Functional Central Limit Theorem for Subgraph Counting Processes

The objective of this study is to investigate the limiting behavior of a subgraph counting process. The subgraph counting process we consider counts the number of subgraphs having a specific shape that exist outside an expanding ball as the sample size increases. As underlying laws, we consider distributions with either a regularly varying tail or an exponentially decaying tail. In both cases, the nature of the resulting functional central limit theorem differs according to the speed at which the ball expands. More specifically, the normalizations in the central limit theorems and the properties of the limiting Gaussian processes are all determined by whether or not an expanding ball covers a region - called a weak core - in which the random points are highly densely scattered and form a giant geometric graph.

FCLT for subgraph counting processes a collection of "rings" around the origin with each ring containing extreme random points which exhibit different geometric and topological behavior. The objective of the current study is to develop a fuller description of this ring-like structure, at least in a geometric graph model, by establishing a variety of FCLTs which describe geometric graph formation between the rings.
By construction, the subgraph counts (1.2) can be viewed as generating a stochastic process in the parameter t ≥ 0, while a process-level extension in (1.1) is much less obvious. Then, while (1.2) captures the dynamic evolution of geometric graphs as t varies, (1.1) only describes the static geometry. Thus, the limits in the FCLT for (1.2) are intrinsically Gaussian processes, rather than one-dimensional Gaussian distributions.
One of the main results of this paper is that the limiting Gaussian processes can be classified into three distinct categories, according to how rapidly R n grows. The most important condition for this classification is whether or not a ball centered at the origin with radius R n , denoted by B(0, R n ), asymptotically covers a weak core. Weak cores are balls, centered at the origin with growing radii as n increases, in which the random points are densely scattered and form a highly connected geometric graph. This notion, along with the related notion of a core, play a crucial role for the classification of the limiting Gaussian processes. Indeed, if B(0, R n ) grows so that it asymptotically covers a weak core, then the geometric graph outside B(0, R n ) is "sparse" with many small disconnected components. In this case, the limit is denoted as the difference between two time-changed Brownian motions. In contrast, if B(0, R n ) is asymptotically covered by a weak core, the geometric graph in the area between the outside of B(0, R n ) and inside of a weak core becomes "dense", and, accordingly, the limit becomes a degenerate Gaussian process with deterministic sample paths. Finally if B(0, R n ) coincides with a weak core, then the limiting Gaussian process possesses more complicated structure and are even non-self-similar. The probabilistic properties of the limiting Gaussian processes are summarized in Tables 1 and 2 at the end of Section 3.
We want to emphasize that the nature of the FCLT depends not only on the growth rate of R n but also the tail property of f . This is in complete contrast to (1.1), because, as seen in Chapter 3 of [23], the proper normalization, limiting Gaussian distribution, etc. of the CLT are all robust to whether f has a heavy or a light tail. In this paper, we particularly deal with the distributions of regularly varying tails and (sub)exponential tails. However, we are not basically concerned with any distribution with a superexponential tail, e.g., a multivariate normal distribution. The details of the FCLT in that case remain for a future study.
The remainder of the paper is organized as follows. First, in Section 2 we provide a formal definition of the subgraph counting process. Section 3 gives an overview of what was shown in the previous work [22] and what will be shown in this paper. Subsequently, in Section 4 we focus on the case in which the underlying density has a regularly varying tail, including power-law tails, and prove the required FCLT. We also investigate the properties of the limiting Gaussian processes, in particular, in terms of self-similarity and sample path continuity. In Section 5, we do the same when the underlying density has an exponentially decaying tail. To distinguish densities via their tail properties, we need basic tools in EVT. In essence, the properties of the limiting Gaussian processes are determined by how rapidly R n grows to infinity, as well as how rapidly the tail of f decays. Finally, Section 6 carefully examines both cores and weak cores for a large class of densities.
Before commencing the main body of the paper, we remark that all the random points in this paper are assumed to be generated by an inhomogeneous Poisson point process on R d with intensity nf . In our opinion, the FCLT in the main theorem can be carried over to a usual i.i.d. random sample setup by a standard "de-Poissonization" argument; see Section 2.5 in [23]. This is, however, a little more technical and challenging, and therefore, we decided to concentrate on the simpler setup of an inhomogeneous Poisson point process. Furthermore we consider only spherically symmetric distributions. Although the spherical symmetry assumption is far from being crucial, we adopt it to avoid unnecessary technicalities.

Subgraph counting process
Let (X i , i ≥ 1) be i.i.d. R d -valued random variables with spherically symmetric probability density f . Given a Poisson random variable N n with mean n, independent of (X i , i ≥ 1), denote by P n = {X 1 , X 2 , . . . , X Nn } a Poisson point process with |P n | := N n . We choose a positive integer k, which remains fixed hereafter. We take k ≥ 2, unless otherwise stated, because many of the functions and objects to follow are degenerate in the case of k = 1.
Moreover h t can be decomposed as follows. Suppose that Γ has k vertices and j edges for some j ∈ k − 1, . . . , k(k − 1)/2 . Letting A be the set of all connected graphs on k vertices and edges (up to graph isomorphism), define for x 1 , . . . , x k ∈ R d , Note that h + t (x 1 , . . . , x k ) = 1 if and only if a geometric graph G {x 1 , . . . , x k }, t either coincides with Γ (up to graph isomorphism) or has more than j edges, while h − t (x 1 , . . . , x k ) = 1 only when G {x 1 , . . . , x k }, t has more than j edges. It is then elementary to check that h ± t are both indicators, taking values 0 or 1, and satisfying, for all x 1 , . . . , x k ∈ R d and 0 ≤ s ≤ t, In addition, since h t is an indicator, it is always the case that The objective of this study is to establish a functional central limit theorem (FCLT) of the subgraph counting process defined by where h t is given in (2.1), m(x 1 , . . . , x k ) = min 1≤i≤k ||x i ||, x i ∈ R d , and (R n , n ≥ 1) is a properly chosen normalizing sequence tending to infinity. Note that (2.7) counts the number of subgraphs in G(P n , t) isomorphic to Γ that lie completely outside of B(0, R n ). More concrete definitions of (R n ) are given in the subsequent sections, where the sequence is shown to be dependent on the tail decay rate of f .
The current work is motivated by extreme value theory (EVT). EVT studies, as its name suggests, the extremal behavior of stochastic processes; in our context, we are interested in the spatial distribution of subgraphs lying far away from the origin. For this reason, we do not treat the case in which the density f has a bounded support with R n tending to a positive and finite constant.
Another important concept related to (2.7) is the number of "components" of G(P n , t) isomorphic to Γ. In this case, the resulting FCLT may partially share the same normalizing constants and limiting Gaussian processes as the FCLT for (2.7), at least when R n grows sufficiently fast. The present paper, however, does not treat this quantity, because the proof of the FCLT, especially that of tightness, involves much more technicalities due to an extra condition that all Γ-subgraphs must be disconnected from the rest of P n .
Subgraph counts are one of the most basic quantities in random geometric graph theory. Nevertheless, the study on subgraph counts is a good starting point for more advanced and general objects. One of the most natural and interesting objects is a randomČech complex, which can be regarded as a higher-dimensional version of a random geometric graph. Then, some variants of (2.7), which counts the number of homological holes, are known as a Betti number in algebraic topology. The relevant article in this direction is [21] which discusses the asymptotic behavior of Betti numbers relating to the tail of probability distributions. In addition, (2.7) can also be seen as a special case of "local" U -statistics, and therefore, for a future work, it would be interesting to construct FCLTs for a general class of U -statistics. The related publications include, e.g., [24], [25], and [9], though these articles do not necessarily focus on the tail of probability distributions.

Annuli structure
The objective of this short section is to clarify what is already known and what is new in this paper. Without any real loss of generality, we will do this via two simple examples, one of which treats a power-law density and the other a density with a (sub)exponential tail. Before this, however, we introduce two important notions.  In other words, a core is a centered ball in which random points are densely scattered, so that placing unit balls around them covers the ball itself. We usually wish to seek the largest possible value of R n such that (3.1) occurs asymptotically with probability 1.
A related notion, the weak core, plays a more decisive role in characterizing the FCLT proven in this paper. It is shown later that a weak core is generally larger but close in size to a core of maximum size. Definition 3.2. Let f be a spherically symmetric density on R d and e 1 = (1, 0, . . . , 0) ∈ R d . A weak core is a centered ball B(0, R for some α > d and normalizing constant C. Using this density, we see how random geometric graphs are formed in all of R d . First, according to [1], there exists a sequence R (c) n , (3.1) occurs asymptotically with probability 1. In addition, as for the radius of a weak core, it suffices to take R (w) n = (Cn) 1/α . Although R (w) n grows faster than R (c) n , they are seen to be "close" to each other in the sense that they have the same regular variation exponent, 1/α.
Beyond a weak core, however, the formation of random geometric graphs drastically varies. In fact, the exterior of a weak core can be divided into annuli of different radii, at which many isolated subgraphs of finite vertices are asymptotically placed in a specific fashion. To be more precise, let us fix connected graphs Γ k with k vertices for k = 2, 3, . . . and let 2,n , and Under this circumstance, [22] considered the subgraph counts given by and showed that (3.3) weakly converges to a Poisson distribution for each fixed t. To be more specific on the geometric side, let Ann(K, L) be an annulus with inner radius K and outer radius L. Then, we have, in an asymptotic sense, 2,n , there are finitely many graphs isomorphic to Γ 2 , but none isomorphic to Γ 3 , Γ 4 , . . . . • Outside B 0, R (p) 3,n , equivalently inside Ann R (p) 3,n , R (p) 2,n , there are infinitely many graphs isomorphic to Γ 2 and finitely many graphs isomorphic to Γ 3 , but none isomorphic to Γ 4 , Γ 5 , . . . . In general, k−1,n , there are infinitely many graphs isomorphic to Γ 2 , . . . , Γ k−1 and finitely many graphs isomorphic to Γ k , but none isomorphic to Γ k+1 , Γ k+2 , . . . etc.
In relation to other radii, they are respectively equivalent to (i) R k,n as n → ∞. We may, therefore, expect that infinitely many subgraphs isomorphic to Γ k appear asymptotically outside B(0, R n ). This in turn implies that, instead of a Poisson limit theorem, the FCLT governs the limiting behavior of the subgraph counting process. More directly, slightly modifying the argument in Theorem 2 of [1], we see that the mean of (3.4) asymptotically behaves as a constant multiple of n k R d n f (R n e 1 ) k , which itself diverges as n → ∞. This also indicates that the subgraph counting process (3.4) obeys a FCLT.
As the analog of the setup for (1.1), when deriving an FCLT, the behavior of (3.4) splits into three different regimes: Specifically, if nf (R n e 1 ) → 0 (i.e., B(0, R n ) contains a weak core), many isolated components of subgraphs isomorphic to Γ k are distributed outside B(0, R n ). If nf (R n e 1 ) → ξ ∈ (0, ∞) (i.e., B(0, R n ) agrees with a weak core), the subgraphs isomorphic to Γ k outside B(0, R n ) begin to be connected to one another. In particular, observing that lim k→∞ R (p) k,n = R (w) n for all n, we see that • Outside of B(0, R (w) n ), there are infinitely many graphs isomorphic to Γ j for every j = 2, 3, . . . . If nf (R n e 1 ) → ∞ (i.e., B(0, R n ) is contained in a weak core), the subgraphs isomorphic to Γ k outside B(0, R n ) are further increasingly connected and form a large component.
In Section 4, we will see that the nature of the FCLT, including the normalizing constants and the properties of the limiting Gaussian processes, differs according to which regime one considers. Combing the results on the FCLT and the Poissonian results in [22] produces a complete picture of the annuli structure formed by heavy tailed random variables. Example 3.4. Next, we turn to a density with a (sub)exponential tail for which the radius of a maximum core is given by Exponential tail Non-SS Non-SS Non-SS Table 1: Self-similarity exponents of the limiting Gaussian processes. A regularly varying tail is a notion generalizing a power-law tail. Non-SS means that the process is non-selfsimilar. A zero limit of nf (R n e 1 ) is equivalent to the case in which a ball B(0, R n ) contains a weak core, and nf (R n e 1 ) → ∞ if and only if B(0, R n ) is contained in a weak core. If nf (R n e 1 ) → ξ ∈ (0, ∞), then B(0, R n ) agrees with a weak core (up to multiplicative constants).
see [1] and [22]. In addition, one can take R for sufficiently large n, but these radii are asymptotically equal. As in the previous example, the exterior of a weak core is characterized by the same kind of layer structure, for which the description in Figure 1 applies, except for the change in the values of R 2,n , and k,n e 1 k → 1 , n → ∞ .
Then, it was shown in [22] that (3.3) converges weakly to a Poisson distribution for each fixed t. In Section 5 of this paper, taking (R n ) such that n k R d−τ n f (R n e 1 ) k → ∞, we establish a FCLT for the subgraph counting process (2.7). To this end, our argument has to be split, once again, into the three different regimes: As in the last example, three different Gaussian limits may appear depending on the regime. This completes the full description of the annuli structure formed by random variables with an exponentially decaying tail, when combined with the Poisson limit theorems in [22].
Before proceeding to the next section, we would like to quickly overview the properties of limiting Gaussian processes in the FCLT in terms of self-similarity and some representation results. The probabilistic features of the limiting Gaussian processes crucially differ whether B(0, R n ) contains a weak core or not. In addition, the tail decay rate of an underlying density plays a decisive role as well. More detailed arguments are presented in the subsequent sections.

Heavy tail case 4.1 The setup
In this section, we explore the case in which the underlying density f on R d has a heavy tail under a more general setup than that in Example 3.3. Let S d−1 be a (d − 1)dimensional unit sphere in R d . We assume that the density has a regularly varying tail nf (R n e 1 ) → 0 nf (R n e 1 ) → ξ nf (R n e 1 ) → ∞  Table 2: Representation results on the limiting Gaussian processes. "New" implies that the limit constitutes a new class of Gaussian processes.
(at infinity) in the sense that for any θ ∈ S d−1 (equivalently, for some θ ∈ S d−1 because of the spherical symmetry of f ), and for some α > d, Denoting by RV −α a collection of regularly varying functions (at infinity) of exponent −α, the above is written as Clearly, a power-law density in Example 3.3 satisfies (4.1). Let k ≥ 2 be an integer that remains fixed throughout this section. We remark that many of the functions and objects are dependent on k, but the dependence may not be stipulated by subscripts (or superscripts). Choosing the sequence R n → ∞ so that we consider the subgraph counting process given in (2.7), whose behavior is, as argued in Example 3.3, expected to be governed by a FCLT.
The scaling constants for the FCLT, denoted by τ n , are shown to depend on the limit value of nf (R n e 1 ) as n → ∞. More precisely, we take The reason for which we need three different normalizations is deeply related to the connectivity of a random geometric graph. To explain this, we need the notion of a weak core; see Definition 3.2 for the formal definition. The main point is that the density of random points between the outside and inside of a weak core is completely different. In essence, random points inside a weak core are highly densely scattered, and the corresponding random geometric graph forms a single giant component. Beyond a weak core, however, random points are distributed sparsely, and as a result, we observe many isolated geometric graphs of smaller size. This disparity between the outside and inside of a weak core requires different normalizations in (τ n ). In Section 6, a more detailed study in this direction is presented.

Limiting Gaussian processes and the FCLT
We introduce a family of Gaussian processes which function as the building blocks for the limiting Gaussian processes in the FCLT. For = 1, . . . , k, let are independent. For = 1, we define G 1 as a Gaussian random variable with zero mean and variance B 1 . We assume that G 1 , . . . , G k are independent.
Note that V 1 is a degenerate Gaussian process with deterministic sample paths. These processes later turn out to be the building blocks of the weak limits in the main theorem.
The covariance function of the process V is given by Using the decomposition (2.4), we can express V as the difference between two Gaussian processes; that is, for = 2, . . . , k − 1, The same decomposition is feasible in an analogous manner for V 1 and V k .
The following proposition shows that the processes V + k and V − k can be represented as a time-changed Brownian motion.
where B is the standard Brownian motion, and K + k : Proof. It is enough to verify that the covariance functions on both sides coincide. It follows from (2.5) that for 0 ≤ s ≤ t, We also claim that the process V is self-similar and has a.s. Hölder continuous sample paths. Recall that a stochastic process for any c > 0, t 1 , . . . , t k ≥ 0, and k ≥ 1.
. . , k and every T > 0, V (t), 0 ≤ t ≤ T has a modification, the sample paths of which are Hölder continuous of any order in [0, 1/2).
Proof. We can immediately prove (i) by the scaling property As for (ii), the statement is obvious for = 1 or = k; therefore, we take ∈ {2, . . . , k − 1}. By Gaussianity, , m = 1, 2, . . . We now show that there exists a constant C > 0, which depends on T , such that By virtue of the decomposition V = V + − V − , showing (4.6) for each of V + and V − suffices. We handle V + only, since V − can be treated in the same manner. We have Because of (2.6), the above integral is not altered if the integral domain is restricted to which verifies (4.6).
FCLT for subgraph counting processes Combining (4.5) and (4.6), we have that for some C 3 > 0, It now follows from the Kolmogorov continuity theorem that there exists a modification of V (t), 0 ≤ t ≤ T , the sample paths of which are Hölder continuous of any order in 0, (m − 1)/(2m) . Since m is arbitrary, we are done by letting m → ∞.
We are now ready to state the FCLT for the subgraph counting process, suitably scaled and centered in such a way that In the following, ⇒ denotes weak convergence. All weak convergence hereafter are in the space D[0, ∞) of right-continuous functions with left limits. The proof of the theorem is deferred to Section 7.1. Since Γ is assumed to be a feasible subgraph, we have that G n (t) > 0 with positive probability for any n ≥ 1 and t > 0, and thus, the resulting FCLTs are always non-trivial.
The processes V 1 , . . . , V k can be viewed as the building blocks of the limiting Gaussian processes; however, how many and which ones contribute to the limit depends on whether the ball B(0, R n ) covers a weak core or not. If B(0, R n ) covers a weak core, equivalently, nf (R n e 1 ) → 0, then V k is the only process remaining in the limit. Although, as seen in Proposition 4.1, V k is generally represented as the difference in two time-changed Brownian motions, it can be denoted as a single time-changed Brownian This is the case when Γ is a complete graph, in which case the negative part h − t is identically zero. In contrast, the process V 1 , a degenerate Gaussian process with deterministic sample paths, only appears in the limit when B(0, R n ) is contained in a weak core, i.e., nf (R n e 1 ) → ∞. Finally, if B(0, R n ) agrees with a weak core (up to multiplicative constants), all of the processes V 1 , . . . , V k contribute to the limit. Interestingly, only in this case, do the weak limits become non-self-similar.

The setup
This section develops the FCLT of the subgraph counting process suitably scaled and centered, when the underlying density on R d possesses an exponentially decaying tail. Typically, in the spirit of extreme value theory, a class of multivariate densities with exponentially decaying tails can be formulated by the so-called von Mises functions. See for example, [3] and [4]. In particular, in the one-dimensional case (d = 1), the von Mises function plays a decisive role in the characterization of the max-domain of attraction of the Gumbel law. See Proposition 1.4 in [26]. We assume that the density f on R d is given by Here, ψ : R + → R is a function of C 2 -class and is referred to as a von Mises function, so In this paper, we restrict ourselves to an unbounded support of the density, i.e., z ∞ ≡ ∞. For notational ease, we introduce the function a(z) = 1/ψ (z), z > 0. Since a (z) → 0 as z → ∞, the Cesàro mean of a converges as well: Suppose that a measurable function L : This condition implies that L behaves as a constant locally in the tail of f , and thus, only ψ plays a dominant role in the characterization of the tail of f . Here, we need to put an extra technical condition on L. Namely, there exist γ ≥ 0, z 0 > 0, and C ≥ 1 such that Since L is negligible in the tail of f , it seems reasonable to classify the density (5.1) in terms of the limit of a. If a(z) → ∞ as z → ∞, we say that f belongs to a class of densities with subexponential tail, because the tail of f decays more slowly than that of an exponential distribution. Conversely, if a(z) → 0 as z → ∞, f is said to have a superexponential tail, and if a(z) → c ∈ (0, ∞), we say that f has an exponential tail.
To be more specific about the difference in tail behaviors, let us consider a slightly more general example than that in Example 3.4, for which f (x) = L ||x|| exp −||x|| τ /τ , τ > 0, x ∈ R d . Clearly, the parameter τ is associated with the speed at which f vanishes in the tail. Observe that a(z) = z 1−τ → ∞ as z → ∞ if 0 < τ < 1, and therefore in this case, f has a subexponential tail. If τ > 1, a(z) decreases to 0, in which case f has a superexponential tail.
An important assumption throughout most of this study is that there exists c ∈ (0, ∞] such that a(z) → c as z → ∞ . In view of the classification described above, (5.6) eliminates the possibility of densities with superexponential tail. As discovered in [22] and [1], random points drawn from a superexponential law hardly form isolated geometric graphs outside a core, whereas random points coming from a subexponential law do constitute a layer of isolated geometric graphs outside a core. Accordingly, it is highly likely that the nature of the FCLT differs according to whether the underlying density has a superexponential or a subexponential tail. The present work focuses on the (sub)exponential tail case, and more detailed studies on a superexponential tail case remain for future work.
To realize a more formal set up, let k ≥ 2 be an integer, which remains fixed for the remainder of this section; however, once again, note that many of the functions and objects are implicitly dependent on k. Define the sequence R n → ∞, so that Defining an alternative sequence R the subgraph counting process using R (p) k,n is known to weakly converge to a Poisson distribution; see [22]. Since R n in (5.7) grows more slowly than R (p) k,n , i.e., R n /R we may expect that an FCLT plays a decisive role in the asymptotic behavior of a subgraph counting process.
As in the last section, we now want to recall the notion of a weak core. Let R (w) n → ∞ be a sequence such that nf (R (w) n e 1 ) → 1 as n → ∞. Then, we say that a ball B 0, R (w) n is a weak core. We have to change, once again, the scaling constants τ n of the FCLT, depending on whether B(0, R n ) covers a weak core or not. More specifically, we define (5.8)

Limiting Gaussian processes and the FCLT
The objective of this subsection is to formulate the limiting Gaussian processes and the FCLT. Let and, further, As we did in Section 4.2, by the decomposition h t = h + t − h − t , one can write the process W as the corresponding difference W = W + − W − for = 1, . . . , k.
It is easy to compute the covariance function of W . We have, for = 1, . . . , k and t, s ≥ 0, This in turn implies that in which case, there is nothing to explore here, because the properties of V have already been studied in Section 4.2.
In contrast, if a(z) → c ∈ (0, ∞) as z → ∞, then M does not directly relate to L as above, and, consequently, the process W exhibits properties different to those of V . For example, although one may anticipate, as the analog of the process V 1 , that W 1 is a degenerate Gaussian process, this is no longer the case. (i) W 1 is a non-degenerate Gaussian process.
Proof. If a(z) → c ∈ (0, ∞) as z → ∞, then M 1 (t, s) cannot be decomposed into a function of t and a function of s, and therefore, W 1 is non-degenerate. As for (ii), M does not match L at all and it loses the scale invariance, meaning that W is non-self-similar.
Similarly to Proposition 4.1, however, the process W k (= W + k − W − k ) can be denoted in law as the difference between two time-changed Brownian motions, regardless of  (ii) For = 1, . . . , k, and every T > 0, W (t), 0 ≤ t ≤ T has a modification, the sample paths of which are Hölder continuous of any order in [0, 1/2).
Proof. The proof of (i) is very similar to that in Proposition 4.1, so we omit it. The proof of (ii) is analogous to that in Proposition 4.2 (ii); we have only to show that for some Because of the decomposition W = W + − W − , it suffices to prove the above for each W + and W − . We check only the case of W + . We see that The rest of the argument is completely the same as Proposition 4.2 (ii). Now, we can state the FCLT of the centered and scaled subgraph counting process where the normalizing sequence (R n ) satisfies (5.7) and (τ n ) is defined in (5.8). The proof of the theorem is presented in Section 7.2.
We have now officially presented all the limiting Gaussian processes of this paper, so it would be beneficial for the readers to return to Tables 1 and 2 in Section 3 to overview their properties once again. These tables indicate that the limiting Gaussian processes are somewhat special when f has an exponential tail. For example, in this case, the limits always lose self-similarity, regardless of the asymptotics of nf (R n e 1 ), whereas, in the regularly varying or the subexponential tail case, the self-similarity is lost only when nf (R n e 1 ) converges to a positive and finite constant. Furthermore, when nf (R n e 1 ) → ∞, a non-degenerate limit appears only in the exponential tail case. Finally, we remark that as in the last section, the resulting FCLTs are necessarily non-trivial, because Γ is a feasible subgraph. (i) If nf (R n e 1 ) → 0 as n → ∞, then

Graph connectivity in weak core
We start this section by recalling the weak core, which was defined as a centered ball B 0, R (w) n such that nf R (w) n e 1 → 1 as n → ∞. In addition, we need the relevant notion, the core, which was defined in Definition 3.1. Recall that, given a Poisson point process P n on R d , a core is a centered ball B(0, R n ) such that B(X, 1) .
In the following, we seek the largest possible sequence R n → ∞ such that the event (6.1) occurs asymptotically with probability 1, and subsequently, it is shown that the largest possible core and a weak core are "close" in size. However, the degree of this closeness depends on the tail of an underlying density f , and therefore, we divide the argument into two cases.
We first assume that the density f on R d is spherically symmetric and has a regularly varying tail, as in (4.1). For increased clarity, we place an extra condition that p(r) := f (re 1 ) is eventually non-increasing in r, that is, p is non-increasing on (r 0 , ∞) for some large r 0 > 0. In this case, the radius of a weak core is, clearly, given by   One can obtain a parallel result when the underlying density has an exponentially decaying tail, as in (5.1). We simplify the situation a bit by assuming f (x) = C exp −ψ ||x|| , x ∈ R d , (6.6) where C is a normalizing constant and ψ : R + → R is of C 2 -class and satisfies ψ ∈ RV v (at infinity) for some v > 0 and ψ > 0. It should be noted that we are permitting the case v > 1, implying that, unlike in the previous section, we do not rule out densities with superexponential tail. Evidently, the radius of a weak core is given by R (w) n = ψ ← (log n + log C) .
The following result is needed as preparation for the proof of these propositions. The proof may be obtained by slightly modifying the proof of Theorem 2.1 in [1], but we repeat the argument in order for this paper to be self-contained. Lemma 6.3. Given a spherically symmetric density f on R d , suppose that p(r) = f (re 1 ) is eventually non-increasing. Let g = 1/(2d 1/2 ). Suppose, in addition, that there exists a sequence R n ∞ such that d log R n − g d nf (R n e 1 ) → −∞ as n → ∞. Then, Proof. For ρ > 0, let Q(ρ) be a collection of cubes with grid g that are contained in B(0, ρ). Then, for all ρ > 0 and n ≥ 1. It now suffices to show that This probability is estimated from above by At the first inequality, we used the fact that p is eventually non-increasing. Clearly, the rightmost term vanishes as n → ∞.
Proof. (proof of Proposition 6.1) Observe that the assumption p ∈ RV −α implies (1/p) ← ∈ RV 1/α , e.g., Proposition 2.6 (v) in [27]. Thus, (6.5) readily follows from the uniform convergence of regularly varying functions; see Proposition 2.4 in [27]. By Lemma 6.3, it suffices to verify that d log R (c) 1 (log n − δ 2 log log n). Using Proposition 2.6 (i) in [27], At the last inequality, we applied the constraint in δ 1 . Therefore, we have d log R Proof. (proof of Proposition 6.2) Since ψ ← ∈ RV 1/v , it is easy to show (6.10), and therefore, we prove only that d log R (c) and that g d nf R (c) n e 1 = g d Ce δ1+δ2 log log n. By virtue of the constraints in δ 1 and δ 2 , we have dv −1 − g d Ce δ1+δ2 < 0; thus, the claim is proved.
Remark 6.4. The proof of Lemma 6.3 merely estimated the probability in (6.11) from below. Therefore, it seems to be possible that in the propositions above, (6.4) and (6.9) may hold for the sequence R n ∞ growing more quickly than R

Proof of main results
This section presents the proof of the main results of this paper. The proof is, however, rather long, and therefore, it is divided into several parts. All the supplemental ingredients necessary are collected in the Appendix, most of which are cited from [23].
Let Ann(K, L) be an annulus of inner radius K and outer radius L. For x 1 , . . . , x k ∈ R d , define Max(x 1 , . . . , x k ) as the function selecting an element with the largest distance from the origin. That is, Max(x 1 , . . . , x k ) = x i if ||x i || = max 1≤j≤k ||x j ||. If multiple x j 's achieve the maximum, we choose an element with the smallest subscript.
In the following, Y, Y , Y i , etc. always represent a finite collection of d-dimensional real vectors. We use the following shorthand notations. That is, for  (7.2) and for ∈ {0, . . . , k}, In particular, we set In Section 7.1, we use, for 1 ≤ K < L ≤ ∞, n ∈ N + and t ≥ 0, The same notations are retained for Section 7.2 to represent, for 0 ≤ K < L ≤ ∞, n ∈ N + and t ≥ 0, Finally, C * denotes a generic positive constant, which may change between lines and does not depend on n.
In the following, we divide the argument into two subsections. Section 7.1 treats the case in which the underlying density has a regularly varying tail; our goal is to prove Theorem 4.3. Subsequently Section 7.2 provides the proof of Theorem 5.3, where the density is assumed to have an exponentially decaying tail. Before the specific subsections, however, we show some preliminary results, which are commonly used for the tightness proof in both subsections.
Proof. We only prove the first inequality. If = 1 or = k, the claim is trivial, and therefore, we can take 2 ≤ ≤ k − 1. It follows from (2.6) that the integral in ( Observe that for i = j, where ω d is the volume of the d-dimensional unit ball. Since the second and the third terms on the rightmost term in (7.5) have the same upper bound, we ultimately obtain FCLT for subgraph counting processes Therefore, the integral in (7.4) is bounded above by An elementary calculation shows that for all i, j, i , j ∈ {1, . . . , − 1} with i > j and i > j , In particular, if i = i and j = j , the integral is identically zero. Applying the same manipulation to the integral of other cross-terms, we can conclude the claim of the lemma.

Regularly varying tail case
Under the setup of Theorem 4.3, we first define the subgraph counting process with restricted domain. For 1 ≤ K < L ≤ ∞, n ∈ N + , and t ≥ 0, let where (R n ) satisfies (4.2). For the special case K = 1 and L = ∞, we simply denote G n (t) = G n,1,∞ (t) and G ± n (t) = G ± n,1,∞ (t). The subgraph counting processes, centered and scaled, for which we prove the FCLT, are given by where (τ n ) is determined by (4.3) according to which regime is considered. The first proposition below computes the covariances of G n,K,L (t) .
(7.9) Substituting (7.8) and (7.9) back into (7.7), while supposing temporarily that the dominated convergence theorem is applicable, we may conclude that Cov G n,K,L (t) , G n,K,L (s) Observe that the limit value of nf (R n e 1 ) completely determines which term on the right hand side of (7.10) is dominant. If nf (R n e 1 ) → 0, then the kth term,i.e., = k, in the sum grows fastest, while the first term, i.e., = 1, grows fastest when nf (R n e 1 ) → ∞. Moreover, if nf (R n e 1 ) → ξ ∈ (0, ∞), then all the terms in the sum grow at the same rate.
This concludes the claim of the proposition. It now remains to establish an integrable upper bound for the application of the dominated convergence theorem. First, condition Next, appealing to Potter's bound ,e.g., Proposition 2.6 (ii) in [27], for every ξ ∈ (0, α − d) and sufficiently large n, we are allowed to apply the dominated convergence theorem.
The next proposition proves the weak convergence of Theorem 4.3 in a finitedimensional sense.  i) − (iii) in the theorem hold in a finite-dimensional sense. Furthermore, let X ± n be the processes defined in (7.6). Then, the following results also hold in a finite-dimensional sense.
Proof. The proofs of (7.11), (7.12), and (7.13) are a bit more technical, but are very similar to the corresponding results in Theorem 4.3; therefore, we check only finitedimensional weak convergences in Theorem 4.3. The argument here is closely related to that in Theorem 3.9 of [23], for which we rely on the so-called Cramér-Wold device.
For 0 ≤ t 1 < · · · < t m < ∞, a 1 , . . . , a m ∈ R and m ≥ 1, define S n := m j=1 a j G n (t j ). For K > 1, S n can be further decomposed into two parts: We define a constant γ K as follows in accordance with the limit of nf (R n e 1 ).
Moreover, γ := lim K→∞ γ K . It directly follows from Proposition 7.2 that, regardless of the regime we consider, For the completion of the proof, we ultimately need to show that τ −1/2 n S n − E{S n } ⇒ N (0, γ) .
Then, T (K) n can be partitioned as follows.
For i, j ∈ V n , we put an edge between i and j (write i ∼ j) if i = j and the distance between Q i and Q j are less than 2kt m . Then, (V n , ∼) gives a dependency graph with FCLT for subgraph counting processes respect to (η ,n , ∈ V n ); that is, for any two disjoint subsets I 1 , I 2 of V n with no edges connecting I 1 and I 2 , (η ,n , ∈ I 1 ) is independent of (η ,n , ∈ I 2 ). Notice that the maximum degree of (V n , ∼) is at most finite.
Writing Φ for a distribution function of the standard normal distribution, it follows from Stein's method for normal approximation (see Theorem 2.4 in [23]) that for all Thus, (7.15) immediately follows if we can show that for p = 3, 4, Since the proof for showing this varies depending on the limit of nf (R n e 1 ), we divide the argument into three different cases. Suppose first that nf (R n e 1 ) → 0 as n → ∞. Let Z ,n denote the number of points in P n lying in Then, Z ,n has a Poisson distribution with mean n Tube(Q ;ktm) f (z)dz. Using Potter's bound, we see that Z ,n is stochastically dominated by another Poisson random variable Z n with mean C * nf (R n e 1 ). Observing that |η ,n | ≤ C * Z ,n k , we have, for q = 1, 2, 3, 4, where in the last step we used the assumption nf (R n e 1 ) → 0. It now follows that for p = 3, 4, where the last convergence follows from (4.2).
In the case of nf (R n e 1 ) → ξ ∈ (0, ∞), the argument for proving (7.16) is very similar to, or even easier than, the previous case, so we omit it.
Finally, suppose that nf (R n e 1 ) → ∞ as n → ∞. We begin by establishing an appropriate upper bound for the fourth moment expectation we see that for every j ∈ {0, . . . , 4}, can be denoted as the expectation of a quadruple sum (7.18) where, for every i = j, either P (i) n is an independent copy of P (j) , there is a pair (Y i , Y j ), i = j having exactly one element in common and no other common elements between Y i 's are present. In this case, (7.18) can be written as In particular, (7.19) appears once in F n (2), 3 2 times in F n (3), and 4 2 times in F n (4).
Thus, the total contribution to (7.17) sums up to We may assume, therefore, that n , |Y 1 ∩ Y 2 | = 2 and no other common elements between Y i 's exist. The argument for the other cases will be omitted because they can be handled in the same manner. Then, by Palm theory, (7.18) is equal to Because of Potter's bound, together with the fact that Q intersects with Ann(R n , KR n ), Similarly, we can obtain E g ,n (Y) ≤ C * f (R n e 1 ) k , and therefore, the absolute value of (7.20), equivalently that of (7.18), is bounded above A similar argument proves that if |Y 1 ∪ Y 2 ∪ Y 3 ∪ Y 4 | = 4k − q for some q ≥ 3, the absolute value of (7.18) is bounded above by C * nf (R n e 1 ) 4k−q . Putting these facts altogether, while recalling nf (R n e 1 ) → ∞ as n → ∞, we may conclude that Now, it is easy to check (7.16).
In terms of the third moment expectation E η ,n − E{η ,n } 3 , we apply Hölder's inequality to obtain Again, it is easy to prove (7.16). Now, we have obtained a CLT in (7.14) as required, regardless of the limit of nf (R n e 1 ).
Proof of Theorem 4.3. The last proposition has justified finite-dimensional weak convergence of the processes (X n ) and (X + n , X − n ). The proof of Theorem 4.3 will be complete, provided that the tightness of (X n ) is verified in the space D[0, ∞) equipped with the Skorohod J 1 -topology; see [8]. To this aim, it suffices to show that (X + n ) and (X − n ) are both tight in D[0, ∞). To see this, suppose that X + n and X − n were tight in D[0, ∞). Then, a joint process (X + n , X − n ) is tight as well in D[0, ∞) × D[0, ∞), which is endowed with the product topology. Because of the already established finite-dimensional weak convergence of (X + n , X − n ), every subsequential limit of (X + n , X − n ) coincides with the limiting process in Proposition 7.3. This in turn implies the weak convergence of (X + n , X − n ) in D[0, ∞) × D[0, ∞). Using the basic fact that the map (x, ∞), while recalling that the limits in Proposition 7.3 all have continuous sample paths, the continuous mapping theorem gives weak convergence of X n = X + n − X − n in D[0, ∞). In the following, we prove the tightness of (X + n ) only, because the argument for (X − n ) is the same as that for (X + n ). By a standard argument for the D-space (see, e.g., Chapter 16 in [8]), it is enough to show the tightness in the space D[0, L] for every L > 0. For notational ease, we omit the superscript "+" from all the functions and objects during the proof. By Theorem 13.5 of [8], it is sufficient to show that there exists B > 0 such for all 0 ≤ r ≤ s ≤ t ≤ L and n ≥ 1.
Note that for every p, q ∈ {0, 1, 2}, F n (p, q) can be represented by where, for every i = j, either P (i) n is an independent copy of P (j) n .
According to the Palm theory given in the Appendix, if |Y 1 ∪ Y 2 ∪ Y 3 ∪ Y 4 | = 4k, i.e., any two of Y i have no common elements, then (7.21) reduces to E{ξ n,t,s } 2 E{ξ n,s,r } 2 .
In the following, we examine the case in which at least one common element exists between Y i 's. First, for = 1, . . . , k, we count the number of times E Y1⊂Pn Y2⊂Pn h n,t,s (Y 1 ) h n,t,s (Y 2 ) 1 |Y 1 ∩ Y 2 | = E{ξ n,s,r } 2 (7.22) appears in each F n (p, q). Indeed, (7.22) appears only once in F n (2, 0), F n (2, 1), and F n (2, 2). Therefore, the total contribution amounts to Similarly, for every = 1, . . . , k, no contribution is made by Subsequently, for = 1, . . . , k, we explore the presence of E{ξ n,s,r } . (7.24) This actually appears once in F n (2, 1) and twice in F n (2, 2); therefore, their overall contribution is 2 2 For the same reason, we can ignore the presence of for all 0 ≤ r ≤ s ≤ t ≤ L and n ≥ 1. We need to check only the following possibilities.
[I] := |Y 1 ∩ Y 2 | ∈ {1, . . . , k}, [III]. Each Y i has at least one common element with at least one of the other three, but neither [I] or [II] is true.
As for the ratio of the densities in the second line of (7.30), we use the basic fact that 1/a is flat for a, that is, as n → ∞, a(R n ) a R n + a(R n )v → 1 , uniformly on bounded v-sets; (7.31) see p142 in [15] for details. Noting that L is also flat for a, we have for every ρ > 0, f R n + a(R n )ρ e 1 f (R n e 1 ) = L R n + a(R n )ρ L(R n ) exp −ψ R n + a(R n )ρ + ψ(R n ) = L R n + a(R n )ρ L(R n ) exp − ρ 0 a(R n ) a R n + a(R n )r dr → e −ρ , as n → ∞ .
To provide an upper bound for the ratio of the densities, let q m (n), m ≥ 0, n ≥ 1 be a sequence defined by q m (n) = a(R n ) −1 ψ ← ψ(R n ) + m − R n , equivalently, ψ R n + a(R n )q m (n) = ψ(R n ) + m .