On central limit theorems in stochastic geometry for add-one cost stabilizing functionals *

We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich ( Ann. Appl. Probab. 11 (4), 1005–1041 (2001)) to the inhomogeneous case. Here functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions. As an application, a central limit theorem for Betti numbers of random geometric complexes in the subcritical regime is derived.


Introduction
The paper introduces a new approach to establish central limit theorems (CLT) for functionals on binomial point processes and Poisson point processes. CLTs in this setting may be found in [18] for general functionals, and in [1,20] for functionals of a specific form. However, the work in [18] only deals with binomial point processes having uniform distribution and homogeneous Poisson point processes. We are going to remove such restrictions in this paper.
Binomial point processes considered here are X n = {X 1 , . . . , X n }, where {X i } ∞ i=1 is an i.i.d. (independent identically distributed) sequence of R d -valued random variables having probability density function f . The function f is assumed to be bounded and to have compact support. Associated with {X n } is the Poissonized version P n = {X 1 , . . . , X Nn } which becomes a Poisson point process with intensity function nf . Here the random variable N n has Poisson distribution with parameter n and is independent of {X i }. By a functional, it means a real-valued measurable function H defined on all finite subsets in R d . We will study CLTs for H(n 1/d X n ) and H(n 1/d P n ) as n tends to infinity, where aX = {ax : x ∈ X} for a ∈ R and X ⊂ R d .
Let us first introduce the results in [18]. Assume that X i is uniformly distributed on some bounded set S, or equivalently f (x) ≡ λ on S. The support S may need some technical assumption. In this case, the point process n 1/d P n has the same distribution with the restriction on n 1/d S of a homogeneous Poisson point process P(λ) with intensity λ. Then a CLT holds for H(n 1/d P n ), that is, n −1/2 (H(n 1/d P n ) − E[H(n 1/d P n )]) converges in distribution to a Gaussian distribution with mean 0 and variance σ 2 ≥ 0, provided that the functional H is weakly stabilizing and satisfies a bounded moment condition.
Here the concept of stabilization is defined via the add-one cost function associated with H, D 0 (·) = H(· ∪ {0}) − H(·), which measures the increment of H by adding a point at the origin. (The precise definition will be given in Section 3.2.) This approach is based on the martingale difference central limit theorem as one may expect due to a spatial independence property of Poisson point processes. A CLT for H(n 1/d X n ) is then derived by a de-Poissonization technique in which a stronger condition, called strong stabilization, is required. Roughly speaking, H is strongly stabilizing if the value of D 0 on P(λ) does not change when adding or removing points far from the origin.
Although some techniques had been developed in [8,12,13], the paper [18] is the first one successfully dealing with general functionals. Since then, it has found many applications.
For the non-uniform distributions case, that martingale-based approach has been shown to work for some specific functionals (eg. the component count in geometric graph [16,Section 13.7] and functionals related to Euclidean minimal spanning trees [13]). To the best knowledge of the author, there is no general result like [18] yet. In this paper, we develop a new fundamental approach to derive CLTs for functionals which are assumed to be strongly stabilizing on P(λ) for all 0 ≤ λ ≤ sup f (x). (Some additional bounded moments conditions are needed.) Note that we impose the strong stabilization on homogeneous Poisson point processes only. This condition is very mild in the sense that it is also a sufficient condition for the well-established de-Poissonization technique in [16,Section 2.5].
Note that for functionals on general Poisson point processes (not necessary point processes on R d ), upper bounds for the normal approximation in the Wasserstein distance and the Kolmogorov distance were established in [9] by using the second order difference operator. Proposition 1.4 and Theorem 6.1 therein are related to our setting. Indeed, it was mentioned in the paragraph following Proposition 1.4 that the crucial condition (1.8) is closely related to the concept of strong stabilization. Of course, the results in [9] can apply to non-stabilizing functionals as well. However, even in case of stabilizing functionals, an additional condition on the radii of stabilization is needed. While CLTs in this paper hold without any further requirement on the stabilization radii. It is worth mentioning another direction in the study of the limiting behavior of H(n 1/d X n ) and H(n 1/d P n ). In this direction, assume that the functional H can be expressed in the following form where ξ(x; X) is a local (or stabilizing) function. (The stabilization of ξ(x; X) has the same meaning with the strong stabilization of D 0 .) Under some more conditions on the tail of stabilization radii, laws of large numbers and central limit theorems have been established [1,17,19,20]. An explicit expression for the limiting variance and a rate of convergence in CLTs have been also known. We need not to compare those results with ours because the scope is different. The paper is organized as follows. Section 2 introduces some probabilistic ingredients. CLTs for homogeneous Poisson point processes, non-homogeneous Poisson point processes and binomial point processes are established in turn in Section 3. A partial result on CLT for Betti numbers in the thermodynamic regime, as an application of the general theory, is discussed in Section 4.

Probabilistic ingredients
This section introduces several useful results needed in this paper.

CLT for triangular arrays
The following is an easy consequence of Lyapunov's central limit theorem. For Lyapunov's central limit theorem, see [2,Theorem 27.3].
Theorem 2.1. For each n, let {ξ n,i } n i=1 be a sequence of independent real random variables. Here we require that n ≤ cn for some constant c > 0. Assume that Here ' d →' denotes the convergence in distribution, and N (0, σ 2 ) denotes the Gaussian distribution with mean zero and variance σ 2 .
The following result is fundamental and is somewhat similar to Theorem 6.3.1 in [15]. For the sake of convenience, a quick proof is provided. (ii) lim k→∞ lim sup n→∞ Var[X n,k − Y n ] = 0.
Then the limit σ 2 = lim k→∞ σ 2 k exists, and as n → ∞, Proof. It follows from the triangular inequality that By letting n → ∞ first, and then let k → ∞ in the above inequalities, we see that Let t ∈ R be fixed. By the assumption (i), for each k,

Poisson point processes
Let f (x) ≥ 0 be a locally integrable function on R d . A Poisson point process with intensity function f is a point process P on R d which satisfies the following conditions (i) for any bounded Borel set A, the number of points inside A, denoted by P(A), has Poisson distribution with parameter ( A f (x)dx); (ii) for disjoint bounded Borel sets A 1 , . . . , A k , the random variables P(A 1 ), . . . , P(A k ) are independent.
A Poisson point process with the intensity function f (x) identically equal to a constant λ ≥ 0 is called a homogeneous Poisson point process with density λ.
We need the following result on the convergence of functionals on Poisson point processes. Recall that a functional H is a real-valued measurable function defined on all finite subsets in R d .
Here P(f n ) (resp. P(f )) denotes a Poisson point process with intensity function f n (resp. f ).
Proof. We use the following coupling. Let Φ be a homogeneous Poisson point process Let P n (resp. P) be the projection of the point process Φ| An (resp. Φ| A ) onto W . Then by using the restriction theorem and the mapping theorem for Poisson point processes (Chapter 5 in [10]), it follows that P n (resp. P) becomes a Poisson point process with intensity function f n (resp. f ).
Consequently, as n → ∞, It follows that on this realization, H(P n ) converges in probability to H(P). Therefore, H(P(f n )) converges in distribution to H(P(f )). The proof is complete.
The functional H is said to be translation-invariant if H(y + X) = H(X) for all finite subsets X ⊂ R d and all y ∈ R d , where y + X = {y + x : x ∈ X}. For translation-invariant functional, Poisson point processes do not need to be defined on the same region. Consequently, we have: ECP 24 (2019), paper 76.

Corollary 2.4.
Let H be a translation-invariant functional. Let W ⊂ R d be a bounded Borel set. Assume that Wn |f n (x) − λ|dx → 0 as n → ∞, where {f n } are non-negative functions defined on W n = y n + W , and λ ≥ 0 is a constant. Then Here P(λ)| W denotes the restriction on W of a homogeneous Poisson point process P(λ) with density λ.
Assume further that for some δ > 0, sup n E[|H(P(f n ))| 2+δ ] < ∞. Then as n → ∞, Proof. Since the functional H is translation-invariant, the first statement follows directly from the previous lemma. The second statement is a standard result in probability theory, (for example, see the corollary following Theorem 25.12 in [2]).
Next, we introduce the so-called Poincaré inequality for the variance of Poisson functional, an essential tool in this paper. Let P be a Poisson point process with intensity function f . Assume that f (x)dx < ∞. Then almost surely, P has finitely many points. For a functional H, define the add-one cost at a point x as (2.1)

Homogeneous Poisson point processes
From now on, assume that the functional H is translation-invariant. Let P(f ) (resp. P(λ)) denote a Poisson point process with intensity function f (resp. homogeneous Poisson point process with density λ).
The functional H is said to be weakly stabilizing on P(λ) if there is a (finite) random variable ∆(λ) such that D 0 (P(λ)| Vn ) → ∆(λ), almost surely, for any sequence {V n 0} ∞ n=1 of cubes which tends to R d as n → ∞. Here a cube means a subset in R d of the form y + [0, a) d .

Remark 3.2. (i) This theorem (with p = 4)
is a special case of Theorem 3.1 in [18] in which the restriction of P(λ) on a general sequence of subsets {B n } was considered.
Thus the weak stabilization and the moment condition (3.1) should be defined in terms of {B n }. Similar to Theorem 3.1 in [18], the above theorem still holds if in the definition of weak stabilization, the almost sure convergence is replaced by the convergence in probability.
(ii) Theorem 3.9 below provides sufficient conditions for the positivity of the limiting varianceσ 2 (λ) in which the strong stabilization (to be defined in the next subsection) is required and the limiting add-one cost ∆(λ) is assumed to be non trivial. In a forthcoming work [5], we derive an explicit expression for σ 2 (λ) in terms of ∆ and show thatσ 2 (λ) > 0, if ∆ is not identically equal to zero, P(∆ = 0) = 1. We may also use [9, Theorem 5.2] to derive a lower bound for the limiting variance.
Proof. For L > 0, and for each n, divide the cube K n := [−n 1/d /2, n 1/d /2) d according to the lattice L 1/d Z d and let {W i } n i=1 be the lattice cubes entirely contained in K n . Let Here for simplicity, we remove λ in formulae. Then X n,L is a (scaled) sum of i.
Here we have used the Poincaré inequality (2.1) for the functional The integrands in the above integrals are uniformly bounded by the assumption (3.1), that is, there is a constant C > 0 such that Thus the first term in (3.3) vanishes as n → ∞.
For the second term, note that the weak stabilization assumption, together with the uniform boundedness assumption (3.1), implies that for any sequence {V n 0} of cubes tending to R d as n → ∞. It follows that for given ε > 0, we can choose a number t > 0 such that for any pair (V, W ) of cubes with Here B r (x) denotes the closed ball of radius r centered at x (with respect to the Euclidean metric). Note that the above inequality still holds if 0 is replaced by any y ∈ R d because of the translation invariance of H and of P(λ).
Here |A| denotes the volume of a set A. Note that W i ⊂ K n . Thus for y ∈ int(W i ), E[|D y (P n ) − D y (P Wi )| 2 ] < ε. Then the second term in (3.3) can be estimated as follows (for n, L > 2t), 2) because ε is arbitrary. The theorem is proved.

Non-homogeneous Poisson point processes
Let f : R d → [0, ∞) be a bounded measurable function with compact support. We are going to establish a central limit theorem for H(n 1/d P(nf )). When f is a probability density function, then P(nf ) has the same distribution with the Poissonized version P n = {X 1 , . . . , X Nn }. However, in this section, f need not be a probability density function. LetP n = n 1/d P(nf ). ThenP n is a Poisson point process with intensity function f (x/n 1/d ).
Let us discuss some terminologies. The functional H is strongly stabilizing on P(λ) if there exist (finite) random variables τ (λ) (a radius of stabilization of H) and ∆(λ) (the limiting add-one cost) such that almost surely,  We claim that this condition on {P n } implies the condition (3.1) on a homogeneous Poisson point process P(λ) with density λ = f (x), provided that x is a Lebesgue point of f . Indeed, by definition, the point x is a Lebesgue point of f if Let W 0 be a cube. Let W n = (n 1/d x + W ) and V n = x + n −1/d W . Then |V n | = n −1 |W |, and hence, Lemma 2.3 applying to the shifted point process (P n | Wn − n 1/d x) and to the add-one cost D 0 implies that Then by Fatou's lemma, Here σ 2 = σ 2 (f (x))dx, withσ 2 (λ) the limiting variance in Theorem 3.1.

Remark 3.4. From the argument following the Poisson bounded moments condition,
we see that the limiting varianceσ 2 (f (x)) is defined at every Lebesgue point x of f . Moreover, the Lebesgue differentiation theorem states that for an integrable function f , almost every point is a Lebesgue point. Thusσ 2 (f (x)) is defined almost everywhere.
We use the same idea as in the proof of Theorem 3.1. Let S be a cube which contains the support of f . For L > 0, divide R d according to the lattice (L/n) 1/d Z d and let {V i } n i=1 be the cubes which intersect with S. Set S n = ∪ i V i . Then it holds that n /n = |S n |/L → |S|/L as n → ∞. Let W i be the image of V i under the map x → n 1/d x. Recall thatP n is a Poisson point process onS n = n 1/d S with intensity function f (x/n 1/d ). Assume that the functional H satisfies all the assumptions in Theorem 3.3.

Lemma 3.5.
There is a constant M > 0 such that for any cube W , Proof. It follows from the Poisson bounded moments condition (3.4) that there is a constant C > 0 such that for all n, all y and all W y, Then the desired estimate is just a direct consequence of the Poincaré inequality (2.1) It follows from Lemma 3.5 that |g n,L (x)| ≤ M . Moreover, when x ∈ S is a Lebesgue point of f , then by Lemma 3.6, as n → ∞, Here V i(x,n) = n −1/d W i(x,n) is the unique cube in {V i } containing x. In addition, it is clear that |S n \ S| → 0 as n → ∞. Recall that almost every x ∈ S is a Lebesgue point. Therefore the convergence of the variance Var[X n,L ] follows by the bounded convergence theorem. The CLT for X n,L then follows from Theorem 2.1 because the locally bounded moments condition (3.5) has been assumed. The proof is complete.
ECP 24 (2019), paper 76. Proof. We begin with a direct application of the Poicaré inequality (2.1) Var It follows from (3.6) that, Let t > 0. Assume that L > 2t. Recall the notations int(W i ) and ∂(W i ) from the proof in the homogeneous case. Then Next, we deal with the case y ∈ int (W i ). Let x = y/n 1/d . Consider a homogeneous Poisson point process P(λ) with density λ = f (x). Let τ (λ) be the stabilization radius of H on P(λ) at y. There is a coupling of P(λ) andP n such that (see the proof of Lemma 2.3) wheret n (y) = t n (x) = Wi |f (y/n 1/d ) − f (z/n 1/d )|dz = n Vi |f (x) − f (z)|dz. On the event A ∩ {τ (λ) ≤ t}, by the definition of the radius of stabilization, D y (P n ) = D y (P n | Wi ). Thus Here C p is a constant which comes from the Poisson bounded moments condition (3.4). We have used Hölder's inequality with q being the Hölder conjugate number of p/2.
Here the bounded convergence theorem has been used in the last estimate. Combining the two estimates (3.7) and (3.8), we arrive at The proof is complete by letting t → ∞.
Proof of Theorem 3.3. Similar to the homogeneous case, the CLT for H(P n ) follows by combining Lemma 3.7 and Lemma 3.8. Indeed, Lemma 3.7 states that for fixed L > 0, as n → ∞, In addition, Lemma 3.8 shows that Therefore, the CLT for Y n holds by taking into account Lemma 2.2, that is, as n → ∞, For the limiting variance, recall that when x is a Lebesgue point of f , Recall also from Lemma 3.6 that L −1 Var[H(P L (λ))] ≤ M . Thus, by the bounded convergence theorem again, it follows that Theorem 3.3 is proved.

Binomial point processes
be an i.i.d. sequence of R d -valued random variables with a common probability density function f . The function f is assumed to be bounded and to have compact support. Let X n = {X 1 , . . . , X n } and P n = {X 1 , . . . , X Nn } be the binomial point processes and the Poisson point processes associated with {X i }, respectively. Assume ECP 24 (2019), paper 76. that the functional H satisfies all the assumptions of Theorem 3.3. Then the CLT for H(n 1/d P n ) holds, that is, as n → ∞, Here recall thatσ 2 (λ) = lim n→∞ n −1 Var[H(P(λ)| [−n 1/d /2,n 1/d /2) d )] is the limiting variance in the homogeneous case.
We now use a de-Poissonization technique to derive a CLT for H(n 1/d X n ). It turns out that we only need two more moments conditions. The first one requires that there is a constant β > 0 such that for any m, n H(n 1/d X m ) ≤ β(m + n) β , almost surely. (3.9) The second one requires for some q > 2, η > 0. When the two conditions are added, the CLT for H(n 1/d X n ) holds, namely, we have Theorem 3.9. Let f be a bounded probability density function with compact support. Let Λ = sup f (x). Assume that the functional H is strongly stabilizing on P(λ) for any λ ∈ [0, Λ], and satisfies conditions (3.4), (3.5), (3.9) and (3.10). Then as n → ∞, Moreover, if the limiting add-one cost ∆(λ) is non-constant for λ ∈ A, where P(X 1 ∈ A) > 0, then τ 2 > 0 and σ 2 > 0.

CLT for Betti numbers
For a finite set of points X = {x 1 , . . . , x n } in R d , theČech complex of radius r > 0, denoted by C(X, r), is defined as the abstract simplicial complex consisting of non-empty subsets of X in the following way {x i0 , . . . , x i k } ∈ C(X, r) ⇔ k j=0 B r (x ij ) = ∅.
The nerve theorem (cf. [4]) tells us that the abstract simplical complex C(X, r) is homotopy equivalent to the union of balls U r (X) = n i=1 B r (x i ).
Cech complexes may be regarded as a generalization of geometric graphs.
Denote by β k (C(X, r)) the kth Betti number, or the rank of the kth homology group of C(X, r), with coefficients from some underlying field. The limiting behavior of ECP 24 (2019), paper 76. β k (C(X n , r n )) has been study intensively, where {r n } is a deterministic sequence tending to zero. It is known that Betti numbers behave differently in three regimes divided according to the limit of {n 1/d r n }: zero, finite, or infinite. Refer to the survey [3] for more details on this topic. Note that the zeroth Betti number β 0 (C(X, r)) just counts the number of connected components in U r (X). Also β k (C(X, r)) = 0, if k ≥ d, as a consequence of the nerve theorem.
We focus now on the thermodynamic regime, also called the critical regime, in which n 1/d r n → r ∈ (0, ∞). Without loss of generality, we may assume that n 1/d r n = r. Define a functional H r as H r (X) = β k (C(X, r)).
The positivity of the limiting variances σ 2 k > τ 2 k > 0 holds because the limiting add-one cost is non-constant for any λ > 0 [22,Theorem 4.7]. Note that β 0 is strongly stabilizing without any restriction on r because the infinite component, when exists, is unique. Thus, Theorem 13.26 and Theorem 13.27 in [16] still hold without the Riemann integrable assumption on f . The regime where n 1/d r n → r ∈ (0, Λ −1/d r c ) is called the subcritical regime. A CLT in the supercritical regime (n 1/d r n → r ∈ [Λ −1/d r c , ∞)) is still open. However, by a duality property, it was shown in the proof of Theorem 4.7 in [22] that H r is strongly stabilizing on P(1), if r / ∈ I d , where In particular, I 2 = ∅, which implies that for d = 2, H r is strongly stabilizing on P(λ) for all λ. Thus in two dimensional case, there is no restriction on r.