication occurring in the application of a series of results in [3], we
have decided not to pursue this direction in the present work.

######  Corollary 2.6.

Recall $\lambda_{k}$ as defined in (2.7).
$(i)$ For $1\leq k\leq d-1$, we have

(2.13) $b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}^{+}\in\cdot)\to\lambda_{k},\ \
\text{ in }\mathcal{M}_{0},\ \ \ n\to\infty.$

$(ii)$ For $1\leq k\leq d-2$, we have

(2.14)
$b_{k,n}^{-1}\mathbb{P}(\eta_{k+1,n}^{(-,\mathsf{r})}\in\cdot)\to\lambda_{k}^{(\mathsf{r})},\
\ \text{ in }\mathcal{M}_{0},\ \ \ n\to\infty,$

where
$\lambda_{k}^{(\mathsf{r})}(\cdot)=D_{k}\int_{\mathbb{R}}{\mathbbm{1}}\\{\delta_{u}\in\cdot\\}\,e^{-u}\operatorname{d\\!}u,$
is the restricted version of $\lambda_{k}$.

$(iii)$ For $1\leq k\leq d-1$, we have

(2.15) $\displaystyle
b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}\in\cdot)\stackrel{{\scriptstyle
v}}{{\to}}D_{k}e^{-u_{0}}\delta_{1},\ \ \text{ in }M_{+}(E),\ \ \ n\to\infty.$

$(iv)$ For $1\leq k\leq d-2$, we have

(2.16) $b_{k,n}^{-1}\mathbb{P}(G_{k+1,n}^{-}\in\cdot)\stackrel{{\scriptstyle
v}}{{\to}}D_{k}e^{-u_{0}}\delta_{1},\ \ \text{ in }M_{+}(E),\ \ \ n\to\infty.$

In particular, (2.15) and (2.16) imply that as $n\to\infty$,

$\displaystyle b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}\geq 1)\to D_{k}e^{-u_{0}},$
$\displaystyle b_{k,n}^{-1}\mathbb{P}(G_{k+1,n}^{-}\geq 1)\to
D_{k}e^{-u_{0}}.$

The proof of the limit theorem for $(\eta_{k,n}^{+})_{n\geq 1}$ is based on
establishing asymptotic equivalence with $(\eta_{k,n})_{n\geq 1}$ by using
estimates in [3]. For the case of $k=1$, we need estimates from [19]. From
this we can deduce convergence of $(G_{k,n}^{+})_{n\geq 1}$ via the continuous
mapping theorem. We then compare $(G_{k,n}^{+})_{n\geq 1}$ with
$(G_{k+1,n}^{-})_{n\geq 1}$ again using estimates in [3]. Finally, by using
the results for $(G_{k+1,n}^{-})_{n\geq 1}$ at many choices of $u_{0}$’s, we
can prove the limit theorem for $(\eta_{k+1,n}^{(-,\mathsf{r})})_{n\geq 1}$.

##  3\. Proofs

For the proof of Theorem 2.3, we need to recall that the _Kantorovich-
Rubinstein distance_ between the laws of point processes $\eta_{1}$ and
$\eta_{2}$ on $\mathbb{Y}$ is defined as

$d_{\mathsf{KR}}\big{(}\mathcal{L}(\eta_{1}),\mathcal{L}(\eta_{2})\big{)}\mathrel{\mathop{\mathchar
58\relax}}=\sup_{h}\big{|}\mathbb{E}[h(\eta_{1})]-\mathbb{E}[h(\eta_{2})]\big{|},$

where $\mathcal{L}(\eta_{i})$ denotes a probability law of $\eta_{i}$, and the
supremum is taken over all $1$-Lipschitz functions
$h\mathrel{\mathop{\mathchar 58\relax}}M_{p}(\mathbb{Y})\to{\mathbb{R}}$, with
respect to the total variation distance on $M_{p}(\mathbb{Y})$. Recall also
that the _total variation distance_ between two measures $\mu_{1}$ and
$\mu_{2}$ on $\mathbb{Y}$ is defined as

$d_{\mathsf{TV}}(\mu_{1},\mu_{2})\mathrel{\mathop{\mathchar
58\relax}}=\sup_{A\subset\mathbb{Y}}\big{|}\mu_{1}(A)-\mu_{2}(A)\big{|}.$

In the previous section, we defined the notion of a critical point
$c({\mathcal{Y}})$, as well as its critical value $\rho({\mathcal{Y}})$, only
when ${\mathcal{Y}}$ forms a critical face; see Definition 2.2. In this
section, however, one needs to extend these concepts, even when
${\mathcal{Y}}$ may not form a critical face. Specifically, let
${\mathcal{Y}}=\\{y_{1},\dots,y_{k+1}\\}$ be a subset of $\mathcal{P}_{n}$,
which is in general position (${\mathcal{Y}}$ does not necessarily form a
critical $k$-face). Let

$E({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}z\in\mathbb{T}^{d}\mathrel{\mathop{\mathchar
58\relax}}\|z-y_{1}\|=\cdots=\|z-y_{k+1}\|\big{\\}}$

be the collection of equidistant points from ${\mathcal{Y}}$. Since
${\mathcal{Y}}$ is in general position, $E({\mathcal{Y}})$ forms a
$(d-k)$-dimensional affine plane, so that there exists a unique point
$c({\mathcal{Y}})\in E({\mathcal{Y}})$ such that
$\|c({\mathcal{Y}})-y_{1}\|=\inf_{z\in E({\mathcal{Y}})}\|z-y_{1}\|$.
Moreover, we define $\rho({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}=\|c({\mathcal{Y}})-y_{1}\|$.

Before commencing the proof of Theorem 2.3, we define a finite and positive
constant

(3.1) $D_{k}\mathrel{\mathop{\mathchar
58\relax}}=\frac{(k!)^{d-k+1}}{(k+1)!d\omega_{d}^{k}}\,\binom{d}{k}\frac{\Omega_{d}}{\Omega_{k}\Omega_{d-k}}\int_{(S^{k-1})^{k+1}}h_{k}({\bm{\theta}})V_{\mathsf{simp}}({\bm{\theta}})^{d-k+1}\operatorname{d\\!}{\bm{\theta}},$

where $\Omega_{j}=\prod_{i=1}^{j}\omega_{i}$ ($\omega_{i}$ is volume of the
$i$-dimensional unit ball), $S^{k-1}$ is the $(k-1)$-dimensional unit sphere,
and $h_{k}({\bm{\theta}})\mathrel{\mathop{\mathchar
58\relax}}={\mathbbm{1}}\\{c({\bm{\theta}})\in\sigma({\bm{\theta}})\\}$
(recall that $\sigma({\bm{\theta}})$ is an open geometric $k$-simplex in
${\mathbb{R}}^{d}$ spanned by ${\bm{\theta}}$). Further,
$V_{\mathsf{simp}}({\bm{\theta}})$ represents the volume of
$\sigma({\bm{\theta}})$.

Throughout this section, denote by $C^{*}$ a generic positive constant, which
is independent of $n$ but may vary between and within the lines.

###  3.1. Proof of Theorem 2.3

###### Proof.

The proof exploits the ideas in Theorem 4.1 of [7]. Recall that
$C_{K}^{+}(\mathbb{Y})$ is the collection of non-negative and continuous
functions on $\mathbb{Y}$ with compact support. Given $H_{1},H_{2}\in
C_{K}^{+}(\mathbb{Y})$ and $\varepsilon_{1},\varepsilon_{2}>0$, define
$F_{H_{1},H_{2},\varepsilon_{1},\varepsilon_{2}}\mathrel{\mathop{\mathchar
58\relax}}M_{p}(\mathbb{Y})\to[0,1]$ by

(3.2)
$F_{H_{1},H_{2},\varepsilon_{1},\varepsilon_{2}}(\xi)=(1-e^{-(\xi(H_{1})-\epsilon_{1})_{+}})(1-e^{-(\xi(H_{2})-\epsilon_{2})_{+}}),$

where $(a)_{+}=a$ if $a\geq 0$ and 0 otherwise, and
$\xi(H_{\ell})=\int_{\mathbb{Y}}H_{\ell}(x)\xi(\operatorname{d\\!}x)$ for
$\ell=1,2$. Denote $\lambda_{k,n}(\cdot)\mathrel{\mathop{\mathchar
58\relax}}=b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\in\cdot)$ and recall the
definition of $\lambda_{k}$ at (2.7). According to Theorem A.2 in [15], (2.6)
immediately follows if one can show that

$\lambda_{k,n}(F_{H_{1},H_{2},\varepsilon_{1},\varepsilon_{2}})\to\lambda_{k}(F_{H_{1},H_{2},\varepsilon_{1},\varepsilon_{2}}),\
\ \text{ as }n\to\infty,$

for all $H_{1},H_{2}\in C_{K}^{+}(\mathbb{Y})$ and
$\varepsilon_{1},\varepsilon_{2}>0$.

Fix $H_{1},H_{2}$ and $\varepsilon_{1},\varepsilon_{2}$ henceforth and write
$F=F_{H_{1},H_{2},\varepsilon_{1},\varepsilon_{2}}$. Let $\zeta_{k,n}$ be a
Poisson point process on $\mathbb{Y}$ with intensity measure
$b_{k,n}D_{k}e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u$ for
$c\in\mathbb{T}^{d}$, $u\in{\mathbb{R}}$. Then, writing
$\lambda_{k,n}(F)=b_{k,n}^{-1}\mathbb{E}\big{[}F(\eta_{k,n})\big{]}$, it
suffices to demonstrate that as $n\to\infty$,

(3.3)
$b_{k,n}^{-1}\big{|}\,\mathbb{E}\big{[}F(\eta_{k,n})\big{]}-\mathbb{E}\big{[}F(\zeta_{k,n})\big{]}\,\big{|}\to
0,$ (3.4)
$b_{k,n}^{-1}\mathbb{E}\big{[}F(\zeta_{k,n})\big{]}\to\lambda_{k}(F).$

We begin with proving (3.3). Since $H_{\ell}$ has compact support on
$\mathbb{Y}$, one can find $u_{0}\in{\mathbb{R}}$ such that

$\text{supp}(H_{1})\bigcup\text{supp}(H_{2})\subset\mathbb{Y}_{0}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{T}^{d}\times[u_{0},\infty],$

where $\text{supp}(H_{\ell})$ denotes the support of $H_{\ell}$. Thus, we may
assume without loss of generality that $\eta_{k,n}$ is a point process with
state space restricted to $\mathbb{Y}_{0}$, i.e.,

(3.5)
$\eta_{k,n}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}g_{k,n}({\mathcal{Y}},\mathcal{P}_{n};u_{0})\delta_{(c({\mathcal{Y}}),\,n\omega_{d}\rho({\mathcal{Y}})^{d}-a_{n})}\in
M_{p}(\mathbb{Y}_{0}).$

Accordingly, $\zeta_{k,n}$ can be viewed as a Poisson point process with mean
measure

$(\mathsf{Leb}\otimes\tau_{k,n})(\operatorname{d\\!}c,\operatorname{d\\!}u)\mathrel{\mathop{\mathchar
58\relax}}=b_{k,n}D_{k}e^{-u}{\mathbbm{1}}\\{u\geq
u_{0}\\}\operatorname{d\\!}c\operatorname{d\\!}u,\ \
c\in\mathbb{T}^{d},\,u\in{\mathbb{R}}.$

Since $0\leq F\leq 1$, it is elementary to see that

$\big{|}F(\mu_{1})-F(\mu_{2})\big{|}\leq 2d_{\mathsf{TV}}(\mu_{1},\mu_{2}),\ \
\ \mu_{1},\mu_{2}\in M_{p}(\mathbb{Y}_{0}).$

This implies that $F$ is $2$-Lipschitz, and hence,

$\big{|}\,\mathbb{E}\big{[}F(\eta_{k,n})\big{]}-\mathbb{E}\big{[}F(\zeta_{k,n})\big{]}\,\big{|}\leq
2d_{\mathsf{KR}}\big{(}\mathcal{L}(\eta_{k,n}),\mathcal{L}(\zeta_{k,n})\big{)}.$

Now, (3.3) will follow if we can prove that

(3.6)
$b_{k,n}^{-1}d_{\mathsf{KR}}\big{(}\mathcal{L}(\eta_{k,n}),\mathcal{L}(\zeta_{k,n})\big{)}\to
0,\ \ \text{as }n\to\infty.$

By virtue of Theorem 4.1 in [7] (see also equ. (6.5) therein), it suffices to
show that as $n\to\infty$,

(3.7)
$b_{k,n}^{-1}d_{\mathsf{TV}}\big{(}\mathbb{E}[\eta_{k,n}(\cdot)],\,\mathsf{Leb}\otimes\tau_{k,n}\big{)}\to
0,$ (3.8)
$b_{k,n}^{-1}\big{\\{}\text{Var}(\eta_{k,n}(\mathbb{Y}_{0}))-\mathbb{E}[\eta_{k,n}(\mathbb{Y}_{0})]\big{\\}}\to
0,$

and

(3.9)
$\displaystyle\begin{split}&\frac{b_{k,n}^{-1}n^{2(k+1)}}{\big{(}(k+1)!\big{)}^{2}}\,\int_{(\mathbb{T}^{d})^{k+1}}\int_{(\mathbb{T}^{d})^{k+1}}{\mathbbm{1}}\big{\\{}\mathcal{B}({\bf
x})\cap\mathcal{B}({\bf z})\neq\emptyset\big{\\}}\\\
&\qquad\qquad\qquad\qquad\qquad\times\mathbb{E}\big{[}g_{k,n}({\bf
x},\mathcal{P}_{n}+\delta_{\bf x};u_{0})\big{]}\mathbb{E}\big{[}g_{k,n}({\bf
z},\mathcal{P}_{n}+\delta_{\bf z};u_{0})\big{]}\operatorname{d\\!}{\bf
z}\operatorname{d\\!}{\bf x}\to 0,\end{split}$

where $\mathcal{B}({\bf x})$ denotes an open ball in ${\mathbb{R}}^{d}$ with
radius $\rho({\bf x})$ centered at $c({\bf x})$. We now prove these equations
in that order.

Proof of (3.7): According to Lemma 2.4 in [8],

(3.10) ${\mathbbm{1}}\\{{\mathcal{Y}}\text{ forms a critical
}k\text{-face}\\}=h_{k}({\mathcal{Y}}){\mathbbm{1}}\big{\\{}\mathcal{B}({\mathcal{Y}})\cap\mathcal{P}_{n}=\emptyset\big{\\}},$

where
$h_{k}({\mathcal{Y}})={\mathbbm{1}}\\{c({\mathcal{Y}})\in\sigma({\mathcal{Y}})\\}$.
Substituting (3.10) into (2.3) with $u=u_{0}$, one can write

(3.11)
$\displaystyle\begin{split}g_{k,n}({\mathcal{Y}},\mathcal{P}_{n};u_{0})&=h_{k}({\mathcal{Y}}){\mathbbm{1}}\big{\\{}\mathcal{B}({\mathcal{Y}})\cap\mathcal{P}_{n}=\emptyset\big{\\}}\times{\mathbbm{1}}\big{\\{}\rho({\mathcal{Y}})\in[r_{n}(u_{0}),R_{n}]\big{\\}}.\end{split}$

Appealing to the multivariate Mecke formula for Poisson point processes (see,
e.g., Chapter 4 in [16]) and using (3.11), we have, for every
$A\subset\mathbb{Y}_{0}$,

(3.12)
$\displaystyle\begin{split}\mathbb{E}[\eta_{k,n}(A)]&=\frac{n^{k+1}}{(k+1)!}\,\int_{(\mathbb{T}^{d})^{k+1}}{\mathbbm{1}}\big{\\{}(c({\bf
x}),n\omega_{d}\rho({\bf x})^{d}-a_{n})\in
A\big{\\}}\mathbb{E}\big{[}g_{k,n}({\bf x},\mathcal{P}_{n}+\delta_{\bf
x};u_{0})\big{]}\operatorname{d\\!}{\bf x}\\\
&=\frac{n^{k+1}}{(k+1)!}\,\int_{(\mathbb{T}^{d})^{k+1}}{\mathbbm{1}}\big{\\{}(c({\bf
x}),n\omega_{d}\rho({\bf x})^{d}-a_{n})\in A\big{\\}}h_{k}({\bf
x}){\mathbbm{1}}\\{\rho({\bf x})\leq R_{n}\\}e^{-n\omega_{d}\rho({\bf
x})^{d}}\operatorname{d\\!}{\bf x}.\end{split}$

By a change of variable based on the Blaschke-Petkantschin-type formula
provided in Lemma C.1 of [3],

(3.13)
$\displaystyle\begin{split}\mathbb{E}[\eta_{k,n}(A)]&=\frac{D_{bp}}{(k+1)!}\,\int_{(S^{k-1})^{k+1}}h_{k}({\bm{\theta}})V_{\mathsf{simp}}({\bm{\theta}})^{d-k+1}\operatorname{d\\!}{\bm{\theta}}\\\
&\quad\times
n^{k+1}\int_{\mathbb{T}^{d}}\int_{0}^{R_{n}}{\mathbbm{1}}\big{\\{}(c,n\omega_{d}\rho^{d}-a_{n})\in
A\big{\\}}\rho^{dk-1}e^{-n\omega_{d}\rho^{d}}\operatorname{d\\!}\rho\operatorname{d\\!}c,\end{split}$

where

$D_{bp}=(k!)^{d-k+1}\binom{d}{k}\frac{\Omega_{d}}{\Omega_{k}\Omega_{d-k}}.$

Performing the change of variable by $u=n\omega_{d}\rho^{d}-a_{n}$,

$\displaystyle
n^{k+1}\int_{\mathbb{T}^{d}}\int_{0}^{R_{n}}{\mathbbm{1}}\big{\\{}(c,n\omega_{d}\rho^{d}-a_{n})\in
A\big{\\}}\rho^{dk-1}e^{-n\omega_{d}\rho^{d}}\operatorname{d\\!}\rho\operatorname{d\\!}c$
$\displaystyle=\frac{b_{k,n}}{d\omega_{d}^{k}}\,\int_{A}{\mathbbm{1}}\big{\\{}u\in(-a_{n},n\omega_{d}R_{n}^{d}-a_{n})\big{\\}}\Big{(}1+\frac{u}{a_{n}}\Big{)}^{k-1}e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u.$

It thus follows that

(3.14)
$\mathbb{E}[\eta_{k,n}(A)]=D_{k}b_{k,n}\int_{A}{\mathbbm{1}}\big{\\{}u\in(-a_{n},n\omega_{d}R_{n}^{d}-a_{n})\big{\\}}\Big{(}1+\frac{u}{a_{n}}\Big{)}^{k-1}e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u,$

where $D_{k}$ is defined in (3.1).

Since

$(\mathsf{Leb}\otimes\tau_{k,n})(A)=b_{k,n}D_{k}\int_{A}e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u,$

we have, as $n\to\infty$,

$\displaystyle
b_{k,n}^{-1}d_{\mathsf{TV}}\big{(}\mathbb{E}[\eta_{k,n}(\cdot)],\,\mathsf{Leb}\otimes\tau_{k,n}\big{)}$
$\displaystyle=b_{k,n}^{-1}\sup_{A\subset\mathbb{Y}_{0}}\big{|}\,\mathbb{E}[\eta_{k,n}(A)]-(\mathsf{Leb}\otimes\tau_{k,n})(A)\big{|}$
$\displaystyle\leq
D_{k}\int_{u_{0}}^{\infty}\Big{|}\Big{(}1+\frac{u}{a_{n}}\Big{)}^{k-1}-1\Big{|}e^{-u}\operatorname{d\\!}u$
$\displaystyle\quad+D_{k}\int_{u_{0}}^{\infty}{\mathbbm{1}}\big{\\{}u\notin(-a_{n},n\omega_{d}R_{n}^{d}-a_{n})\big{\\}}e^{-u}\operatorname{d\\!}u\to
0.$

The last convergence follows from the dominated convergence theorem and the
assumption that $a_{n}/(nR_{n}^{d})\to 0$, $n\to\infty$.

Proof of (3.8): By a simple calculation,

$\displaystyle\mathbb{E}\big{[}\eta_{k,n}(\mathbb{Y}_{0})^{2}\big{]}$
$\displaystyle=\mathbb{E}\Big{[}\Big{(}\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}g_{k,n}({\mathcal{Y}},\mathcal{P}_{n};u_{0})\Big{)}^{2}\Big{]}$
$\displaystyle=\sum_{\ell=0}^{k+1}\mathbb{E}\Big{[}\sum_{\begin{subarray}{c}{\mathcal{Y}}\subset\mathcal{P}_{n},\\\
|{\mathcal{Y}}|=k+1\end{subarray}}\sum_{\begin{subarray}{c}{\mathcal{Y}}^{\prime}\subset\mathcal{P}_{n},\\\
|{\mathcal{Y}}^{\prime}|=k+1\end{subarray}}g_{k,n}({\mathcal{Y}},\mathcal{P}_{n};u_{0})g_{k,n}({\mathcal{Y}}^{\prime},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\\{|{\mathcal{Y}}\cap{\mathcal{Y}}^{\prime}|=\ell\\}\Big{]}$
$\displaystyle=\mathrel{\mathop{\mathchar
58\relax}}\sum_{\ell=0}^{k+1}I_{\ell,n}.$

One can compute $I_{\ell,n}$ explicitly using Palm theory for Poisson
processes; see Section 8 in [3]. We will now use some of the bounds therein.
Since $I_{k+1,n}=E[\eta_{k,n}(\mathbb{Y}_{0})]$, it follows that

$\text{Var}(\eta_{k,n}(\mathbb{Y}_{0}))-\mathbb{E}[\eta_{k,n}(\mathbb{Y}_{0})]=\sum_{\ell=1}^{k}I_{\ell,n}+(I_{0,n}-I_{k+1,n}^{2}).$

Now, for the proof of (3.8), one needs to show that for every
$\ell\in\\{1,\dots,k\\}$,

(3.15) $b_{k,n}^{-1}I_{\ell,n}\to 0,\ \ \ n\to\infty,$

and

(3.16) $b_{k,n}^{-1}(I_{0,n}-I_{k+1,n}^{2})\to 0,\ \ \ n\to\infty.$

For the upper bound of (3.15), we use equ. (8.15) in [3], while replacing
$\Lambda\mathrel{\mathop{\mathchar 58\relax}}=n\omega_{d}r_{n}^{d}(u_{0})$
therein with $a_{n}+u_{0}$; this yields that for every
$\ell\in\\{1,\dots,k\\}$,

(3.17) $\displaystyle\begin{split}b_{k,n}^{-1}I_{\ell,n}&\leq
C^{*}b_{k,n}^{-1}n(a_{n}+u_{0})^{k-1}e^{-(a_{n}+u_{0})}\Big{\\{}\varepsilon_{\ell}^{k+2-\ell}(a_{n}+u_{0})^{k+1-\ell}\\\
&\qquad+\Big{(}\frac{\delta_{\ell}}{\varepsilon_{\ell}}\Big{)}^{d-k}\big{(}\delta_{\ell}(a_{n}+u_{0})\big{)}^{k-\ell+1}+(a_{n}+u_{0})^{k+1-\ell}e^{-C_{1}\delta_{\ell}(a_{n}+u_{0})}\Big{\\}},\end{split}$

where $C_{1}\in(0,\infty)$ is some constant and

(3.18) $\varepsilon_{\ell}\mathrel{\mathop{\mathchar
58\relax}}=(a_{n}+u_{0})^{\frac{1}{2(k+2-\ell)}-1};\ \ \
\delta_{\ell}\mathrel{\mathop{\mathchar
58\relax}}=\frac{k+2-\ell}{C_{1}}\cdot\frac{\log(a_{n}+u_{0})}{a_{n}+u_{0}}.$

Substituting (3.18) into (3.17), we obtain that

$b_{k,n}^{-1}I_{\ell,n}\leq C^{*}\big{\\{}a_{n}^{-1/2}+C^{*}(\log
a_{n})^{d-\ell+1}a_{n}^{-\frac{d-k}{2(k+2-\ell)}}+a_{n}^{-1}\big{\\}}\to 0,\ \
\ n\to\infty.$

For the proof of (3.16), we use equ. (8.17) in [3] and replace
$\Lambda\mathrel{\mathop{\mathchar 58\relax}}=n\omega_{d}r_{n}^{d}(u_{0})$
therein with $a_{n}+u_{0}$, to obtain that

$\displaystyle b_{k,n}^{-1}(I_{0,n}-I_{k+1,n}^{2})$ $\displaystyle\leq
C^{*}b_{k,n}^{-1}n(a_{n}+u_{0})^{k-1}e^{-(a_{n}+u_{0})}$
$\displaystyle\quad\times\big{\\{}\varepsilon_{0}^{d+2}(a_{n}+u_{0})^{k+1}+(a_{n}+u_{0})^{k+1}e^{-C_{1}\varepsilon_{0}(a_{n}+u_{0})}\big{\\}},$

where $\varepsilon_{0}\mathrel{\mathop{\mathchar
58\relax}}=(a_{n}+u_{0})^{-(k+2)/(d+2)}$. It is then easy to show that

$b_{k,n}^{-1}(I_{0,n}-I_{k+1,n}^{2})\leq
C^{*}\big{\\{}a_{n}^{-1}+a_{n}^{k+1}e^{-C_{1}a_{n}^{(d-k)/(d+2)}}\big{\\}}\to
0,\ \ \text{as }n\to\infty.$

Proof of (LABEL:e:cond3.thm4.1): First fix ${\bf x}\in(\mathbb{T}^{d})^{k+1}$.
For any ${\bf z}\in(\mathbb{T}^{d})^{k+1}$ with $\mathcal{B}({\bf
x})\cap\mathcal{B}({\bf z})\neq\emptyset$, we have $\|c({\bf x})-c({\bf
z})\|\leq\rho({\bf x})+\rho({\bf z})\leq 2R_{n}$. Thus,

${\mathbbm{1}}\big{\\{}\mathcal{B}({\bf x})\cap\mathcal{B}({\bf
z})\neq\emptyset\big{\\}}\leq{\mathbbm{1}}\big{\\{}c({\bf z})\in
B_{2R_{n}}(c({\bf x}))\big{\\}}.$

Applying this inequality, while proceeding as in (3.12) and (3.13),

$\displaystyle\frac{b_{k,n}^{-1}n^{k+1}}{(k+1)!}\,\int_{(\mathbb{T}^{d})^{k+1}}{\mathbbm{1}}\big{\\{}\mathcal{B}({\bf
x})\cap\mathcal{B}({\bf z})\neq\emptyset\big{\\}}\,\mathbb{E}[g_{k,n}({\bf
z},\mathcal{P}_{n}+\delta_{\bf z};u_{0})]\operatorname{d\\!}{\bf z}$
$\displaystyle\leq\frac{b_{k,n}^{-1}n^{k+1}}{(k+1)!}\,\int_{(\mathbb{T}^{d})^{k+1}}{\mathbbm{1}}\big{\\{}c({\bf
z})\in B_{2R_{n}}(c({\bf x}))\big{\\}}h_{k}({\bf
z}){\mathbbm{1}}\big{\\{}\rho({\bf
z})\in[r_{n}(u_{0}),R_{n}]\big{\\}}e^{-n\omega_{d}\rho({\bf
z})^{d}}\operatorname{d\\!}{\bf z}$
$\displaystyle=\frac{b_{k,n}^{-1}n^{k+1}}{(k+1)!}\,D_{bp}\int_{(S^{k-1})^{k+1}}h_{k}({\bm{\theta}})V_{\mathsf{simp}}({\bm{\theta}})^{d-k+1}\operatorname{d\\!}{\bm{\theta}}\int_{B_{2R_{n}}(c({\bf
x}))}\operatorname{d\\!}c\int_{r_{n}(u_{0})}^{R_{n}}\rho^{dk-1}e^{-n\omega_{d}\rho^{d}}\operatorname{d\\!}\rho$
$\displaystyle=b_{k,n}^{-1}(2R_{n})^{d}\omega_{d}\mathbb{E}[\eta_{k,n}(\mathbb{Y}_{0})],$

where the last equality is due to (3.13) with $A=\mathbb{Y}_{0}$. Because of
(3.7), the last term above is further bounded by
$C^{*}b_{k,n}^{-1}R_{n}^{d}(\mathsf{Leb}\otimes\tau_{k,n})(\mathbb{Y}_{0})=C^{*}R_{n}^{d}$.
Referring the obtained bound back to (LABEL:e:cond3.thm4.1), one can
eventually bound (LABEL:e:cond3.thm4.1) by

$C^{*}R_{n}^{d}\,\frac{n^{k+1}}{(k+1)!}\,\int_{(\mathbb{T}^{d})^{k+1}}\mathbb{E}[g_{k,n}({\bf
x},\mathcal{P}_{n}+\delta_{\bf x};u_{0})]\operatorname{d\\!}{\bf
x}=C^{*}R_{n}^{d}\mathbb{E}[\eta_{k,n}(\mathbb{Y}_{0})]=o(R_{n}^{d})\to 0,\ \
\ n\to\infty,$

as desired.

We will proceed to show (3.4). Notice that one may express
$\zeta_{k,n}=\sum_{i=1}^{N_{k,n}}\delta_{(C_{i},U_{i})}$, where $N_{k,n}$ is a
Poisson random variable with mean $b_{k,n}D_{k}e^{-u_{0}}$ and $(C_{i},U_{i})$
are i.i.d. random vectors on $\mathbb{Y}$ with density
$e^{-(u-u_{0})}{\mathbbm{1}}\\{u\geq
u_{0}\\}\operatorname{d\\!}c\operatorname{d\\!}u$. Besides, $N_{k,n}$ can be
taken to be independent of $(C_{i},U_{i})$. With this construction now
available, we have

$\displaystyle b_{k,n}^{-1}\mathbb{E}[F(\zeta_{k,n})]$
$\displaystyle=b_{k,n}^{-1}\mathbb{E}\Big{[}\prod_{\ell=1}^{2}\Big{(}1-e^{-\big{(}\sum_{i=1}^{N_{k,n}}H_{\ell}(C_{i},U_{i})-\varepsilon_{\ell}\big{)}_{+}}\Big{)}\Big{]}$
$\displaystyle=b_{k,n}^{-1}\mathbb{E}\Big{[}\prod_{\ell=1}^{2}\Big{(}1-e^{-(H_{\ell}(C_{1},U_{1})-\varepsilon_{\ell})_{+}}\Big{)}{\mathbbm{1}}\\{N_{k,n}=1\\}\Big{]}$
$\displaystyle\qquad\qquad+b_{k,n}^{-1}\mathbb{E}\Big{[}\prod_{\ell=1}^{2}\Big{(}1-e^{-\big{(}\sum_{i=1}^{N_{k,n}}H_{\ell}(C_{i},U_{i})-\varepsilon_{\ell}\big{)}_{+}}\Big{)}{\mathbbm{1}}\\{N_{k,n}\geq
2\\}\Big{]}$ $\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}A_{n}+B_{n}.$

By an elementary calculation,

$B_{n}\leq b_{k,n}^{-1}\mathbb{P}(N_{k,n}\geq 2)\leq C^{*}b_{k,n}\to 0,\ \ \
n\to\infty.$

By the independence of $N_{k,n}$ and $(C_{1},U_{1})$,

$\displaystyle A_{n}$
$\displaystyle=b_{k,n}^{-1}\mathbb{E}\Big{[}\prod_{\ell=1}^{2}\Big{(}1-e^{-(H_{\ell}(C_{1},U_{1})-\varepsilon_{\ell})_{+}}\Big{)}\Big{]}\mathbb{P}(N_{k,n}=1)$
$\displaystyle=D_{k}e^{-u_{0}}\cdot
e^{-b_{k,n}D_{k}e^{-u_{0}}}\int_{\mathbb{Y}}\prod_{\ell=1}^{2}\Big{(}1-e^{-(H_{\ell}(c,u)-\varepsilon_{\ell})_{+}}\Big{)}e^{-(u-u_{0})}{\mathbbm{1}}\\{u\geq
u_{0}\\}\operatorname{d\\!}c\operatorname{d\\!}u$ $\displaystyle\to
D_{k}\int_{\mathbb{Y}}\prod_{\ell=1}^{2}\Big{(}1-e^{-(H_{\ell}(c,u)-\varepsilon_{\ell})_{+}}\Big{)}e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u=\lambda_{k}(F),$

as required. ∎

###  3.2. Proof of Theorem 2.5

###### Proof.

By restricting the state space from $\mathbb{Y}$ to
$\mathbb{Y}_{0}=\mathbb{T}^{d}\times[u_{0},\infty]$, we can establish
$\mathcal{M}_{0}$-convergence analogous to Theorem 2.3. Namely, as
$n\to\infty$,

$b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\in\cdot)\to\lambda_{k},\ \ \text{ in
}\mathcal{M}_{0}.$

Here, $\eta_{k,n}$ is defined as in (3.5) due to the restriction of the state
space, while the limiting measure is written as

$\lambda_{k}(\cdot)=D_{k}\int_{\mathbb{Y}_{0}}{\mathbbm{1}}\\{\delta_{(c,u)}\in\cdot\\}e^{-u}\operatorname{d\\!}c\operatorname{d\\!}u.$

Define a continuous map $V\mathrel{\mathop{\mathchar
58\relax}}M_{p}(\mathbb{Y}_{0})\to{\mathbb{N}}\mathrel{\mathop{\mathchar
58\relax}}=\\{0,1,2,\dots\\}$ by $V(\xi)=\xi(\mathbb{Y}_{0})$, where
$M_{p}(\mathbb{Y}_{0})$ is equipped with vague topology, and ${\mathbb{N}}$ is
equipped with the discrete topology. It then follows from Theorem 2.5 in [14]
that

$\lambda_{k,n}\circ V^{-1}\to\lambda_{k}\circ V^{-1},\ \ \text{ in
}\mathcal{M}_{0}(M_{p}({\mathbb{N}})),\ \ \ n\to\infty;$

equivalently,

(3.19) $b_{k,n}^{-1}\mathbb{P}(G_{k,n}\in\cdot)\to D_{k}e^{-u_{0}}\delta_{1},\
\ \text{ in }\mathcal{M}_{0}(M_{p}({\mathbb{N}})),\ \ \ n\to\infty.$

For every $x>0$, the indicator
${\mathbbm{1}}_{[x,\infty)}(\cdot)\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{N}}\to\\{0,1\\}$ is bounded and continuous with respect to
the discrete topology. Moreover, it vanishes in the neighborhood of $0$ (i.e.,
the origin of ${\mathbb{N}}$). In conclusion,
${\mathbbm{1}}_{[x,\infty)}(\cdot)\in\mathcal{C}_{0}$, and thus, (3.19)
implies that for every $x>0$,

$b_{k,n}^{-1}\mathbb{P}(G_{k,n}\geq x)\to
D_{k}e^{-u_{0}}\delta_{1}([x,\infty)),\ \ \ n\to\infty.$

Now, Lemma 6.1 in [20] concludes the desired vague convergence in Theorem 2.5.
∎

###  3.3. Proof of Corollary 2.6

###### Proof of (2.13).

Part I: $2\leq k\leq d-1$.
It suffices to show that as $n\to\infty$,

$b_{k,n}^{-1}\big{|}\,\mathbb{E}\big{[}F(\eta_{k,n})\big{]}-\mathbb{E}\big{[}F(\eta_{k,n}^{+})\big{]}\,\big{|}\to
0,$

where $F$ is defined at (3.2) (subscripts are all omitted). Since
$\bigcup_{\ell=1}^{2}\text{supp}(H_{\ell})\subset\mathbb{Y}_{0}=\mathbb{T}^{d}\times[u_{0},\infty]$
for some $u_{0}\in{\mathbb{R}}$, we can reformulate $\eta_{k,n}^{\pm}$ as

(3.20)
$\eta_{k,n}^{\pm}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+1}g_{k,n}^{\pm}({\mathcal{Y}},\mathcal{P}_{n};u_{0})\delta_{(c({\mathcal{Y}}),\,n\omega_{d}\rho({\mathcal{Y}})^{d}-a_{n})}\in
M_{p}(\mathbb{Y}_{0}).$

As $0\leq F\leq 1$, we have that

$b_{k,n}^{-1}\big{|}\,\mathbb{E}\big{[}F(\eta_{k,n})\big{]}-\mathbb{E}\big{[}F(\eta_{k,n}^{+})\big{]}\,\big{|}\leq
b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\neq\eta_{k,n}^{+}).$

By definition, $\eta_{k,n}=\eta_{k,n}^{+}+\eta_{k,n}^{-}$; hence,
$\eta_{k,n}\neq\eta_{k,n}^{+}$ implies that
$G_{k,n}^{-}=\eta_{k,n}^{-}(\mathbb{Y}_{0})\geq 1$. Now, by Markov’s
inequality,

$b_{k,n}^{-1}\mathbb{P}(\eta_{k,n}\neq\eta_{k,n}^{+})\leq
b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{-}\geq 1)\leq
b_{k,n}^{-1}\mathbb{E}[G_{k,n}^{-}].$

By the proof of Proposition 5.1 in [3] (see the equation at page 744, line 3
therein),

(3.21) $b_{k,n}^{-1}\mathbb{E}[G_{k,n}^{-}]\leq
C^{*}b_{k,n}^{-1}n(a_{n}+u_{0})^{k-2}e^{-(a_{n}+u_{0})}\leq C^{*}a_{n}^{-1}\to
0,\ \ \text{as }n\to\infty,$

for which we have replaced “$\Lambda$” in [3] with $a_{n}+u_{0}$. This
completes the proof of Part I.

Part II: $k=1$.
Our proof uses ideas from Chapter 13 of [19]. As in Part I, $\eta_{1,n}$ and
$\eta_{1,n}^{+}$ can be given in the form of (3.5) and (3.20) for a fixed
$u_{0}\in{\mathbb{R}}$. For $r>0$, let $G(\mathcal{P}_{n},r)$ be a random
geometric graph on a Poisson point process $\mathcal{P}_{n}$ with edges
$\\{x,y\\}$ for all pairs $x,y\in\mathcal{P}_{n}$ with $\|x-y\|\leq r$. Since
$\eta_{1,n}^{-}$ corresponds to the edges of the minimal spanning tree on
$\mathcal{P}_{n}$, $G_{1,n}^{-}\geq 1$ implies that
$G(\mathcal{P}_{n},2r_{n}(u_{0}))$ is not connected, and thus, by arguing as
in Part I we have that

$\mathbb{P}(\eta_{1,n}\neq\eta_{1,n}^{+})\leq\mathbb{P}\big{(}G_{1,n}^{-}\geq
1\big{)}\leq\mathbb{P}\big{(}G(\mathcal{P}_{n},2r_{n}(u_{0}))\text{ is not
connected}\big{)}.$

Hence, it suffices to demonstrate that

(3.22) $b_{1,n}^{-1}\mathbb{P}\big{(}G(\mathcal{P}_{n},2r_{n}(u_{0}))\text{ is
not connected}\big{)}\to 0,\ \ \ n\to\infty.$

Before continuing, we shall introduce a few required notions. Given a graph
$G$ with vertex set $V$, a non-empty subset $U\subset V$ is said to be a
separating set for $G$ if none of the vertices in $V\setminus U$ are adjacent
to $U$. Moreover, a pair of non-empty disjoint sets $U\subset V$, $W\subset V$
is called a separating pair for $G$, if $(i)$ the subgraph of $G$ induced by
$U$ is connected and the same holds for $W$, $(ii)$ none of the elements in
$U$ are adjacent to any element in $W$, and $(iii)$ none of the elements of
$V\setminus(U\cup W)$ are adjacent to $U\cup W$. Using these notions and given
$K>0$, we define $E_{n}(K)$ as the event that there exists a separating set
$U$ for $G(\mathcal{P}_{n},2r_{n}(u_{0}))$ with at least two elements, such
that the diameter of $U$ is less than $2Kr_{n}(u_{0})$. Further, denote by
$F_{n}(K)$ the event that there exists a separating pair $(U,W)$ for
$G(\mathcal{P}_{n},2r_{n}(u_{0}))$, so that the diameters of $U$ and $W$ both
exceed $2Kr_{n}(u_{0})$.

Now, one can upper bound (3.22) as

(3.23) $b_{1,n}^{-1}\mathbb{P}\big{(}G(\mathcal{P}_{n},2r_{n}(u_{0}))\text{ is
not connected}\big{)}\leq
b_{1,n}^{-1}\mathbb{P}(A_{n})+b_{1,n}^{-1}\mathbb{P}(B_{n}),$

where $A_{n}$ is the event that there exists at least one isolated vertex in
$G(\mathcal{P}_{n},2r_{n}(u_{0}))$, and $B_{n}$ is the event for which there
are no isolated vertices but $G(\mathcal{P}_{n},2r_{n}(u_{0}))$ contains
_multiple_ connected components (of size at least $2$). Now Markov’s
inequality and the Mecke formula for Poisson point processes yield that

$b_{1,n}^{-1}\mathbb{P}(A_{n})\leq
b_{1,n}^{-1}n\int_{\mathbb{T}^{d}}\mathbb{P}\big{(}\mathcal{P}_{n}(B_{2r_{n}(u_{0})}(x))=0\big{)}\operatorname{d\\!}x=e^{a_{n}-2^{d}(a_{n}+u_{0})}\to
0,\ \ \text{as }n\to\infty.$

In order to handle the remaining term in (3.23), it is sufficient to prove the
following results.
$(i)$ There exists $K>0$ such that

(3.24) $b_{1,n}^{-1}\mathbb{P}(F_{n}(K))\to 0,\ \ \ n\to\infty.$

$(ii)$ For $K$ as above, there exists $K_{1}\in(0,K)$ such that

(3.25) $\displaystyle b_{1,n}^{-1}\mathbb{P}(E_{n}(K_{1}))\to 0,\ \ \
n\to\infty,$ (3.26) $\displaystyle
b_{1,n}^{-1}\mathbb{P}\big{(}E_{n}(K)\setminus E_{n}(K_{1})\big{)}\to 0,\ \ \
n\to\infty.$

Indeed, if one can show $(i)$ and $(ii)$ above, it follows from the proof of
Theorem 13.10 in [19] and Lemma 13.1 therein that as $n\to\infty$,

$\displaystyle b_{1,n}^{-1}\mathbb{P}(B_{n})$ $\displaystyle\leq
b_{1,n}^{-1}\mathbb{P}\big{(}E_{n}(K)\big{)}+b_{1,n}^{-1}\mathbb{P}(F_{n}(K))$
$\displaystyle=b_{1,n}^{-1}\mathbb{P}(E_{n}(K_{1}))+b_{1,n}^{-1}\mathbb{P}\big{(}E_{n}(K)\setminus
E_{n}(K_{1})\big{)}+b_{1,n}^{-1}\mathbb{P}(F_{n}(K))\to 0.$

Proof of (3.24): By the proof of Proposition 13.13 in [19], there exist
constants $\varepsilon>0$, $\gamma>0$ such that we have, for $K>0$ and large
enough $n$,

$\displaystyle\mathbb{P}(F_{n}(K))$ $\displaystyle\leq C^{*}\sum_{i\geq
K/\varepsilon}(2r_{n}(u_{0}))^{-2d}ie^{\gamma i}\cdot
e^{-i\varepsilon^{d}(2r_{n}(u_{0}))^{d}n}\leq
C^{*}\frac{n^{2}}{a_{n}^{2}}\sum_{i\geq
K/\varepsilon}e^{-((2\varepsilon)^{d}(a_{n}+u_{0})/\omega_{d}-\gamma^{\prime})i},$

for some $\gamma^{\prime}>0$ with $ie^{\gamma i}\leq e^{\gamma^{\prime}i}$ for
all $i$. The last expression is further bounded by
$C^{*}n^{2}a_{n}^{-2}e^{-2^{d}\varepsilon^{d-1}Ka_{n}/\omega_{d}}$; thus,

$b_{1,n}^{-1}\mathbb{P}(F_{n}(K))\leq
C^{*}ne^{-(2^{d}\varepsilon^{d-1}K/\omega_{d}-1)a_{n}}\leq
C^{*}n^{2-2^{d}\varepsilon^{d-1}K/\omega_{d}}.$

If one chooses $K>\omega_{d}/(2\varepsilon)^{d-1}$, then (3.24) follows as
desired.

Proof of (3.25): Having fixed $K$ as above, we next show (3.25). It follows
from equ. at page 297, line -3 in [19] and the proof of Lemma 13.5 in [19]
that there exists $K_{1}\in(0,K)$ such that

$\displaystyle\mathbb{P}(E_{n}(K_{1}))$ $\displaystyle\leq
C^{*}n(nr_{n}(u_{0})^{d})^{1-d}\int_{\mathbb{T}^{d}}\mathbb{P}\big{(}\mathcal{P}_{n}(B_{2r_{n}(u_{0})}(x))=0\big{)}\operatorname{d\\!}x$
$\displaystyle=C^{*}n\Big{(}\frac{a_{n}+u_{0}}{\omega_{d}}\Big{)}^{1-d}e^{-2^{d}(a_{n}+u_{0})}\leq
C^{*}na_{n}^{1-d}e^{-2^{d}a_{n}}.$

Therefore, $b_{1,n}^{-1}\mathbb{P}(E_{n}(K_{1}))\leq
C^{*}a_{n}^{1-d}e^{-(1+2^{d})a_{n}}\to 0$ as $n\to\infty$.

Proof of (3.26): By equ. (13.43) in [19] and the last line of the proof of
Lemma 13.16 therein, there exists $\xi>0$ such that

$\mathbb{P}\big{(}E_{n}(K)\setminus E_{n}(K_{1})\big{)}\leq
C^{*}n\int_{\mathbb{T}^{d}}\mathbb{P}\big{(}\mathcal{P}_{n}(B_{2r_{n}(u_{0})}(x))=0\big{)}e^{-\frac{\xi
n}{2}(2r_{n}(u_{0}))^{d}}\operatorname{d\\!}x\leq C^{*}ne^{-2a_{n}-2^{d-1}\xi
a_{n}/\omega_{d}}.$

This implies that as $n\to\infty$,

$b_{1,n}^{-1}\mathbb{P}\big{(}E_{n}(K)\setminus E_{n}(K_{1})\big{)}\leq
C^{*}e^{-a_{n}-2^{d-1}\xi a_{n}/\omega_{d}}\to 0.$

∎

###### Proof of (2.15).

One can exploit the same proof strategy as in Theorem 2.5, with $\eta_{k,n}$
replaced by $\eta^{+}_{k,n}$ and (2.6) replaced by (2.13).

∎

###### Proof of (2.16).

By Lemma 6.1 in [20] and (2.15), we only need to prove that for every $x>0$,

$b_{k,n}^{-1}\big{\\{}\mathbb{P}(G_{k+1,n}^{-}\geq
x)-\mathbb{P}(G_{k,n}^{+}\geq x)\big{\\}}\to 0,\ \ \ n\to\infty.$

It thus suffices to show that as $n\to\infty$,

(3.27) $\displaystyle b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}>G_{k+1,n}^{-})\to 0,$
(3.28) $\displaystyle b_{k,n}^{-1}\mathbb{P}(G_{k+1,n}^{-}>G_{k,n}^{+})\to 0.$

We begin with proving (3.27). Let $i_{*}\mathrel{\mathop{\mathchar
58\relax}}H_{k}\big{(}\bigcup_{p\in\mathcal{P}_{n}}B_{r_{n}(u_{0})}(p)\big{)}\to
H_{k}(\mathbb{T}^{d})$ be a map induced by the inclusion
$i\mathrel{\mathop{\mathchar
58\relax}}\bigcup_{p\in\mathcal{P}_{n}}B_{r_{n}(u_{0})}(p)\hookrightarrow\mathbb{T}^{d}$,
where $H_{k}(\cdot)$ represents the $k$th homology group. By the Nerve Lemma
(see, e.g., Theorem 10.7 of [1]), there exists an isomorphism
$f_{*}\mathrel{\mathop{\mathchar
58\relax}}H_{k}\big{(}\check{C}(\mathcal{P}_{n},r_{n}(u_{0}))\big{)}\to
H_{k}\big{(}\bigcup_{p\in\mathcal{P}_{n}}B_{r_{n}(u_{0})}(p)\big{)}$. Now, we
can define the map

$j_{*}\mathrel{\mathop{\mathchar 58\relax}}=i_{*}\circ
f_{*}\mathrel{\mathop{\mathchar
58\relax}}H_{k}\big{(}\check{C}(\mathcal{P}_{n},r_{n}(u_{0}))\big{)}\to
H_{k}(\mathbb{T}^{d}).$

Suppose that $j_{*}$ is surjective; that is,
$\text{Im}(j_{*})=H_{k}(\mathbb{T}^{d})$. Then, all the positive critical
$k$-faces will be eventually terminated by a “matching” negative critical
$(k+1)$-face, which in turn means that $G_{k,n}^{+}\leq G_{k+1,n}^{-}$.

For $0<r_{1}<r_{2}<\infty$ and a subset ${\mathcal{Y}}\subset\mathcal{P}_{n}$
with $|{\mathcal{Y}}|=d+1$, let $A_{r_{1},r_{2}}({\mathcal{Y}})$ be the
closure of an annulus $B_{r_{2}}(c({\mathcal{Y}}))\setminus
B_{r_{1}}(c({\mathcal{Y}}))$, i.e., the closed annulus centered at the
critical point $c({\mathcal{Y}})$. We then define

$\hat{G}_{d,n}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=d+1}g_{d,n}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\Big{\\{}A_{r_{n}(u_{0}),4r_{n}(u_{0})}({\mathcal{Y}})\not\subset\bigcup_{p\in\mathcal{P}_{n}}B_{r_{n}(u_{0})}(p)\Big{\\}}.$

According to the proof of Lemma 5.7 in [2], as well as Lemma 5.9 therein, one
can see that if $\hat{G}_{d,n}=0$, then $j_{*}$ becomes surjective. Combining
these observations, it now follows that

$b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}>G_{k+1,n}^{-})\leq
b_{k,n}^{-1}\mathbb{P}(\hat{G}_{d,n}\geq 1)\leq
b_{k,n}^{-1}\mathbb{E}[\hat{G}_{d,n}].$

Finally, appealing to the last equation in the proof of Lemma 5.8 in [2],

$b_{k,n}^{-1}\mathbb{E}[\hat{G}_{d,n}]\leq
C^{*}b_{k,n}^{-1}n(a_{n}+u_{0})^{d-1}e^{-(a_{n}+u_{0})(1+C^{*})}\leq
C^{*}a_{n}^{d-k}e^{-C^{*}a_{n}}\to 0,\ \ \ n\to\infty,$

as desired.

Next, we focus our attention to (3.28). To estimate the probability in (3.28),
we need to refer to the detailed discussion on the structure of critical
$k$-faces, provided in Section 7 of [3]. Below, we introduce some additional
concepts and notation, which we try to keep as consistent as possible with
those in [3].

Suppose in the sequel that a $(k+1)$-point subset ${\mathcal{Y}}$ of
$\mathcal{P}_{n}$ is in general position. Let $\Pi({\mathcal{Y}})$ be the
unique linear $k$-plane centered at $c({\mathcal{Y}})$ containing
${\mathcal{Y}}$, and $S^{k-1}$ be the unit sphere in $\Pi({\mathcal{Y}})$
(centered at $c({\mathcal{Y}})$). Denote by
$\theta({\mathcal{Y}})=\\{\theta_{1}({\mathcal{Y}}),\dots,\theta_{k+1}({\mathcal{Y}})\\}$
the spherical coordinates of ${\mathcal{Y}}$ in $S^{k-1}$, and define
$\hat{\theta}_{i}({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}=\theta({\mathcal{Y}})\setminus\\{\theta_{i}({\mathcal{Y}})\\}$. Let

$\phi_{1}({\mathcal{Y}})\mathrel{\mathop{\mathchar 58\relax}}=\min_{1\leq
i\leq k+1}\big{\|}c(\hat{\theta}_{i}({\mathcal{Y}}))\big{\|}$

be the scaled distance via $\rho({\mathcal{Y}})$, between the center
$c({\mathcal{Y}})$ and the nearest $(k-1)$-face of ${\mathcal{Y}}$. Moreover,
denote by $\hat{\mathcal{Y}}_{\text{min}}$ such a nearest $(k-1)$-face of
${\mathcal{Y}}$ from the center $c({\mathcal{Y}})$. Now, let
$\hat{\mathcal{Y}}_{i}$, $i=1,\dots,k$ be the remaining $(k-1)$-faces of
${\mathcal{Y}}$ _except for_ $\hat{\mathcal{Y}}_{\text{min}}$. We then define

$\phi_{2}({\mathcal{Y}})\mathrel{\mathop{\mathchar 58\relax}}=\min_{1\leq
i\leq
k}\rho({\mathcal{Y}})^{-1}\inf_{z\in\Pi(\hat{\mathcal{Y}}_{i})}\big{\|}c(\hat{\mathcal{Y}}_{\text{min}})-z\big{\|},$

which is the distance scaled by $\rho({\mathcal{Y}})$, between
$c(\hat{\mathcal{Y}}_{\text{min}})$ and the nearest $(k-1)$-face of
${\mathcal{Y}}$ (except for $\hat{\mathcal{Y}}_{\text{min}}$). Next, let

$\hat{\rho}({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}=\rho({\mathcal{Y}})+\|c(\hat{\mathcal{Y}}_{\text{min}})-c({\mathcal{Y}})\|,$

and

$\hat{\mathcal{B}}({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}=B_{\hat{\rho}({\mathcal{Y}})}\big{(}c(\hat{\mathcal{Y}}_{\text{min}})\big{)}\setminus\mathcal{B}({\mathcal{Y}}),$

where $\mathcal{B}({\mathcal{Y}})$ is defined at (LABEL:e:cond3.thm4.1). Let
$\hat{\Pi}({\mathcal{Y}})$ be the $(d-1)$-dimensional affine plane containing
$c(\hat{\mathcal{Y}}_{\text{min}})$ and orthogonal to the line through
$c({\mathcal{Y}})$ and $c(\hat{\mathcal{Y}}_{\text{min}})$. Given $\alpha>0$,
define

(3.29) $\hat{\mathcal{B}}_{\alpha}({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}y\in\hat{\mathcal{B}}({\mathcal{Y}})\mathrel{\mathop{\mathchar
58\relax}}\inf_{z\in\hat{\Pi}({\mathcal{Y}})}\|y-z\|\geq\alpha\rho({\mathcal{Y}})\big{\\}}.$

Since $\Pi(\hat{\mathcal{Y}}_{\text{min}})\subset\hat{\Pi}({\mathcal{Y}})$,
the set (3.29) contains points in $\hat{\mathcal{B}}({\mathcal{Y}})$ that are
distant at least $\alpha\rho({\mathcal{Y}})$ from the plane
$\Pi(\hat{\mathcal{Y}}_{\text{min}})$.

Before proceeding, we define

$G_{k+1,n}^{(-,+)}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}g_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\\{\hat{\mathcal{Y}}_{\text{min}}\text{
is a positive critical }k\text{-face}\\},$

which is the number of negative critical $(k+1)$-faces whose critical values
are between $r_{n}(u_{0})$ and $R_{n}$, so that
$\hat{\mathcal{Y}}_{\text{min}}$ forms a positive critical $k$-face. Because
of equ. (7.11) in [3], it holds that

$G_{k+1,n}^{-}-G_{k+1,n}^{(-,+)}\leq\sum_{i=1}^{4}G_{k+1,n}^{(i)},$

where $G_{k+1,n}^{(i)}$ above are defined respectively as

$\displaystyle G_{k+1,n}^{(1)}$
$\displaystyle=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}g_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\\{\phi_{1}({\mathcal{Y}})>\varepsilon_{1}\\},$
$\displaystyle G_{k+1,n}^{(2)}$
$\displaystyle=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}g_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\\{\phi_{1}({\mathcal{Y}})\leq\varepsilon_{1},\,\phi_{2}({\mathcal{Y}})\leq\varepsilon_{2}\\},$
$\displaystyle G_{k+1,n}^{(3)}$
$\displaystyle=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}g_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\big{\\{}\phi_{1}({\mathcal{Y}})\leq\varepsilon_{1},\,\big{(}\hat{\mathcal{B}}({\mathcal{Y}})\setminus\hat{\mathcal{B}}_{\varepsilon_{3}}({\mathcal{Y}})\big{)}\cap\mathcal{P}_{n}\neq\emptyset\big{\\}},$
$\displaystyle G_{k+1,n}^{(4)}$
$\displaystyle=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}g_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\\{\phi_{1}({\mathcal{Y}})\leq\varepsilon_{1},\,\phi_{2}({\mathcal{Y}})>\varepsilon_{2},\,\hat{\mathcal{B}}_{\varepsilon_{3}}({\mathcal{Y}})\cap\mathcal{P}_{n}\neq\emptyset\\},$

with

$\varepsilon_{1}\mathrel{\mathop{\mathchar
58\relax}}=\frac{2}{D_{k,2}}\cdot\frac{\log(a_{n}+u_{0})}{a_{n}+u_{0}};\ \
\varepsilon_{2}\mathrel{\mathop{\mathchar
58\relax}}=\big{(}\log(a_{n}+u_{0})\big{)}^{-4};\ \
\varepsilon_{3}\mathrel{\mathop{\mathchar 58\relax}}=\varepsilon_{1}^{2/3}$

($D_{k,2}$ is a positive constant introduced in the proof of Lemma 5.6 in
[3]). Although $\varepsilon_{1}$ and $\varepsilon_{3}$ above are defined
analogously to equ. (7.12) in [3], the definition of $\varepsilon_{2}$ is
different from equ. (7.12) of [3].

Furthermore, define

$G_{k+1,n}^{(5)}=\sum_{{\mathcal{Y}}\subset\mathcal{P}_{n},\,|{\mathcal{Y}}|=k+2}g_{k+1,n}^{-}({\mathcal{Y}},\mathcal{P}_{n};u_{0}){\mathbbm{1}}\\{\phi_{1}({\mathcal{Y}})\leq\varepsilon_{1},\,\rho({\mathcal{Y}})>r_{n}(u_{0}),\,\rho(\hat{\mathcal{Y}}_{\text{min}})<r_{n}(u_{0})\big{\\}}.$

From the discussion in Part II of the proof of Proposition 7.1 in [3], it is
known that $G_{k+1,n}^{(i)}=0$ for $i=1,\dots,5$ implies that
$G_{k+1,n}^{(-,+)}\leq G_{k,n}^{+}$. In conclusion, one can see that

$\displaystyle b_{k,n}^{-1}\mathbb{P}(G_{k+1,n}^{-}>G_{k,n}^{+})$
$\displaystyle\leq
b_{k,n}^{-1}\mathbb{P}\Big{(}\sum_{i=1}^{4}G_{k+1,n}^{(i)}+G_{k+1,n}^{(-,+)}>G_{k,n}^{+}\Big{)}$
$\displaystyle\leq
b_{k,n}^{-1}\mathbb{P}\Big{(}G_{k+1,n}^{(-,+)}>G_{k,n}^{+},\,\sum_{i=1}^{5}G_{k+1,n}^{(i)}=0\Big{)}+b_{k,n}^{-1}\mathbb{P}\Big{(}\sum_{i=1}^{5}G_{k+1,n}^{(i)}\geq
1\Big{)}$
$\displaystyle\leq\sum_{i=1}^{5}b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(i)}].$

From this analysis, it is enough to show that for every $i\in\\{1,\dots,5\\}$,

(3.30) $b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(i)}]\to 0,\ \ \text{ as
}n\to\infty.$

The proof of (3.30) is highly related to Lemmas 7.9–7.13 in [3]. First, by
replacing “$\Lambda$” in [3] with $a_{n}+u_{0}$, the proof of Lemma 7.9 in [3]
ensures that

$b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(1)}]\leq
C^{*}b_{k,n}^{-1}n(a_{n}+u_{0})^{k-2}e^{-(a_{n}+u_{0})}\leq C^{*}a_{n}^{-1}\to
0,\ \ \ n\to\infty.$

Subsequently, proceeding as in the proof of Lemma 7.10 in [3],

$\displaystyle b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(2)}]$ $\displaystyle\leq
C^{*}b_{k,n}^{-1}n(a_{n}+u_{0})^{k}e^{-(a_{n}+u_{0})}\Big{\\{}\frac{\varepsilon_{1}\varepsilon_{12}}{\sqrt{1-\varepsilon_{1}^{2}}}+\frac{\varepsilon_{1}(\varepsilon_{1}+\varepsilon_{2})}{\varepsilon_{12}}\Big{\\}}$
$\displaystyle\leq
C^{*}a_{n}\Big{\\{}\frac{\varepsilon_{1}\varepsilon_{12}}{\sqrt{1-\varepsilon_{1}^{2}}}+\frac{\varepsilon_{1}(\varepsilon_{1}+\varepsilon_{2})}{\varepsilon_{12}}\Big{\\}},$

where $\varepsilon_{12}\mathrel{\mathop{\mathchar
58\relax}}=(\log(a_{n}+u_{0}))^{-2}$. Since $\varepsilon_{1}\to 0$ and
$\varepsilon_{1}/\varepsilon_{2}\to 0$ as $n\to\infty$, one can see that
$b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(2)}]$ $\leq C^{*}(\log a_{n})^{-1}\to 0$
as $n\to\infty$. Next, it follows from the proof of Lemma 7.11 in [3] that

$\displaystyle b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(3)}]\leq
C^{*}b_{k,n}^{-1}\varepsilon_{3}^{4}n(a_{n}+u_{0})^{k+1}e^{-(a_{n}+u_{0})}\leq
C^{*}(\log a_{n})^{8/3}a_{n}^{-2/3}\to 0,\ \ \ n\to\infty.$

Furthermore, by the proof of Lemma 7.12 in [3],

$b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(4)}]\leq
C^{*}b_{k,n}^{-1}\varepsilon_{0}^{-d}\varepsilon_{1}n(a_{n}+u_{0})^{k}e^{-(a_{n}+u_{0})(1+C^{*}\varepsilon_{0})},$

where $\varepsilon_{0}\mathrel{\mathop{\mathchar
58\relax}}=(a_{n}+u_{0})^{-3/4}$. Thus, we have

$b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(4)}]\leq C^{*}a_{n}^{3d/4}(\log
a_{n})e^{-C^{*}a_{n}^{1/4}}\to 0,\ \ \ n\to\infty.$

Finally, the proof of Lemma 7.13 in [3] concludes that as $n\to\infty$,

$\displaystyle b_{k,n}^{-1}\mathbb{E}[G_{k+1,n}^{(5)}]$ $\displaystyle\leq
C^{*}b_{k,n}^{-1}\varepsilon_{1}n(a_{n}+u_{0})^{k}e^{-(a_{n}+u_{0})}\big{\\{}d\varepsilon_{1}^{2}(a_{n}+u_{0})+o(\varepsilon_{1}^{2}(a_{n}+u_{0}))\big{\\}}$
$\displaystyle\leq
C^{*}b_{k,n}^{-1}\varepsilon_{1}^{3}n(a_{n}+u_{0})^{k+1}e^{-(a_{n}+u_{0})}\leq
C^{*}(\log a_{n})^{3}a_{n}^{-1}\to 0.$

Now, (3.30) has been established. ∎

###### Proof of (2.14).

Analogously to (2.12), we define the process
$(\eta_{k,n}^{(+,\mathsf{r})})_{n\geq 1}$ by dropping the locational
coordinate $c({\mathcal{Y}})$ from (2.11). Note that by continuous mapping
theorem, the restricted process $(\eta_{k,n}^{(+,\mathsf{r})})_{n\geq 1}$
satisfies $\mathcal{M}_{0}$ convergence analogous to (2.13) with the limit
measure $\lambda_{k}^{(r)}$ defined at (2.14). Define
$F=F_{H_{1},H_{2},\epsilon_{1},\epsilon_{2}}$ as in (3.2), for which the
domain of $F$ is restricted to $M_{p}\big{(}(-\infty,\infty]\big{)}$. Then, to
complete the proof, we need to show that

(3.31)
$b_{k,n}^{-1}\big{|}\mathbb{E}[F(\eta_{k,n}^{(+,\mathsf{r})})]-\mathbb{E}[F(\eta_{k+1,n}^{(-,\mathsf{r})})]\big{|}\to
0.$

Again the compact support of $H_{\ell}$ on $(-\infty,\infty]$ gives us
$u_{0}\in{\mathbb{R}}$ such that

$\text{supp}(H_{1})\bigcup\text{supp}(H_{2})\subset[u_{0},\infty].$

From (2.15) and (2.16), we have $b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}(u_{0})\geq
2)\to 0$ and $b_{k,n}^{-1}\mathbb{P}(G_{k+1,n}^{-}(u_{0})\geq 2)\to 0$ as
$n\to\infty$. Because of $0\leq F\leq 1$, (3.27), (3.28), one can obtain
(3.31) if we show that

(3.32)
$b_{k,n}^{-1}\mathbb{E}\big{[}|F(\eta_{k,n}^{(+,\mathsf{r})})-F(\eta_{k+1,n}^{(-,\mathsf{r})})|{\mathbbm{1}}\\{G_{k,n}^{+}(u_{0})=G_{k+1,n}^{-}(u_{0})=1\\}\big{]}\to
0,\ \ \ n\to\infty.$

Fix $\delta>0$. Again from (2.15) and (2.16), we can choose $u_{1}>u_{0}$ so
large that

$\lim_{n\to\infty}b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}(u_{1})\geq 1)\leq\delta,\
\ \text{ and }\ \
\lim_{n\to\infty}b_{k,n}^{-1}\mathbb{P}(G_{k+1,n}^{-}(u_{1})\geq
1)\leq\delta.$

Define $A_{n}\mathrel{\mathop{\mathchar
58\relax}}=\\{G_{k,n}^{+}(u_{0})=G_{k+1,n}^{-}(u_{0})=1,\,G_{k,n}^{+}(u_{1})=G_{k+1,n}^{-}(u_{1})=0\\}$;
then, (3.32) follows if we can show that

(3.33)
$\lim_{n\to\infty}b_{k,n}^{-1}\mathbb{E}\big{[}|F(\eta_{k,n}^{(+,\mathsf{r})})-F(\eta_{k+1,n}^{(-,\mathsf{r})})|{\mathbbm{1}}_{A_{n}}\big{]}=0.$

Observing that

$\big{|}F(\eta_{k,n}^{(+,\mathsf{r})})-F(\eta_{k+1,n}^{(-,\mathsf{r})})\big{|}\leq
2\sum_{\ell=1}^{2}|\eta_{k,n}^{(+,\mathsf{r})}(H_{\ell})-\eta_{k+1,n}^{(-,\mathsf{r})}(H_{\ell})|,$

(3.33) is implied by

$\lim_{n\to\infty}b_{k,n}^{-1}\mathbb{E}\big{[}|\eta_{k,n}^{(+,\mathsf{r})}(H_{\ell})-\eta_{k+1,n}^{(-,\mathsf{r})}(H_{\ell})|{\mathbbm{1}}_{A_{n}}\big{]}=0,$

for each $\ell=1,2$.

Fix $\ell\in\\{1,2\\}$. If $A_{n}$ holds, we may assume, without loss of
generality, that $H_{\ell}$ is supported on $[u_{0},u_{1}]$. Moreover, under
$A_{n}$, there exist random variables $X_{n},X_{n}^{\prime}\in[u_{0},u_{1}]$
such that $\eta_{k,n}^{(+,\mathsf{r})}(H_{\ell})=H_{\ell}(X_{n})$ and
$\eta_{k+1,n}^{(-,\mathsf{r})}(H_{\ell})=H_{\ell}(X_{n}^{\prime})$. Since
$H_{\ell}$ is uniformly continuous on $[u_{0},u_{1}]$, for every $\delta>0$,
there exists $\delta_{0}>0$ such that
$\big{|}H_{\ell}(X_{n})-H_{\ell}(X_{n}^{\prime})\big{|}\leq\delta$ whenever
$|X_{n}-X_{n}^{\prime}|\leq\delta_{0}$. Applying the above observations,

$b_{k,n}^{-1}\mathbb{E}\big{[}|\eta_{k,n}^{(+,\mathsf{r})}(H_{\ell})-\eta_{k+1,n}^{(-,\mathsf{r})}(H_{\ell})|{\mathbbm{1}}_{A_{n}}\big{]}\leq\delta
b_{k,n}^{-1}\mathbb{P}(A_{n})+C^{*}b_{k,n}^{-1}\mathbb{P}\big{(}A_{n}\cap\big{\\{}|X_{n}-X_{n}^{\prime}|>\delta_{0}\big{\\}}\big{)}.$

By (2.15),

$\limsup_{n\to\infty}\delta
b_{k,n}^{-1}\mathbb{P}(A_{n})\leq\delta\lim_{n\to\infty}b_{k,n}^{-1}\mathbb{P}(G_{k,n}^{+}(u_{0})\geq
1)=\delta D_{k}e^{-u_{0}}.$

As $\delta>0$ is arbitrary, it now remains to show that for any
$\delta_{0}>0$,

$\lim_{n\to\infty}b_{k,n}^{-1}\mathbb{P}\big{(}A_{n}\cap\big{\\{}|X_{n}-X_{n}^{\prime}|>\delta_{0}\big{\\}}\big{)}=0.$

Let $\delta_{0}>0$ be fixed. For the proof, let us partition
$[u_{0},u_{1})=\bigcup_{i=1}^{m}[v_{i},v_{i+1})$ such that
$|v_{i+1}-v_{i}|\leq\delta_{0}$ for all $i\in\\{1,\dots,m\\}$. For $1\leq
i\leq m$, set

$\displaystyle G_{k,n}^{+}(i)\mathrel{\mathop{\mathchar
58\relax}}=G_{k,n}^{+}(v_{i})-G_{k,n}^{+}(v_{i+1}),\ \ \text{ and }\ \
G_{k+1,n}^{-}(i)\mathrel{\mathop{\mathchar
58\relax}}=G_{k+1,n}^{-}(v_{i})-G_{k+1,n}^{-}(v_{i+1}).$

By construction, one can see that

$A_{n}\cap\big{\\{}|X_{n}-X_{n}^{\prime}|>\delta_{0}\big{\\}}\subset\bigcup_{i=1}^{m}\big{\\{}G_{k,n}^{+}(i)\neq
G_{k+1,n}^{-}(i)\big{\\}}.$

Indeed, if $|X_{n}-X_{n}^{\prime}|>\delta_{0}$ under $A_{n}$, then $X_{n}$ and
$X_{n}^{\prime}$ must fall into distinct subsets of the partition. Since $m$
is finite, it suffices to demonstrate that for all $1\leq i\leq m$, as
$n\to\infty$,

$b_{k,n}^{-1}\mathbb{P}\big{(}G_{k,n}^{+}(i)\neq G_{k+1,n}^{-}(i)\big{)}\to
0.$

This is however a direct consequence of (3.27) and (3.28) by restricting the
state space properly. ∎

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# A drug classification pipeline for Medicaid claims using RxNorm

Nicholas Williams
Department of Epidemiology
Columbia University
New York, NY 10032
<EMAIL_ADDRESS>
Kara E. Rudolph
Department of Epidemiology
Columbia University
New York, NY 10032
<EMAIL_ADDRESS>

###### Abstract

Objective: Freely preprocess drug codes recorded in electronic health records
and insurance claims to drug classes that may then be used in biomedical
research.

Materials and Methods: We developed a drug classification pipeline for linking
National Drug Codes to the World Health Organization Anatomical Therapeutic
Chemical classification. To implement our solution, we created an R package
interface to the National Library of Medicine’s RxNorm API.

Results: Using the classification pipeline, 59.4% of all unique NDC were
linked to an ATC, resulting in 95.5% of all claims being successfully linked
to a drug classification. We identified 12,004 unique NDC codes that were
classified as being an opioid or non-opioid prescription for treating pain.

Discussion: Our proposed pipeline performed similarly well to other NDC
classification routines using commercial databases. A check of a small, random
sample of non-active NDC found the pipeline to be accurate for classifying
these codes.

Conclusion: The RxNorm NDC classification pipeline is a practical and reliable
tool for categorizing drugs in large-scale administrative claims data.

_Keywords_ Medicaid claims $\cdot$ Electronic health record $\cdot$ Drug class
$\cdot$ RxNorm $\cdot$ ATC

##  1 Background and Significance

Electronic health record (EHR) and administrative insurance claims data have
become an increasingly important asset in conducting observational
epidemiological and pharmacological research [1, 2]. Much of this research
requires classifying drugs in terms of mechanism of action, therapeutic
intent, or chemical structure; commonly referred to as drug class. However,
drug data in the EHR are typically recorded using codes that do not indicate
the general drug class, and so require preprocessing before they can be used
for biomedical research. In the United States (US), EHR drug data are recorded
as national drug codes (NDCs), which are unique three-segment, 11-digit
identifiers assigned to all human drugs. The first segment is a manufacturer
identifier, the second segment is a product identifier, and the third is a
package identifier. Although an NDC uniquely identifies a specific drug
product, it does not directly identify the drug class. Thus, preprocessing
these data by converting NDC to useful study-specific drug classes is a key
step in applied research.

One option for classifying NDC is to pay for a commercial database[3], such as
IBM Micromedex Red Book [4] or Multum from Oracle Health [5]. However, paying
for one of these services limits access to researchers with large enough
budgets, and introduces a roadblock for reproducibility. Another approach
would be to manually create a list of all the NDCs that belong to a given drug
class of interest [6, 7]. While free, this approach introduces variability
between research teams, is time-consuming, and may have poor performance as
the number of NDC that belong to a certain class may be well into the
thousands (see Table 1).

We address this gap by providing a freely available user-friendly software
that performs this preprocessing step, linking NDCs with drug class codes. In
performing this linkage, we use the free-to-use US National Library of
Medicine’s (NLM’s) RxNorm, which links many products to different drug
classification systems. We use RxNorm’s linkage to the World Health
Organization’s (WHO’s) Anatomical Therapeutic Chemical classification (ATC),
which classifies substances "according to the organ or system on which they
act and their therapeutic, pharmacological and chemical properties." [8] In
the ATC system, drugs are assigned a code that indicates a hierarchical
subdivision into five different levels with increasing levels of specificity.
We are only interested in the first four ATC levels as the fifth level
indicates the specific substance. For example the ATC code, up to the fourth
level, for fentanyl is N01AH: "N" indicates fentanyl operates on the nervous
system, "01" indicates fentanyl is an anesthetic, the "A" indicates it’s part
of the "general" subgroup of anesthetics, and the "H" indicates it’s an opioid
anesthetic.

##  2 Objective

We were motivated to develop this software by the need to identify opioid and
non-opioid drugs prescribed for pain in a large cohort of Medicaid claims
data. We use our motivating example to demonstrate the software’s utility and
performance. Our software uses the NLM RxNorm API and an R package interface.
We evaluate its performance by the proportion of prescription claims that are
linked to a drug class.

##  3 Materials and methods

###  3.1 Classification pipeline

Our approach retrieves drug information using a set of free-to-use application
program interfaces (APIs) provided by NLM via representational state transfer
(REST) to access the RxNorm and RxTerms datasets: the RxNorm and RxTerms [9],
and RxClass [10] APIs. To avoid sending potentially thousands of requests to
the NLM servers, we also make use of RxNav-in-a-Box, a locally installable
version of the APIs, in combination with Docker [11]. To query the APIs, we
created an R [12] package interface which is available for download on GitHub.

First, we retrieve an NDC’s status. In RxNorm, an NDC is classified as either
"active", "obsolete", "alien", or "unknown". We ignore NDCs with a status of
"unknown" as they are items that have never appeared in RxNorm. Often these
products are medical supplies, vitamins or minerals, dietary drugs, and other
over-the-counter medications.

If an NDC is considered "active", we then query RxNorm for the RxCUI assigned
to the NDC and then submit a query for the associated product-level ATC. If a
product-level ATC is not found, we instead submit a query for the ingredient-
level ATC of all active-ingredients in the drug. Note that product-level ATC
are assigned by NLM, while ingredient-level ATC are assigned to a substance by
the WHO.

If an NDC is considered "obsolete" or "alien", we again query RxNorm for the
RxCUI assigned to the NDC, but then perform an additional query for the RxCUI
status to see if the RxCUI has been "remapped", "quantified", or considered
"not current" or "obsolete". If the RxCUI is "remapped" or "not current", we
attempt to replace the original RxCUI with the current RxCUI, and again search
for the product or ingredient level ATC. If the RxCUI status is "obsolete" we
attempt to link the NDC to a new RxCUI through the semantic clinical drug
(SCD) concept which itself is a mapping of a drug product into its ingredient,
strength, and dose form; again, then searching for the product or ingredient
level ATC of the RxCUI associated with the SCD. If the RxCUI status is
"quantified", it indicates that the RxCUI is missing a "quantity factor" but
has been linked to another related concept that may be quantified, which we
then again attempt to link to a product or ingredient-level ATC.

The R script used for running the above classification pipeline is available
on GitHub.

Table 1: ATC codes of interest ATC |  Description | No. of NDC linked
---|---|---
Opioids for pain |
N02A |  Opioid analgesics | 2264
N01AH |  Opioid anesthetics | 117
N07BC |  Drugs used in opioid dependence | 169
Non-opioids for pain |
A03D |  Antispasmodics in combination with analgesics | 0
A03EA |  Antispasmodics, psycholeptics and analgesics in combination | 0
M01 |  Anti-inflammatory and anti-rheumatic products | 2529
M02A |  Topical products for joint and muscular pain | 295
M03 |  Muscle relaxants | 848
N02B |  Other analgesics and anti-pyretics | 3154
N06A |  Anti-depressants | 2628
Total: | 12004

###  3.2 Data

Our dataset includes prescription drug claims in the Medicaid T-MSIS Analytic
Files (TAF) pharmacy and other services files from non-pregnant adults aged
35-64 years who were non-dual-eligible Medicaid beneficiaries enrolled
2016-2019 from the following 26 states that implemented Medicaid expansion
under the Affordable Care Act in or prior to 2014: ND, VT, NH, CA, OR, MI, IA,
NV, OH, IL, NY, MD, MA, RI, HI, WV, WA, KY, DE, AZ, NJ, MN, NM, CT, CO, AR
[13]. We limited our dataset to valid NDC that were 11-digits long and only
contained values 0-9. We refer the reader to Hoffman et al. [14] for further
background.

##  4 Results

Our goal was to identify opioids and non-opioids typically prescribed for
treating pain. The classification pipeline was run on a 2021 MacBook Pro with
an Apple M1 Pro chip and 32GB of RAM. Using parallel processing, the pipeline
ran in approximately 12.5 minutes. The ATC codes used to identify these
categories and the number of NDC that were found to belong to an ATC class are
shown in Table 1. In total, we identified 12,004 NDC codes that belonged to
one of the opioid and non-opioid ATC categories of interest.

The dataset consisted of 809,484,945 claims and 126,604 unique NDC. Of the
unique NDC, 50,142 (39.6%) had an NDC status of "active"; 23,520 (18.6%) were
"obsolete"; 34,267 were "unknown" (27.1%); 18,675 were "alien" (14.8%). Among
the "active" NDC, 49,861 (99.4%) were successfully mapped to a product-level
or ingredient-level ATC; 20,946 (89.1%) of the "obsolete" NDC were mapped to
an ATC; 4,484 of the "alien" NDC were linked to an ATC. Of all NDC, 75,270
(59.4%) were successfully mapped to an ATC. However, the classified NDCs
accounted for 95.5% of all the non-unique NDCs in the dataset.

Among the NDC that did have a linked RxCUI but did not have an ATC associated
with the RxCUI 13,289 (77.4%) had an RxCUI status of "not current", 1,974
(11.5%) had an RxCUI status of "obsolete", 1,195 (7%) had an RxCUI status of
"remapped", and 75 (0.4%) had an RxCUI status of "quantified". 635 "active"
RxCUI were unsuccessfully mapped to an ATC. Of the "obsolete" RxCUI, 1,078
(54.6%) were matched to an RxCUI with an associated ATC; 947 (79.2%) of the
"remapped" RxCUI and 74 (99%) of the "quantified" RxCUI were matched to a new
RxCUI with a corresponding ATC. As an audit for accuracy, we took a 0.01%
sample of the non-"active" NDC that were linked to an ATC code and did an ad-
hoc search for their appropriate ATC category. The results of this check are
shown in Table 2. There were two NDC incorrectly classified NDC; both NDC were
classified as "R05D" (cough suppressants, excl. combinations with
expectorants) while the correct classification was "R05F" (cough suppressants
and expectorants, combinations).

Table 3 shows the concept names for the five most common "active", "alien", or
"obsolete" NDC and the five most common "unknown" NDC that were not linked to
an ATC code. Of these, six were glucose testing strips, two were condoms, one
was needle and one was a chamber used for an inhaler.

Table 2: Random sample of "alien" or "obsolete" NDC that were linked to an ATC code and checked for ad-hoc accuracy. NDC | NDC Status | Pipeline ATC | Ad-hoc ATC
---|---|---|---
00173024955 | Obsolete | C01AA | C01AA
11917014765 | Alien | A11CC | A11CC
57770005500 | Obsolete | S01XA | S01XA/S01KA
00781286531 | Obsolete | A02BA | A02BA
66870050512 | Alien | A02AA | A02AA
51079012220 | Obsolete | A02BD/P01AB | P01AB
51079078719 | Obsolete | C10AB | C10AB
00904526161 | Obsolete | A02BA | A02BA
11822074050 | Obsolete | N02BE | N02BE
50428322435 | Alien | D06AX | D06AX
40986001765 | Alien | B03BA | B03BA
43063005202 | Obsolete | A04AA | A04AA
43292055802 | Alien | A11GA | A11GA
79854001535 | Alien | A11HA | A11HA
00093213193 | Obsolete | J01XE | J01XE
60432004504 | Obsolete | R05DA | R05FA
52544093628 | Obsolete | G03AA/G03AB | G03AB
00440632530 | Obsolete | C02AC | C02AC
50428034138 | Obsolete | N05CH | N05CH
76439013004 | Obsolete | M05BA | M05BA
60687029825 | Obsolete | N06BA | N06BA
70030014843 | Alien | P03AC | P03AC
50428689594 | Obsolete | A02AF | A02AF
54569317700 | Obsolete | R05DA | R05FA
45802091334 | Obsolete | D10AE | D10AE
Table 3: Examples of NDC and their concept names that were not matched to an ATC code. NDC |  Concept Name
---|---
NDC with a status of "active", "obsolete", or "alien"
99073070822 |  FREESTYLE LITE (GLUCOSE) TEST STRIP
99073070827 |  FREESTYLE LITE (GLUCOSE) TEST STRIP
99073013001 |  LANCET,FREESTYLE
56151146004 |  TRUE METRIX (GLUCOSE) TEST STRIP
53885024450 |  ONE TOUCH ULTRA (GLUCOSE) TEST STRIP
"unknown" NDC
94046000124 |  ADVOCATE REDI-CODE TEST STRIP
26893010002 |  CONDOMS, LATEX, LUBRICATED
08373917700 |  PROCHAMBER HOLDING CHAMBER
26893010102 |  ATLAS CONDOM MIS LUB/COLR
53885085401 |  ONETOUCH KIT VERIO IQ

##  5 Discussion

EHR data are increasingly common and valuable source for health research.
However, drug data in the EHR often require classification before being
usable. Motivated by the need to classify opioid and non-opioid prescription
claims for pain medications in Medicaid, we developed a free-to-use, open-
source, and reliable drug classification pipeline using the NLM RxNorm API and
R.

Applying our proposed pipline to our motivating example of classifying
prescription pain medications in Medicaid claims data, we were able to link
59.4% of all valid unique NDC to an ATC code. Limiting to NDC with an NDC
status of "active", "obsolete", or "alien", we linked 81.5% of NDC to an ATC
code. These results are similar to Homer et al. [3] who found that RxNorm
linked 60% of NDC to an ATC, Peters and Bodenreider [15] who found that 84.2%
of NDC could be linked to a drug class when including historical versions of
RxNorm, and Ostropolets et al. [16] who linked 79.6% of clinical drugs in
RxNorm to ATC. The classified NDC accounted for 95.5% of all the claims in the
cohort which is on par with the 98.2% coverage from Homer et al. [3] when
using a commercial database. We evaluated the accuracy of the pipeline for
classifying "obsolete" and "alien" NDC using a small random sample of the
classified codes and found its performance to be reliable.

A limitation of our pipeline is that we limited drug classification to ATC.
RxNorm can link to other class sources, such as VA class or a classification
for diseases that may be treatable with a drug product from the Medication
Reference Terminology, both produced by the Veterans Health Administration. A
future improvement to the pipeline could be to attempt to link the
unclassified NDC to one of these other classes as a way to increase the claims
coverage. While we used the pipeline to classify Medicaid claims, we expect it
to be easily adaptable to other data as it only depends on NDC. Another
limitation of approach is that it is limited to classifying drugs in the
United States. Recently, however, Reich et al. [17] developed RxNorm Extension
which extends RxNorm to drugs outside of the United States. Future work could
examine the performance of classifying drugs not used in the United States
with ATC using this tool.

##  6 Conclusion

We developed a pipeline to link NDC to ATC in R using the National Library of
Medicine’s RxNorm API. Applying the classification pipeline to large-scale
Medicaid claims dataset, we found it to be a practical and reliable tool for
categorizing NDC.

## References

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