2. Set-up

To begin, we start by defining some essential concepts towards proving the results in this paper. We look at point clouds generated by $\mathcal{P}_n$ , a Poisson process on ${\mathbb{R}}^d$ , $d\geq 2$ . We take $\mathcal{P}_n$ to have the intensity measure $n\int_A f(x)\,\text{d}{x}$ for all measurable A in ${\mathbb{R}}^d$ , where f is a probability density that is almost everywhere bounded and continuous with respect to Lebesgue measure. Throughout the paper, Lebesgue measure on ${\mathbb{R}}^{d(k+1)}$ is denoted by $m_k$ and for convenience we let $m \,:\!= m_0$ .

As an aside, we have a few definitions to mention before we begin. First, let $\lVert f \rVert_{\infty}$ be the essential supremum of the aforementioned f, which is finite as f is almost everywhere bounded. Furthermore, define $\theta_d \,:\!= m(B(0;\, 1))$ to be the volume of the unit ball in ${\mathbb{R}}^d$ . The constant $C_{f,k}$ is mentioned frequently in the study and is defined as

\[C_{f,k} \,:\!= \dfrac{1}{(k+2)!} \int_{{\mathbb{R}}^d} f(x)^{k+2} \text{d}{x}.\]

Furthermore, we let ${\mathbb{R}}_+ \,:\!= [0, \infty)$ and $\mathbb{N}$ be the positive integers and $\mathbb{N}_0 \,:\!= \mathbb{N} \cup \{0\}$ – the non-negative integers, and ${{\bf 1}} \{ \cdot \}$ denotes an indicator function.

It is useful to define the notion of a finite point cloud throughout the study. We let $\mathcal X_m\,:\!= \{X_1, X_2, \ldots, X_m\}$ , where $X_i$ are i.i.d. with density f as mentioned before, though it may represent an arbitrary subset of ${\mathbb{R}}^d$ of cardinality m as needed. Thus, if $N_n$ is a Poisson random variable with parameter n, which is independent of the $X_i$ , then we can represent the Poisson process $\mathcal{P}_n$ as

\[\mathcal{P}_n(A) = \sum_{i = 1}^{N_n} \delta_{X_i}(A),\]

for all measurable $A \subset {\mathbb{R}}^d$ , with $\delta_x$ a Dirac measure at $x \in {\mathbb{R}}^d$ .

With this definition in tow, we turn towards the study of Betti numbers. Fixing $1\leq k <d$ , we define $h_t(x_1, \ldots, x_{k+2})$ , $x_i \in {\mathbb R}^d$ , to be the indicator that ${\skew4\check{C}}(\{x_1, x_2, \ldots, x_{k+2}\}, t)$ contains an empty $(k+1)$ -simplex. This means that ${\skew4\check{C}}(\{x_1, x_2, \ldots, x_{k+2}\}, t)$ does not contain a $(k+1)$ -simplex but does contain all possible k-simplices.

With this in mind, we see that $h_t$ can be represented as

\[h_t(x_1,\ldots, x_{k+2}) = h_t^+(x_1,\ldots,x_{k+2}) - h_t^-(x_1,\ldots,x_{k+2}),\]

where we define

\begin{align*} h_t^+(x_1,\ldots,x_{k+2}) &\,:\!= \prod_{i=1}^{k+2} {{\bf 1}} \bigg\{ \bigcap_{j=1, \, j \neq i}^{k+2} B(x_j;\, t/2) \neq \emptyset \bigg\}, \\[3pt] h_t^-(x_1,\ldots,x_{k+2}) &\,:\!= {{\bf 1}} \bigg\{\, \bigcap_{j=1}^{k+2} B(x_j;\, t/2) \neq \emptyset \bigg\}.\end{align*}

It is important to note that $h_t^\pm$ is non-decreasing in t. That is,

\[h_s^\pm (x_1,\ldots,x_{k+2}) \leq h_t^\pm (x_1,\ldots,x_{k+2})\]

for all $0 \leq s \lt t$ and $x_i \in {\mathbb R}^d$ .

Throughout the paper we interest ourselves in the kth Betti number for ${\skew4\check{C}}(\mathcal{P}_n, r_n(t))$ where $r_n(t) \,:\!= s_nt$ . Recall that the nature of how $s_n$ decays to 0 as $n \to \infty$ is the object of our study. We let $S_{k,n}(t)$ denote the number of empty $(k+1)$ -simplex components of ${\skew4\check{C}}(\mathcal{P}_n, r_n(t))$ . In other words, $S_{k,n}(t)$ represents the number of connected components C on $k+2$ points such that $\beta_k(C)=1$ . More generally, for integers $i \geq k + 2$ and $j \gt 0$ , we define $U_{i,j,n}(t)$ as the number of connected components C of $\skew4\check{C}(\mathcal{P}_n, r_n(t))$ such that $|C| = i$ and $\beta_k(C) = j$ . Then the kth Betti number of $\skew4\check{C}(\mathcal{P}_n, r_n(t))$ can be represented as

(2.1) \begin{equation}\beta_{k, n}(t) = \sum_{i \geq k+2} \sum_{j \gt 0} j U_{i,j,n}(t), \quad t \gt 0.\end{equation}

Since $S_{k,n}(t) = U_{k+2,1,n}(t)$ and one cannot form multiple empty $(k+1)$ -simplices from $k+2$ points, (2.1) can also be represented as

(2.2) \begin{equation}\beta_{k,n}(t) = S_{k,n}(t) + \sum_{i \gt k+2} \sum_{j \gt 0} jU_{i,j,n}(t), \quad t \gt 0.\end{equation}

In this setting it is instructive to introduce the following indicator functions to formalize these concepts for an arbitrary collection of points ${\mathcal{Y}} \subset \mathcal{X} \subset {\mathbb{R}}^d$ :

1. • $J_{i,t}({\mathcal{Y}}, \mathcal{X}) \,:\!= {{\bf 1}} \{ \skew4\check{C} ({\mathcal{Y}}, t) \text{ is a connected component of } \skew4\check{C}(\mathcal X, t) \}\, {{\bf 1}} \{ |{\mathcal{Y}}|=i \}$ ,

2. • $b_{j,t}({\mathcal{Y}}) \,:\!= {{\bf 1}} \{ \beta_k ( \skew4\check{C} ({\mathcal{Y}}, t) )=j \}$ ,

3. • $g_t^{(i,j)}({\mathcal{Y}}, \mathcal{X}) \,:\!= b_{j,t}({\mathcal{Y}}) J_{i, t}({\mathcal{Y}}, \mathcal{X})$ .

In particular, denote

\[g_t({\mathcal{Y}}, \mathcal X) \,:\!= g_t^{(k+2,1)}({\mathcal{Y}}, \mathcal{X}) = b_{1,t}({\mathcal{Y}}) J_{k+2, t}({\mathcal{Y}}, \mathcal{X}) = h_t({\mathcal{Y}}) J_{k+2, t}({\mathcal{Y}}, \mathcal{X}).\]

Further, for $A \subset \mathbb{R}^d$ , let

1. • $h_{t,A}({\mathcal{Y}}) \,:\!= h_t({\mathcal{Y}}){{\bf 1}} \{ \text{LMP} ({\mathcal{Y}}) \in A \}$ ,

2. • $g_{t, A}^{(i,j)}({\mathcal{Y}}, \mathcal{X}) \,:\!= g_t^{(i,j)}({\mathcal{Y}}, \mathcal{X}){{\bf 1}} \{ \text{LMP} ({\mathcal{Y}}) \in A \}$ ,

where $\textrm{LMP}({\mathcal{Y}})$ is the leftmost point, in dictionary order, of the set ${\mathcal{Y}}$ .

With the above indicators now available, it is clear that

\[S_{k,n}(t) = \sum_{{\mathcal{Y}} \subset \mathcal{P}_n} {g_{r_n(t)}}({\mathcal{Y}}, \mathcal{P}_n), \quad U_{i,j,n}(t) = \sum_{{\mathcal{Y}} \subset \mathcal{P}_n} {g_{r_n(t)}^{(i,j)}} ({\mathcal{Y}}, \mathcal{P}_n).\]

As a final bit of notation, let

\[\beta_{k,n,A}(t) = \sum_{i \geq k+2} \sum_{j \gt 0} jU_{i,j,n,A}(t) = S_{k,n,A}(t) + \sum_{i \gt k+2} \sum_{j \gt 0} jU_{i,j,n,A}(t),\]

where, in all functions above, we require the leftmost point of every subset ${\mathcal{Y}}$ to be an element of A. When brevity is paramount, we occasionally shorten $ \sum_{i \gt k+2} \sum_{j \gt 0} jU_{i,j,n}(t)$ to $R_{k,n}(t)$ and $ \sum_{i> k+2} \sum_{j \gt 0} jU_{i,j,n,A}(t)$ to $R_{k,n, A}(t)$ respectively.

3. Central limit theorem in the sparse regime

Throughout this section we assume that $ns_n^d \to 0$ and $\rho_n \,:\!= n^{k+2} s_n^{d(k+1)} \to \infty$ as $n \to \infty$ . The essence is that Čech complexes are distributed sparsely with many separate connected components, because of a fast decay of $s_n$ as a result of $ns_n^d\to0$ . Consequently, all k-cycles are asymptotically supported on $k+2$ points. From a more analytic viewpoint, the behavior of the kth Betti number (2.2) is completely determined by $S_{k,n}(t)$ , whereas $R_{k,n}(t) = \beta_{k,n}(t) - S_{k,n}(t)$ is asymptotically negligible in the sense that ${\mathbb{E}}[S_{k,n}(t)]/\rho_n$ tends to a finite positive constant, but ${\mathbb{E}}[R_{k,n}(t)]/\rho_n \to 0$ as $n\to\infty$ .

The study most relevant to this section is [Reference Kahle and Meckes21], in which the central limit theorem for the sparse regime is discussed. We have extended [Reference Kahle and Meckes21] (with the erratum paper [Reference Kahle and Meckes22]) in two directions. First, we develop the process-level central limit theorem. This highlights the chief contribution of this paper. Whereas [Reference Kahle and Meckes21] and [Reference Kahle and Meckes22], as well as [Reference Yogeshwaran, Subag and Adler34] in the ensuing section, treat the ‘static’ topology of random Čech complexes (i.e. no time parameter t involved), the main focus of this paper is ‘dynamic’ topology of the same complex, treating Betti numbers as a stochastic process. Second, our central limit theorem is for the entirety of the sparse regime, without requiring that $s_n = o(n^{-1/d -\delta})$ for some $\delta \gt 0$ as assumed in [Reference Kahle and Meckes22].

Before presenting the main result we define the limiting stochastic process

(3.1) \begin{equation}\mathcal{G}_k(t) \,:\!= \int_{{\mathbb{R}}^{d(k+1)}} h_t(0, \mathbf{y}) \, G_k(\text{d}{\mathbf{y}}),\end{equation}

where $G_k$ is a Gaussian random measure such that $G_k(A) \sim \mathcal N(0, C_{f, k}m_{k}(A))$ for all measurable A in ${\mathbb{R}}^{d(k+1)}$ . Furthermore, for $A_1, \ldots, A_m$ disjoint, $G_k(A_1), \ldots, G_k(A_m)$ are independent. As defined, ${\mathcal G}_k(t)$ depends on the indicator $h_t$ , meaning that due to sparsity of the Čech complex in this regime, the k-cycles affecting ${\mathcal G}_k(t)$ must always be formed by connected components on $k+2$ points (i.e. components of the smallest size).

The significance of the characterization of the process at (3.1) is that if we define

\[\mathcal{G}^{\pm}_k(t) \,:\!= \int_{{\mathbb{R}}^{d(k+1)}} h^{\pm}_t(0, {\bf y}) \, G_k(\text{d}{{\bf y}}),\]

then $\mathcal{G}^{\pm}_k(t)$ becomes a time-changed Brownian motion; see Proposition 3.1 below. Hence ${\mathcal G}_k(t) = \mathcal{G}^{+}_k(t) - \mathcal{G}^{-}_k(t)$ is a difference of two dependent time-changed Brownian motions, where dependence is due to the same Gaussian random measure $G_k$ shared by ${\mathcal G}_k^+(t)$ and ${\mathcal G}_k^-(t)$ . Those wishing to examine this characterization in more detail should refer to [Reference Owada24]. For example, it is proved in [Reference Owada24] that the process ${\mathcal G}_k(t)$ is self-similar with exponent $H=d(k-1)/2$ and is Hölder-continuous of any order in $[0,1/2)$ .

Proposition 3.1 The process $\mathcal{G}^{\pm}_k(t)$ can be expressed as

\[(\mathcal{G}^{\pm}_k(t), t \geq 0) \overset{d}{=} (B(C_{f,k}m_k(D_1^{\pm})t^{d(k+1)}), t \geq 0),\]

where B is a standard Brownian motion and $D_t^{\pm} = \{{\bf y} \in {\mathbb{R}}^{d(k+1)}\,\colon h^{\pm}_t(0, {\bf y}) =1\}$ .

Proof. We prove only the result for $\mathcal{G}^{+}_k$ , as the proof for $\mathcal{G}^{-}_k$ is the same. It is elementary to show that $\mathcal{G}^{+}_k(t)$ has mean zero. Thus, it only remains to demonstrate the covariance result. Since $h_t^+$ is non-decreasing in t, we have $D_{t_1}^+ \subset D_{t_2}^+$ for $0 \leq t_1 \leq t_2$ ; therefore,

\begin{equation*}{\mathbb{E}}[\mathcal{G}^{+}_k(t_1)\mathcal{G}^{+}_k(t_2)] = {\mathbb{E}}[G_k(D_{t_1}^+) G_k(D_{t_2}^+)] = {\mathbb{E}}[G_k(D_{t_1}^+)^2]= C_{f,k} m_k(D_{t_1}^+) = C_{f,k}m_k(D_1^{+})t_1^{d(k+1)}.\,\, \qed\end{equation*}

Our main result can be seen below. The proof is briefly presented in Section 6.2 as a straightforward variant of the proof for the critical regime. For the proof we need to examine the asymptotic growth rate of expectations and covariances of $\beta_{k,n}(t)$ . The detailed results are presented in Proposition 6.2, where it is seen that the expectation and covariance both grow at the rate $\rho_n$ .

Theorem 3.1. Suppose that $ns_n^d \to 0$ and $\rho_n = n^{k+2}s_n^{d(k+1)} \to \infty$ . Assume that f is an almost everywhere bounded and continuous density function. Then, we have the following weak convergence in the finite-dimensional sense, namely

\begin{equation*}\rho_n^{-1/2} ( \beta_{k,n}(t) - {\mathbb{E}}[\beta_{k,n}(t)] ) \overset{\text{fidi}}{\Rightarrow} \mathcal{G}_k(t),\end{equation*}

meaning that for every $m \in \mathbb{N}$ and $0 \lt t_1 \lt t_2 \lt \cdots \lt t_m \lt \infty$ we have

\begin{equation*}\rho_n^{-1/2} ( \beta_{k,n}(t_i) - {\mathbb{E}}[\beta_{k,n}(t_i)], \, i=1,\ldots,m ) \Rightarrow ( \mathcal{G}_k(t_i),\, i=1,\ldots,m )\end{equation*}

weakly in ${\mathbb{R}}^m$ .

4. Central limit theorem in critical regime

We now expand on the results of [Reference Yogeshwaran, Subag and Adler34] (see also [Reference Trinh31]) by offering an explicit limit of appropriately scaled moments and a central limit theorem for $\beta_{k,n}(t)$ . In the critical regime, the connectivity radius $s_n$ isdefined to be $s_n = n^{-1/d}$ . This sequence decays more slowly than that in the previous section; hence, Čech complexes become highly connected with many topological holes of any dimension $k< d$ . More analytically, all terms in the sum (2.1) contribute to the kth Betti number, unlike in the sparse regime. This implies that the k-cycles in the limit could be supported not only on $k+2$ points but also on i points for all possible $i \gt k+2$ .

Yogeshwaran, Subag, and Adler [Reference Yogeshwaran, Subag and Adler34] established the central limit theorem for the first time for the critical regime (though they referred to it as the ‘thermodynamic’ regime). There are two key differences between their paper and ours. The first is that the Poisson process they consider is homogeneous with unit intensity, restricted to a set $B_n$ such that $m(B_n) = n$ . The second difference between the two, and equivalent to the contrast indicated in the sparse regime, is again that [Reference Yogeshwaran, Subag and Adler34] treats the static topology of random Čech complexes whereas we treat the dynamic topology. As a consequence, while the weak limit in [Reference Yogeshwaran, Subag and Adler34] is a simple Gaussian distribution with unknown variance, our limit is a Gaussian process having a structure similar to that of the Betti number (2.1).

The other relevant article to our study is [Reference Trinh31], which generalizes [Reference Yogeshwaran, Subag and Adler34] to an inhomogeneous Poisson process case, but again only deals with static topology. We would like to emphasize that our proof techniques are significantly different from those in [Reference Yogeshwaran, Subag and Adler34] and [Reference Trinh31]. In fact, our proof is highly analytic in nature, borrowing machinery from [Reference Penrose26] and [Reference Kahle and Meckes21], whereas the proofs of [Reference Yogeshwaran, Subag and Adler34] and [Reference Trinh31] rely more on the topological nature of the objects, including weakly/strongly stabilizing properties of Betti numbers, the notion of critical radius of percolation, and the theory of geometric functionals as in [Reference Penrose and Yukich27]; see also Remark 4.2. By virtue of our analytic approach, we can fully specify the structure of the limiting Gaussian process as in (4.1) below. This is actually the main objective of this study.

We now define the aforementioned limiting Gaussian process by

(4.1) \begin{equation}\mathcal H_k(t)= \sum_{i \geq k+2} \sum_{j>0} j \mathcal H_k^{(i,j)}(t), \quad t \gt 0,\end{equation}

where $( \mathcal H_k^{(i,j)}, \, i \geq k+2, j>0 )$ is a family of centered Gaussian processes with inter-process dependence between $\mathcal H_k^{(i_1,j_1)}$ and $\mathcal H_k^{(i_2,j_2)}$ determined by

(4.2) \begin{equation}\text{Cov} ( \mathcal H_k^{(i_1,j_1)}(t_1), \mathcal H_k^{(i_2,j_2)}(t_2) ) = \dfrac{1}{i_1!}\, \eta_{k, \mathbb R^d}^{(i_1, j_1, j_2)}(t_1, t_2) \delta_{i_1, i_2} + \dfrac{1}{i_1! i_2!}\, \nu_{k, \mathbb R^d}^{(i_1, i_2, j_1, j_2)} (t_1, t_2).\end{equation}

Here $\delta_{i_1, i_2}$ is the Kronecker delta, and the functions $\eta_{k, \mathbb R^d}^{(i_1, j_1, j_2)}$ , $\nu_{k, \mathbb R^d}^{(i_1, i_2, j_1, j_2)}$ are explicitly defined during the proof of the main theorem (see (6.2) and (6.3)). From (4.2), the covariance of $\mathcal H_k^{(i,j)}$ is given by

\begin{equation*} \text{Cov} ( \mathcal H_k^{(i,j)}(t_1), \mathcal H_k^{(i,j)}(t_2) ) = \dfrac{1}{i!}\, \eta_{k, \mathbb R^d}^{(i, j, j)}(t_1, t_2) + \dfrac{1}{(i!)^2}\, \nu_{k, \mathbb R^d}^{(i, i, j, j)} (t_1, t_2).\end{equation*}

The main point here is that the Betti number (2.1) and the limit (4.1) are represented in a very similar fashion. In fact, the process $U_{i,j,n}(t)$ in (2.1) and $\mathcal H_k^{(i,j)}(t)$ in (4.1) both capture the spatial distribution of connected components C with $|C| = i$ and $\beta_k(C) = j$ . In particular, $\mathcal H_k^{(k+2,1)}(t)$ represents the distribution of components C on $k+2$ points with $\beta_k(C)=1$ (i.e. components of the smallest size) as does ${\mathcal G}_k(t)$ in the sparse regime. In the present regime, however, many of the Gaussian processes in (4.1) beyond $\mathcal H_k^{(k+2,1)}(t)$ do contribute to the limit.

As a bit of a technical remark, note that for every $i \geq k+2$ , there exists $j_0 \gt 0$ such that $b_{j,t}({{\bf x}})=0$ for all $j \geq j_0$ , $t>0$ , and ${{\bf x}} \in {\mathbb R}^{di}$ . In this case,

\[\eta_{k, \mathbb R^d}^{(i,j,j)}(t,t) = \nu_{k, \mathbb R^d}^{(i,i,j,j)}(t,t) = 0,\]

and thus $\mathcal H_k^{(i,j)}$ becomes an identically zero process. For example, $\mathcal H_k^{(k+2,j)} \equiv 0$ for all $j \geq 2$ , since one cannot create multiple k-cycles from $k+2$ points.

All proofs are collected in Section 6.1. In particular we will see that the growth rate of the expectation and variance of $\beta_{k,n}(t)$ is of order n; see Proposition 6.1. This indicates that the scaling constant for the central limit theorem must be of order $n^{1/2}$ .

Theorem 4.1. Suppose that $ns_n^d = 1$ and f is an almost everywhere bounded and continuous density function. If $0 \lt t_1 \lt t_2 \lt \cdots \lt t_m \lt (e \lVert f \rVert_{\infty} \theta_d)^{-1/d}$ , and $\mathcal H_k(t)$ is the centered Gaussian process defined above, then we have the following weak convergence in the finite-dimensional sense, namely

\[n^{-1/2} (\beta^{}_{k,n}(t) - {\mathbb{E}}[\beta_{k,n}(t)]) \overset{\text{fidi}}{\Rightarrow} \mathcal{H}_k(t).\]

This means that for every $m \in \mathbb{N}$ we have

\begin{equation*} n^{-1/2} ( \beta_{k,n}(t_i) - {\mathbb{E}}[\beta_{k,n}(t_i)], \, i=1,\ldots,m ) \Rightarrow (\mathcal{H}_k(t_i), \, i=1,\ldots,m),\end{equation*}

weakly in ${\mathbb{R}}^m$ .

Remark 4.2. Although Theorem 4.1 imposes a restriction on the range of $t_i$ , the central limit theorem holds for every $t>0$ in the case ofthe ‘truncated’ Betti number

\[\beta_{k,n}^{(M)}(t) = \sum_{i=k+2}^M \sum_{j \gt0} j U_{i,j,n}(t), \quad M \in {\mathbb N},\]

which itself is useful for the approximation arguments in our proof.

The need for the restriction on $t_i$ seems to be a delicate issue. First, according to [Reference Yogeshwaran, Subag and Adler34], if the Poisson process is homogeneous, the non-functional central limit theorem holds for any fixed $t>0$ . However, in the case of an inhomogeneous Poisson process, [Reference Trinh31] has put a restriction on the value of t, despite significant difference in proof techniques with this paper. More specifically, in the notation of Theorem 4.1, the result of [Reference Trinh31] indicates that $t_m$ must be less than $r_c\|\, f \|_\infty^{-1/d}$ , where $r_c$ is the critical radius for percolation in a Boolean model in ${\mathbb{R}}^d$ ([Reference Meester and Roy23]).

The restriction on $t_i$ in Theorem 4.1 looks a little artificial; indeed, it has only been used to demonstrate that

\[\lim_{M\to\infty} \limsup_{n\to\infty} \text{Var}( \beta_{k,n}(t) - \beta_{k,n}^{(M)}(t) )=0.\]

Nevertheless, removing this restriction is not easy, because of the ‘global’ nature of Betti numbers. In fact, the Betti number consists of (infinitely many) component counts as in (2.1), and their moment calculations always involve an entire point set $\mathcal P_n$ (see the definition of $J_{i,t}({\mathcal{Y}}, \mathcal X)$ after (2.2)). This property of Betti numbers seems to make it highly difficult to estimate the probability of large components, unless the radius (i.e. the value of t) is restricted in some way or other. For example, under our restriction ${\mathbb{E}} [\mathcal P_n ( B(x;\, r_n(t)) ) ]$ is bounded by $e^{-1}$ for all $x\in {\mathbb R}^d$ , while it can be as large as possible if t is unrestricted. Furthermore, as mentioned above, if the Betti number is truncated we no longer need to deal with large components, hence we do not need a restriction on the range of t.

Before concluding this section we shall exploit Theorem 4.6 in [Reference Yogeshwaran, Subag and Adler34] and present the strong law of large numbers of $\beta_{k,n}(t)$ . Although the strong law of large numbers has already been proved in [Reference Goel, Trinh and Tsunoda17] and [Reference Trinh30] for any fixed $t>0$ , we shall state the result to highlight the novelty of our representation of the limit as the sum of contributions to the Betti number for variously sized components. The proof is short and is given at the end of Section 6.1.

Corollary 4.2. Under the condition of Theorem 4.1, we assume moreover that f has a compact, convex support such that $\inf_{x \in {supp} (\kern0.7ptf)}f(x)>0$ . Then we have, as $n\to\infty$ ,

\[\dfrac{\beta_{k,n}(t)}{n} \to \sum_{i=k+2}^\infty \sum_{j>0} \dfrac{j}{i!}\, \eta_{k, \mathbb R^d}^{(i,j,j)}(t,t) \quad {a.s.}\]

5. Poisson limit theorem in the sparse regime

Before concluding this paper we shall explore the random topology of Čech complexes when the complex is even more sparse than that in Section 3, so that k-cycles hardly ever occur. Then, the kth Betti number no longer follows a central limit theorem. Nevertheless, it does obey a Poisson limit theorem. In terms of the connectivity radii, we assume $\rho_n = n^{k+2}s_n^{d(k+1)} = 1$ , equivalently $s_n = n^{-(k+2)/d(k+1)}$ , so that $s_n$ converges to 0 more rapidly than in Section 3.

For the definition of a ‘Poissonian’ type limiting process, we let $M_k$ be a Poisson random measure with mean measure $C_{f,k} m_k$ . Namely, it is defined by

\[ M_k(A) \sim {\textrm{Poi}}(C_{f,k} m_{k}(A))\]

for all measurable A in ${\mathbb{R}}^{d(k+1)}$ . Further, if $A_1,\ldots,A_m$ are disjoint, $M_k(A_1), \ldots, M_k(A_m)$ are independent. We are now ready to define the stochastic process

\[\mathcal{V}_k(t) = \int_{{\mathbb{R}}^{d(k+1)}} h_t(0, \mathbf{y}) \, M_k(\text{d}{\mathbf{y}}),\]

which appears below as a weak limit in the main theorem. What is interesting about this is that if we define

\[\mathcal{V}_k^{\pm}(t) \,:\!= \int_{{\mathbb{R}}^{d(k+1)}} h_t^{\pm}(0, \mathbf{y}) \, M_k(\text{d}{\mathbf{y}}),\]

then $\mathcal{V}_k(t) = \mathcal{V}_k^+(t) - \mathcal{V}_k^-(t)$ is the difference of two dependent (time-changed) Poisson processes on ${\mathbb{R}}_+$ . Interestingly, this treatment is analogous to the statement of the Gaussian process limit in Section 3, and those wanting a deeper exploration of this in a similar setting should refer to [Reference Owada25]. What is precisely meant by this can be seen in the following proposition.

Proposition 5.1. The process $\mathcal{V}_k^{\pm}$ can be expressed as

\[(\mathcal{V}^{\pm}_k(t), t \geq 0) \overset{d}{=} (N_k^{\pm}(t^{d(k+1)}), t \geq 0),\]

where $N_k^{\pm}$ is a (homogeneous) Poisson process with intensity $C_{f,k}m_k(D_1^{\pm})$ with $D_t^\pm = \{ {\bf y} \in {\mathbb R}^{d(k+1)}\,\colon h_t^\pm (0,{\bf y}) = 1 \}$ .

Proof. As with Proposition 3.1, we prove only the result for $\mathcal{V}^{+}_k$ , as the proof for $\mathcal{V}^{-}_k$ is the same. We can see that if $0=t_0 \lt t_1 \lt \cdots \lt t_k \lt\infty$ and $\lambda_i \gt0$ , $i=1,\ldots,k$ , then by the non-decreasingness of $h_t^+$ ,

\begin{align*}{\mathbb{E}}\bigg[\! \exp\bigg({-}\sum_{i=1}^k \lambda_i (\mathcal{V}_k^+(t_i) - \mathcal{V}_k^+(t_{i-1}))\bigg)\bigg] = {\mathbb{E}}\bigg[\! \exp\bigg({-}\sum_{i=1}^k \lambda_i M_k (D_{t_i}^+ \setminus D_{t_{i-1}}^+) \bigg)\bigg],\end{align*}

where $D_{t_i}^+ \setminus D_{t_{i-1}}^+$ are disjoint and $M_k (D_{t_i}^+ \setminus D_{t_{i-1}}^+)$ , $i=1,\ldots,k$ , are independent. Moreover, $M_k (D_{t_i}^+ \setminus D_{t_{i-1}}^+)$ is Poisson-distributed with parameter

\[C_{f,k} m_k (D_{t_i}^+ \setminus D_{t_{i-1}}^+) = C_{f,k} m_k (D_1^+) (t_i^{d(k+1)} - t_{i-1}^{d(k+1)})\]

by a change of variable. Hence we have

\[{\mathbb{E}}\bigg[\! \exp\bigg({-}\sum_{i=1}^k \lambda_i M_k (D_{t_i}^+ \setminus D_{t_{i-1}}^+) \bigg)\bigg] = \prod_{i=1}^k \exp ( C_{f,k} m_k (D_1^+) (t_i^{d(k+1)} - t_{i-1}^{d(k+1)}) (e^{-\lambda_i} - 1) ),\]

which implies that the process ${\mathcal V}_k^+(t^{1/d(k+1)})$ has independent increments and

\[{\mathcal V}_k^+((t+s)^{1/d(k+1)}) - {\mathcal V}_k^+(s^{1/d(k+1)})\]

is Poisson with parameter $C_{f,k} m_k (D_1^+)t$ .

In what follows we assume $\rho_n = 1$ , though we could easily modify this to suppose that $\rho_n \to 1$ as $n \to \infty$ . For simplicity in our proofs we assert the former. The proof is again given in Section 6 and the main techniques there are those in [Reference Decreusefond, Schulte and Thäle12].

Theorem 5.1. Suppose that $\rho_n = 1$ and f is an almost everywhere bounded and continuous density function. Then, we have the following weak convergence in the finite-dimensional sense, namely

\[\beta_{k,n}(t) \overset{\text{fidi}}{\Rightarrow} \mathcal{V}_k(t),\]

meaning that for every $m \in \mathbb{N}$ and $0 \lt t_1 \lt t_2 \lt \cdots \lt t_m \lt \infty$ we have

(5.1) \begin{equation}( \beta_{k,n}(t_i), \, i=1,\ldots,m ) \Rightarrow (\mathcal{V}_k(t_i), \, i=1,\ldots,m ),\end{equation}

weakly in ${\mathbb{R}}^m$ .

6. Proofs

In this section we prove the theorems seen in the sections above, with the exposition focused on Sections 4 and 5. We only briefly discuss the proof of Section 3, since the proof is considerably similar to (or even easier than) the critical regime case.

Henceforth, we write $x + {\bf y} = (x + y_1, \ldots, x+y_m)$ for $x\in {\mathbb R}^d$ and ${\bf y} = (y_1,\ldots,y_m) \in {\mathbb R}^{dm}$ .

6.1. Central limit theorem in critical regime

The first step towards the required central limit theorem is to examine the asymptotic moments as follows. Before proceeding with the proof, let us define the ‘truncated’ Betti numbers

(6.1) \begin{equation}\beta_{k,n,A}^{(M)}(t) \,:\!= \sum_{i=k+2}^M \sum_{j>0} j U_{i,j,n,A}(t), \quad M \in {\mathbb N} \cup \{ \infty \}\end{equation}

for any measurable $A \subset {\mathbb R}^d$ . Clearly $\beta_{k,n,A}(t) = \beta_{k,n,A}^{(\infty)}(t)$ .

Let us introduce a few items useful for specifying the limiting covariances. In the following $i, i_1, i_2, j_1$ , and $j_2$ are positive integers, $t_1, t_2$ are non-negative reals, A is an open subset of ${\mathbb R}^d$ with $m(\partial A) = 0$ , and $a \wedge b \,:\!= \min \{ a,b \}$ with $a\vee b \,:\!= \max \{ a,b \}$ . Further, we define the two functions

(6.2) \begin{align}\eta_{k,A}^{(i,j_1,j_2)} (t_1,t_2) &\,:\!= \int_{{\mathbb R}^{d(i-1)}} \int_{{\mathbb R}^d} {{\bf 1}} \{ \skew4\check{C} ( \{ 0,{\bf y} \}, t_1 \wedge t_2 ) \text{ is connected} \} \prod_{\ell=1}^2 b_{j_\ell, t_\ell}(0,{\bf y}) \notag\\[3pt] &\quad\, \times \exp ({-}(t_1 \vee t_2)^d f(x) m ( \mathcal{B} (\{ 0,{\bf y} \};\, 1) ) )\,f(x)^i {{\bf 1}}_A (x)\,\text{d}x\, \text{d}{\bf y},\end{align}

and

(6.3) \begin{align}\nu_{k,A}^{(i_1,i_2,j_1,j_2)}(t_1,t_2) &\,:\!= \int_{{\mathbb R}^d}\text{d}x \int_{{\mathbb R}^{d(i_1-1)}} \text{d}{\bf y}_1 \int_{{\mathbb R}^{di_2}} \text{d}{\bf y}_2 \, {{\bf 1}} \{ \skew4\check{C} ( \{ 0,{\bf y}_1 \}, t_1 ) \text{ is connected} \}\, \notag \\[4pt] &\quad\, \times {{\bf 1}} \{ \skew4\check{C} ( {\bf y}_2 , t_2 ) \text{ is connected} \} b_{j_1,t_1}(0,{\bf y}_1)\, b_{j_2, t_2}({\bf y}_2) \notag \\[4pt] &\quad\, \times [ ( \alpha_{t_1,t_2} ( \{ 0,{\bf y}_1 \}, {\bf y}_2 ) - \alpha_{(t_1 \vee t_2) / 2} ( \{ 0,{\bf y}_1 \}, {\bf y}_2 ) )e^{-f(x) m ( \mathcal{B}( \{ 0,{\bf y}_1 \};\, t_1 ) \cup \mathcal{B}({\bf y}_2;\, t_2) )} \notag \\[4pt] &\quad\, \quad\, - \alpha_{t_1,t_2} ( \{ 0,{\bf y}_1 \}, {\bf y}_2 ) e^{-f(x) \{ m( \mathcal{B}(\{0,{\bf y}_1 \};\, t_1) ) + m( \mathcal{B}({\bf y}_2;\, t_2) ) \}}]\, f(x)^{i_1 + i_2} {{\bf 1}}_A(x),\end{align}

where

(6.4) \begin{equation}\mathcal{B}(\mathcal X;\, r) \,:\!= \bigcup_{y \in \mathcal X} B(y;\, r)\end{equation}

for a collection $\mathcal X$ of ${\mathbb R}^d$ -valued vectors and $r>0$ . Moreover,

\[\alpha_{r,s} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}})\,:\!= {{\bf 1}} \{ \mathcal{B}({\mathcal X_{i_1}};\, r)\cap \mathcal{B}({\mathcal X_{i_2}};\, s) \neq \emptyset \},\]

and $\alpha_r({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) \,:\!= \alpha_{r,r}({\mathcal X_{i_1}}, {\mathcal X_{i_2}})$ . Finally, for $M \in {\mathbb N} \cup \{ \infty \}$ we define

\[\Phi_{k,A}^{(M)} (t_1,t_2) \,:\!= \sum_{i_1=k+2}^M \sum_{i_2=k+2}^M \sum_{j_1 \gt 0} \sum_{j_2 \gt 0} j_1\, j_2\, \bigg( \dfrac{\eta_{k, \mathbb R^d}^{(i_1, j_1, j_2)}(t_1, t_2) \delta_{i_1, i_2}}{i_1!}+ \dfrac{\nu_{k, \mathbb R^d}^{(i_1, i_2, j_1, j_2)}(t_1, t_2)}{i_1! i_2!} \bigg),\]

where $\delta_{i_1, i_2}$ is again the Kronecker delta and we define $\Phi_{k,A} (t_1,t_2) \,:\!= \Phi_{k,A}^{(\infty)}(t_1,t_2)$ .

Proposition 6.1. Let f be an almost everywhere bounded and continuous density function. Let $ns_n^d = 1$ , and $A \subset \mathbb{R}^d$ is open with $m(\partial A) = 0$ .

1. (i) If $M \lt \infty$ , then for $t, t_1,t_2>0$ ,

   \begin{align*}& n^{-1} \, {\mathbb{E}}[\beta_{k,n,A}^{(M)}(t)] \to \sum_{i=k+2}^M \sum_{j>0} \dfrac{j}{i!}\, \eta^{(i,j,j)}_{k,A}(t, t), \quad n \to \infty,\\[3pt] & n^{-1}{\textrm{Cov}}(\beta_{k,n,A}^{(M)}(t_1), \beta_{k,n,A}^{(M)}(t_2)) \to \Phi_{k, A}^{(M)}(t_1, t_2), \quad n \to\infty.\end{align*}

2. (ii) If $M = \infty$ , then for $0 \lt t, t_1, t_2 \lt ( e \|\, f \|_\infty \theta_d )^{-1/d}$ ,

   \begin{align*}& n^{-1} \, {\mathbb{E}}[\beta_{k,n,A}(t)] \to \sum_{i=k+2}^{\infty} \sum_{j>0} \dfrac{j}{i!}\, \eta^{(i,j,j)}_{k,A}(t, t), \quad n \to \infty,\\[3pt] & n^{-1}{\textrm{Cov}}(\beta_{k,n,A}(t_1), \beta_{k,n,A}(t_2)) \to \Phi_{k, A}(t_1, t_2), \quad n \to\infty,\end{align*}
   so that the limits above are finite non-zero constants.

Proof. We only establish the statements in (ii). We aim to demonstrate the convergence of the expectation in Part 1 and then in Part 2, the convergence of the covariance to $\Phi_{k,A}(t_1, t_2)$ . For ease of description we treat only the case when $A = {\mathbb{R}}^d$ . The argument for a general A will be the same except for obvious minor changes.

Part 1. The definition in (2.1), the Palm theory for Poisson processes in [Reference Penrose26], and the monotone convergence theorem provide

(6.5) \begin{equation}n^{-1} \, {\mathbb{E}}[\beta_{k,n}(t)] = \sum_{i=k+2}^{\infty} \sum_{j>0} j\, \dfrac{n^{i-1}}{i!}\, {\mathbb{E}}[{g_{r_n(t)}^{(i,j)}}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n)],\end{equation}

where ${\mathcal{X}_{i}} = (X_1,\ldots,X_i) \in {\mathbb R}^{di}$ is a collection of i.i.d. random points in ${\mathbb R}^d$ with common density f. By conditioning on ${\mathcal{X}_{i}}$ , we have

(6.6) \begin{align}&n^{i-1} \, {\mathbb{E}}[{g_{r_n(t)}^{(i,j)}}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n)] \notag \\[3pt] &\quad = n^{i-1} \, {\mathbb{E}} [ b_{j,r_n(t)}({\mathcal{X}_{i}}) \, {\mathbb{E}} [ J_{i,r_n(t)}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n)\mid {\mathcal{X}_{i}} ] ] \notag \\[3pt] &\quad = n^{i-1} \int_{{\mathbb R}^{di}} {{\bf 1}} \{ \skew4\check{C} ({{\bf x}}, r_n(t)) \text{ is connected} \}\, b_{j,r_n(t)} ({{\bf x}}) \exp ({-}nI_{r_n(t)} ({{\bf x}})) \prod_{j=1}^i f(x_j) \text{d}{{\bf x}},\end{align}

where

\[I_{r_n(t)}({{\bf x}}) = I_{r_n(t)}(x_1,\ldots,x_i) = \int_{\mathcal{B}({{\bf x}};\, r_n(t))} f(z)\, \text{d}{z}.\]

Subsequently we perform the change of variables $x_1 = x$ and $x_j = x + s_ny_{j-1}$ for $j=2,\ldots, i$ , to find that (6.6) is equal to

\begin{align*}&(ns_n^d)^{i-1} \int_{{\mathbb R}^{d(i-1)}}\int_{{\mathbb R}^{d}} {{\bf 1}} \{ \skew4\check{C} (\{ x,x+s_n{\bf y} \}, r_n(t)) \text{ is connected}\} b_{j,r_n(t)}(x,x+s_n {\bf y}) \\[3pt] &\quad\quad\, \times \exp ({-}nI_{r_n(t)}(x,x+s_n{\bf y}) ) f(x) \prod_{j=1}^{i-1} f(x+s_n y_j)\, \text{d}x\, \text{d}{\bf y} \\[3pt] &\quad = \int_{{\mathbb R}^{d(i-1)}}\int_{{\mathbb R}^{d}} {{\bf 1}} \{ \skew4\check{C} (\{ 0,{\bf y} \}, t) \text{ is connected}\} b_{j,t}(0,{\bf y}) \\[3pt] &\quad\quad\, \times \exp ({-}nI_{r_n(t)}(x,x+s_n{\bf y}) ) f(x) \prod_{j=1}^{i-1} f(x+s_n y_j)\, \text{d}x\, \text{d}{\bf y},\end{align*}

where the equality follows from the location and scale invariance of both of the indicator functions. By the continuity of f we have $\prod_{j=1}^{i-1} f(x+s_n y_j) \to f(x)^{i-1}$ a.e. as $n\to\infty$ . As for the convergence of the exponential term, we have

\[nI_{r_n(t)}(x, x + s_n{\bf y}) = n\int_{\mathcal{B}(\{ x, x+s_n {\bf y} \};\, r_n(t))} f(z)\, \text{d}z,\]

which after the change of variable $z = x + s_n v$ gives us

\[n\int_{\mathcal{B}(\{ x, x+s_n {\bf y} \};\, r_n(t))} f(z)\, \text{d}z \to t^d f(x) m ( \mathcal{B}( \{ 0,{\bf y} \};\, 1 ) ).\]

It then follows from the dominated convergence theorem that

\[n^{i-1} \, {\mathbb{E}}[{g_{r_n(t)}^{(i,j)}}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n)] \to \eta_{k, \mathbb R^d}^{(i,j,j)}(t,t), \quad n\to\infty.\]

It remains to find a summable upper bound for (6.5) to apply the dominated convergence theorem for sums. To this end we use the inequality $j \leq \binom{i}{k+1}$ , which is the result of the fact that there must be a k-simplex in $\skew4\check{C} (\mathcal X_i, r_n(t))$ whenever $\beta_k ( \skew4\check{C} (\mathcal X_i, r_n(t)) )>0$ . In addition, using an obvious inequality

(6.7) \begin{equation}J_{i,r_n(t)}(\mathcal X_i, \mathcal X_i \cup \mathcal{P}_n) \leq {{\bf 1}} \{ {\skew4\check{C}} (\mathcal X_i, r_n(t)) \text{ is connected} \},\end{equation}

we obtain

(6.8) \begin{align}n^{-1} \, {\mathbb{E}} [\beta_{k,n}(t)] &\leq \sum_{i=k+2}^\infty \binom{i}{k+1} \dfrac{n^{i-1}}{i!} \sum_{j=1}^{\binom{i}{k+1}} \, {\mathbb{E}} [ {{\bf 1}} \{ {\skew4\check{C}} (\mathcal X_i, r_n(t)) \text{ is connected} \}\, b_{j,r_n(t)} (\mathcal X_i) ]\notag \\[3pt] &\leq \sum_{i=k+2}^\infty \binom{i}{k+1} \dfrac{n^{i-1}}{i!}\, {\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_i, r_n(t)) \text{ is connected} ).\end{align}

For further analysis we claim that

(6.9) \begin{equation}{\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_i, r_n(t)) \text{ is connected} ) \leq i^{i-2} ( r_n(t)^d \|\, f \|_\infty \theta_d )^{i-1}.\end{equation}

Indeed, this can be derived from

(6.10) \begin{align}&{\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_i, r_n(t)) \text{ is connected} ) \notag \\[3pt] &\quad = \int_{{\mathbb R}^{di}} {{\bf 1}} \{ {\skew4\check{C}} ({{\bf x}}, r_n(t)) \text{ is connected} \} \prod_{j=1}^i f(x_j) \text{d}{{\bf x}} \notag \\[3pt] &\quad = r_n(t)^{d(i-1)} \int_{{\mathbb R}^{di}} {{\bf 1}} \{ {\skew4\check{C}} (\{ 0,{\bf y} \}, 1) \text{ is connected} \}\, f(x) \prod_{j=1}^{i-1} f(x+r_n(t) y_j)\, \text{d} x\, \text{d}{\bf y} \notag \\[3pt] &\quad \leq (r_n(t)^d \|\, f \|_\infty )^{i-1} \int_{{\mathbb R}^{d(i-1)}} {{\bf 1}} \{ {\skew4\check{C}} (\{ 0,{\bf y} \}, 1) \text{ is connected} \} \text{d}{\bf y} \notag \\[3pt] &\quad \leq i^{i-2} ( r_n(t)^d \|\, f \|_\infty \theta_d )^{i-1}.\end{align}

The last inequality comes from the basic fact that there are $i^{i-2}$ spanning trees on i vertices. Combining (6.8), (6.9), and $ns_n^d=1$ we conclude that

\begin{align*}n^{-1} \, {\mathbb{E}} [\beta_{k,n}(t)] &\leq \dfrac{1}{(k+1)!}\sum_{i=k+2}^{\infty} \dfrac{i^{i-2}}{(i-k-1)!} (t^d \|\, f \|_{\infty} \theta_d)^{i-1} =\!:\, \dfrac{1}{(k+1)!}\sum_{i=k+2}^{\infty}a_i.\end{align*}

It is easy to check that $a_{i+1}/a_i \to et^d \|\, f \|_\infty \theta_d$ as $i\to\infty$ , where the limit is less than 1 by our assumption. So the ratio test has shown that $\sum_{i=k+2}^\infty a_i$ converges as required.

Part 2. We assume $0 \lt t_1 \leq t_2 \lt (e \lVert f \rVert_{\infty} \theta_d)^{-1/d}$ and proceed with the fact that

\begin{align*} &{\mathbb{E}}[\beta_{k,n}(t_1)\beta_{k,n}(t_2)] \\[3pt] &\quad = \sum_{i_1=k+2}^\infty \sum_{i_2=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\, j_2 \,{\mathbb{E}}\bigg[ \sum_{{\mathcal{Y}}_1 \subset \mathcal{P}_n} \sum_{{\mathcal{Y}}_2 \subset \mathcal{P}_n} {g_{r_n(t_1)}^{(i_1, j_1)}}({\mathcal{Y}}_1, \mathcal{P}_n)\, {g_{r_n(t_2)}^{(i_2, j_2)}}({\mathcal{Y}}_2, \mathcal{P}_n) \bigg] \\[3pt] &\quad = \sum_{i=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\, j_2 \, {\mathbb{E}}\bigg[ \sum_{{\mathcal{Y}} \subset \mathcal{P}_n} g_{r_n(t_1)}^{(i,j_1)}({\mathcal{Y}}, \mathcal{P}_n)\, g_{r_n(t_2)}^{(i,j_2)}({\mathcal{Y}}, \mathcal{P}_n) \bigg] \\[3pt] &\quad\quad\, + \sum_{i_1=k+2}^\infty \sum_{i_2=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\, j_2\, \\[3pt] &\quad\qquad\,\, \times{\mathbb{E}} \bigg[ \sum_{{\mathcal{Y}}_1 \subset \mathcal{P}_n} \sum_{{\mathcal{Y}}_2 \subset \mathcal{P}_n} {g_{r_n(t_1)}^{(i_1, j_1)}}({\mathcal{Y}}_1, \mathcal{P}_n)\, {g_{r_n(t_2)}^{(i_2, j_2)}}({\mathcal{Y}}_2, \mathcal{P}_n) {{\bf 1}} \{ |{\mathcal{Y}}_1 \cap {\mathcal{Y}}_2| = 0\} \bigg].\end{align*}

The second equality comes from an observation that if ${\mathcal{Y}}_1 \neq {\mathcal{Y}}_2$ and the intersection of ${\mathcal{Y}}_1$ and ${\mathcal{Y}}_2$ is non-empty, then $\skew4\check{C} ( {\mathcal{Y}}_2, r_n(t_2))$ cannot be an isolated component of $\skew4\check{C}(\mathcal{P}_n, r_n(t_2))$ , so these terms are zero. Appealing to Palm theory again, as seen in [Reference Kahle and Meckes21], we obtain

\begin{align*}&{\mathbb{E}}[\beta_{k,n}(t_1)\beta_{k,n}(t_2)] \\[3pt] & \quad = \sum_{i=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\, j_2 \dfrac{n^i}{i!}\, {\mathbb{E}}\big[ g_{r_n(t_1)}^{(i,j_1)}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n)\, g_{r_n(t_2)}^{(i,j_2)}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n) \big] \\[3pt] &\quad\quad\, + \sum_{i_1=k+2}^\infty \sum_{i_2=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\, j_2\, \dfrac{n^{i_1 + i_2}}{i_1! i_2!}\, \\[3pt] &\quad\quad\,\quad \, \times {\mathbb{E}} \big[ {g_{r_n(t_1)}^{(i_1, j_1)}}({\mathcal X_{i_1}}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n)\, {g_{r_n(t_2)}^{(i_2, j_2)}}({\mathcal X_{i_2}}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n) \big],\end{align*}

where ${\mathcal{X}_{i}}$ and $\mathcal{P}_n$ are independent, and ${\mathcal X_{i_1}}$ , ${\mathcal X_{i_2}}$ , and $\mathcal{P}_n$ are also mutually independent such that ${\mathcal X_{i_1}}$ and ${\mathcal X_{i_2}}$ are disjoint.

Applying (6.5) to each ${\mathbb{E}}[\beta_{k,n}(t_i)]$ , $i=1,2$ , and utilizing the independence of ${\mathcal X_{i_1}}$ and ${\mathcal X_{i_2}}$ , we see that the covariance function can be written as

(6.11) \begin{equation}{\textrm{Cov}}(\beta_{k, n}(t_1), \beta_{k,n}(t_2)) = A_{1,n} + A_{2,n},\end{equation}

with

(6.12) \begin{align}A_{1,n} \,:\!= \sum_{i=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\,j_2\dfrac{n^i}{i!}\, {\mathbb{E}} \big[ g_{r_n(t_1)}^{(i, j_1)} (\mathcal X_i, \mathcal X_i \cup \mathcal{P}_n) g_{r_n(t_2)}^{(i, j_2)} (\mathcal X_i, \mathcal X_i \cup \mathcal{P}_n) \big],\\[-35pt]\nonumber \end{align}

(6.13) \begin{align} A_{2,n} &\,:\!= \sum_{i_1 = k+2}^{\infty} \sum_{i_2 = k+2}^{\infty} \sum_{j_1 \gt0} \sum_{j_2 \gt0} j_1\, j_2\, \dfrac{n^{i_1 + i_2}}{i_1! i_2!} \nonumber \\[3pt] &\quad\,\times {\mathbb{E}}\big[{g_{r_n(t_1)}^{(i_1, j_1)}}({\mathcal X_{i_1}}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n){g_{r_n(t_2)}^{(i_2, j_2)}}({\mathcal X_{i_2}}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n) \nonumber\\[3pt] &\quad\quad\,\phantom{\mathbb{E}} - {g_{r_n(t_1)}^{(i_1, j_1)}}({\mathcal X_{i_1}}, {\mathcal X_{i_1}} \cup \mathcal{P}_n){g_{r_n(t_2)}^{(i_2, j_2)}}({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P'}_{\!\!n})\big],\end{align}

where $\mathcal{P'}_{\!\!n}$ is an independent copy of $\mathcal{P}_n$ and is also independent of ${\mathcal X_{i_1}}$ and ${\mathcal X_{i_2}}$ .

Let us denote the expectation portions of $A_{1,n}$ and $A_{2,n}$ as ${E_{1,n}^{(i,\mathbf j)}}$ and ${E_{2,n}^{(\mathbf i,\mathbf j)}}$ , with $\mathbf{i} = (i_1,i_2)$ , and $\mathbf{j} = (j_1,j_2)$ respectively. Our goal is to show that $n^{-1} (A_{1,n} + A_{2,n})$ tends to $\Phi_{k,{\mathbb R}^d}(t_1,t_2)$ as $n\to\infty$ . For now we shall compute the limits of $n^{i-1} {E_{1,n}^{(i,\mathbf j)}}$ and $n^{i_1+i_2-1} {E_{2,n}^{(\mathbf i,\mathbf j)}}$ for each $i, i_1, i_2, j_1$ , and $j_2$ , while temporarily assuming that the dominated convergence theorem for sums is applicable for both $n^{-1}A_{1,n}$ and $n^{-1}A_{2,n}$ . By mirroring the argument from Part 1 with the same change of variables and recalling $t_1 \leq t_2$ ,

\begin{align*}n^{i-1} {E_{1,n}^{(i,\mathbf j)}} &= n^{i-1} \, {\mathbb{E}} \bigg[ {{\bf 1}} \{ \skew4\check{C} ({\mathcal{X}_{i}}, r_n(t_1)) \text{ is connected} \} \prod_{\ell=1}^2 b_{j_\ell, r_n(t_\ell)} ({\mathcal{X}_{i}}) \exp ({-}nI_{r_n(t_2)} ({\mathcal{X}_{i}}) )\bigg] \\&=\int_{{\mathbb R}^{d(i-1)}}\int_{{\mathbb R}^d} {{\bf 1}} \{ \skew4\check{C} ( \{ 0,{\bf y} \}, t_1 ) \text{ is connected} \}\, \prod_{\ell=1}^2 b_{j_\ell, t_\ell} (0,{\bf y}) \\&\quad\, \times \exp ({-}nI_{r_n(t_2)}(x,x + s_n {\bf y}) )\,f(x) \prod_{j=1}^{i-1} f(x+s_n y_j)\, \text{d}x\, \text{d}{\bf y} \\*&\to \eta_{k, \mathbb R^d}^{(i,j_1,j_2)}(t_1, t_2) \quad \text{as } n\to\infty.\end{align*}

Hence the assumed dominated convergence theorem for sums concludes that

(6.14) \begin{equation}n^{-1}A_{1,n} \to \sum_{i=k+2}^\infty \sum_{j_1 \gt0}\sum_{j_2 \gt0} \dfrac{j_1\,j_2}{i!}\, \eta_{k, \mathbb R^d}^{(i, j_1, j_2)}(t_1, t_2)\, \quad n \to \infty.\end{equation}

To demonstrate convergence for $n^{i_1+i_2-1} {E_{2,n}^{(\mathbf i,\mathbf j)}}$ , let us shorten ${g_{r_n(t_1)}^{(i_1, j_1)}}$ to $g_1$ and ${g_{r_n(t_2)}^{(i_2, j_2)}}$ to $g_2$ and decompose ${E_{2,n}^{(\mathbf i,\mathbf j)}}$ into two terms:

\begin{align*}{E_{2,n}^{(\mathbf i,\mathbf j)}} &= {\mathbb{E}} [ g_1({\mathcal X_{i_1}}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n) g_2({\mathcal X_{i_2}}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n) \\[3pt] &\quad\,\phantom{\mathbb{E}} - g_1({\mathcal X_{i_1}}, {\mathcal X_{i_1}} \cup \mathcal{P}_n) g_2({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}_n) ] \notag \\[3pt] &\quad\, + {\mathbb{E}}[g_1({\mathcal X_{i_1}}, {\mathcal X_{i_1}} \cup \mathcal{P}_n)( g_2({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}_n) - g_2({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}^{\prime}_n))] \notag \\[3pt] &\,:\!= B_{1,n} + B_{2,n}. \notag\end{align*}

Note that for $\ell=1,2,$

\[g_\ell (\mathcal X_{i_\ell}, {\mathcal X_{i_1} \cup \mathcal X_{i_2}} \cup \mathcal{P}_n) = g_\ell (\mathcal X_{i_\ell}, \mathcal X_{i_\ell} \cup \mathcal{P}_n)\, {{\bf 1}} \{ \mathcal{B}({\mathcal X_{i_1}};\, r_n(t_\ell)/2)\cap \mathcal{B}({\mathcal X_{i_2}};\, r_n(t_\ell)/2) = \emptyset \},\]

where $\mathcal{B}(\mathcal X;\, r)$ is defined in 6.4. Hence we have

\begin{equation*}B_{1,n} = - {\mathbb{E}} [ g_1({\mathcal X_{i_1}},{\mathcal X_{i_1}} \cup \mathcal{P}_n)\, g_2({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}_n)\, \alpha_{r_n(t_2)/2} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) ].\end{equation*}

At the same time, the spatial independence of $\mathcal{P}_n$ justifies that

\begin{align*}B_{2,n} &= {\mathbb{E}}[g_1({\mathcal X_{i_1}}, {\mathcal X_{i_1}} \cup \mathcal{P}_n)(g_2({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}_n) - g_2({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}^{\prime}_n))\, \alpha_{r_n(t_1), r_n(t_2)} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}})].\end{align*}

Consequently, we can rewrite ${E_{2,n}^{(\mathbf i,\mathbf j)}}$ as

(6.15) \begin{align}{E_{2,n}^{(\mathbf i,\mathbf j)}}&= {\mathbb{E}} [ g_1 ({\mathcal X_{i_1}}, {\mathcal X_{i_1}} \cup \mathcal{P}_n) g_2 ({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P}_n) \, ( \alpha_{r_n(t_1), r_n(t_2)}({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) - \alpha_{r_n(t_2)/2}({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) ) ] \notag \\[3pt] &\quad\, - {\mathbb{E}} [ g_1 ({\mathcal X_{i_1}}, {\mathcal X_{i_1}} \cup \mathcal{P}_n) g_2 ({\mathcal X_{i_2}}, {\mathcal X_{i_2}} \cup \mathcal{P'}_{\!\!n}) \alpha_{r_n(t_1), r_n(t_2)} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) ] \notag \\[3pt] &\,:\!= C_{1,n} - C_{2,n}.\end{align}

After conditioning on ${\mathcal X_{i_1} \cup \mathcal X_{i_2}}$ , the customary change of variable yields

\begin{align*}n^{i_1 + i_2 - 1}C_{1,n} &= n^{i_1 + i_2 - 1} \, {\mathbb{E}} \bigg[ \prod_{\ell=1}^2 {{\bf 1}} \{ \skew4\check{C} (\mathcal X_{i_\ell}, r_n(t_\ell)) \text{ is connected} \}\, b_{j_\ell, r_n(t_\ell)} (\mathcal X_{i_\ell}) \\[3pt] &\quad\, \times ( \alpha_{r_n(t_1), r_n(t_2)}({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) - \alpha_{r_n(t_2)/2} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) ) \\[3pt] &\quad\, \times \exp \bigg({-}n\int_{\mathcal{B}({\mathcal X_{i_1}};\, r_n(t_1)) \cup \mathcal{B}({\mathcal X_{i_2}};\, r_n(t_2))} f(z)\,\text{d}z \bigg)\bigg] \\[3pt] &= \int_{{\mathbb R}^d} \text{d}x \int_{{\mathbb R}^{d(i_1 -1)}} \text{d}{\bf y}_1 \int_{{\mathbb R}^{di_2}} \text{d}{\bf y}_2 \, {{\bf 1}} \{ \skew4\check{C} ( \{ 0,{\bf y}_1 \}, t_1 ) \text{ is connected} \}\, \\[3pt] &\quad\, \times {{\bf 1}} \{ \skew4\check{C} ( {\bf y}_2, t_2 ) \text{ is connected} \} b_{j_1,t_1}(0,{\bf y}_1)\, b_{j_2, t_2}({\bf y}_2) \\[3pt] &\quad\, \times ( \alpha_{t_1,t_2} (\{ 0,{\bf y}_1 \}, {\bf y}_2) - \alpha_{t_2/2} (\{0,{\bf y}_1 \}, {\bf y}_2) ) \\[3pt] &\quad\, \times \exp \bigg({-}n\int_{\mathcal{B}(\{x,x+s_n {\bf y}_1 \};\, r_n(t_1)) \cup \mathcal{B}(x + s_n {\bf y}_2;\, r_n(t_2)) }f(z)\,\text{d}z \bigg) \\[3pt] &\quad\, \times f(x) \prod_{j=1}^{i_1-1} f(x + s_n y_{1,j})\prod_{j=1}^{i_2} f(x + s_n y_{2,j}) \\[3pt] &\to\int_{{\mathbb R}^d} \text{d}x \int_{{\mathbb R}^{d(i_1 -1)}} \text{d}{\bf y}_1 \int_{{\mathbb R}^{di_2}} \text{d}{\bf y}_2 \, {{\bf 1}} \{ \skew4\check{C} ( \{ 0,{\bf y}_1 \}, t_1 ) \text{ is connected} \} \\[3pt] &\quad\, \times {{\bf 1}} \{ \skew4\check{C} ( {\bf y}_2, t_2 ) \text{ is connected} \} b_{j_1,t_1}(0,{\bf y}_1)\, b_{j_2, t_2}({\bf y}_2) \\[3pt] &\quad\, \times ( \alpha_{t_1,t_2} (\{ 0,{\bf y}_1 \}, {\bf y}_2 ) - \alpha_{t_2/2} (\{0,{\bf y}_1 \}, {\bf y}_2) ) \\[3pt] &\quad\, \times e^{-f(x) m ( \mathcal{B}(\{ 0,{\bf y}_1 \};\, t_1) \cup \mathcal{B}({\bf y}_2;\, t_2) )} f(x)^{i_1 + i_2},\end{align*}

where ${\bf y}_1 = (y_{1,1}, \ldots, y_{1,i_1-1}) \in {\mathbb R}^{d(i_1-1)}$ and ${\bf y}_2 = (y_{2,1}, \ldots, y_{2,i_2}) \in {\mathbb R}^{di_2}$ .

Similarly, one can see that

\begin{align*}n^{i_1 + i_2 - 1}C_{2,n} &\to \int_{{\mathbb R}^d} \text{d}x \int_{{\mathbb R}^{d(i_1 -1)}} \text{d}{\bf y}_1 \int_{{\mathbb R}^{di_2}} \text{d}{\bf y}_2 \, {{\bf 1}} \{ \skew4\check{C} ( \{ 0,{\bf y}_1 \}, t_1 ) \text{ is connected} \} \\[4pt] &\quad\, \times {{\bf 1}} \{ \skew4\check{C} ( {\bf y}_2, t_2 ) \text{ is connected} \} b_{j_1,t_1}(0,{\bf y}_1)\, b_{j_2, t_2}({\bf y}_2) \alpha_{t_1, t_2} ( \{ 0,{\bf y}_1 \}, {\bf y}_2 )\\[4pt] &\quad\, \times e^{-f(x) \{ m(\mathcal{B}(\{ 0,{\bf y}_1 \};\, t_1)) + m (\mathcal{B}({\bf y}_2;\, t_2)) \}} f(x)^{i_1 + i_2}.\end{align*}

Therefore,

\[n^{i_1 + i_2 - 1} {E_{2,n}^{(\mathbf i,\mathbf j)}} = n^{i_1 + i_2 - 1}( C_{1,n} - C_{2,n} ) \to \nu_{k, \mathbb R^d}^{(i_1,i_2,j_1,j_2)} (t_1,t_2), \quad n\to\infty.\]

Assuming convergence under summation, we have

(6.16) \begin{equation}n^{-1}A_{2,n} \to \sum_{i_1=k+2}^\infty \sum_{i_2 = k+2}^\infty \sum_{j_1>0} \sum_{j_2 \gt 0} \dfrac{j_1\, j_2}{i_1! i_2!}\, \nu_{k, \mathbb R^d}^{(i_1, i_2, j_1, j_2)}(t_1, t_2), \quad n\to\infty.\end{equation}

From (6.14) and (6.16), it follows that $n^{-1} (A_{1,n} + A_{2,n}) \to \Phi_{k,{\mathbb R}^d}(t_1, t_2)$ as $n\to\infty$ .

Now we would like to show that both $n^{i-1} {E_{1,n}^{(i,\mathbf j)}}$ and $n^{i_1 + i_2-1}| {E_{2,n}^{(\mathbf i,\mathbf j)}}|$ are bounded by a summable quantity, so that application of the dominated convergence theorem for sums is valid for both $n^{-1} A_{1,n}$ and $n^{-1} A_{2,n}$ . Using the bounds (6.7), (6.9) together with $ns_n^d=1$ , we have

(6.17) \begin{align}n^{-1} A_{1,n} &\leq \sum_{i=k+2}^\infty \sum_{j_1>0} \sum_{j_2>0}j_1\, j_2\, \dfrac{n^{i-1}}{i!}\, {\mathbb{E}}\bigg[ {{\bf 1}} \{ {\skew4\check{C}} (\mathcal{X}_i, r_n(t_1)) \text{ is connected} \} \prod_{\ell=1}^{2} b_{j_{\ell}, r_n(t_{\ell})}(\mathcal{X}_i) \bigg] \notag \\[3pt] &\leq \sum_{i = k+2}^{\infty} \binom{i}{k+1}^2 \dfrac{n^{i-1}}{i!}\, {\mathbb{P}}( {\skew4\check{C}} (\mathcal X_{i}, r_n(t_1)) \text{ is connected} ) \notag \\[3pt] &\leq \dfrac{1}{((k+1)!)^2} \sum_{i=k+2}^\infty \dfrac{i! i^{i-2}}{( (i-k-1)! )^2}\, ( t_1^d \|\, f \|_\infty \theta_d )^{i-1}.\end{align}

The last term is convergent by appealing to the assumption $t_1 \lt (e \|\, f \|_\infty \theta_d)^{-1/d}$ and the ratio test for sums.

Subsequently we turn our attention to $n^{-1}A_{2,n}$ . Returning to (6.15) and using obvious relations

\[\alpha_{r_n(t_1), r_n(t_2)} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) \leq \alpha_{r_n(t_2)} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}), \quad \alpha_{r_n(t_2)/2} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) \leq \alpha_{r_n(t_2)} ({\mathcal X_{i_1}}, {\mathcal X_{i_2}}),\]

we obtain

\[|C_{1,n} - C_{2,n}| \leq 3 \, {\mathbb{E}} \bigg[ \prod_{\ell=1}^2 {{\bf 1}} \{ \skew4\check{C} (\mathcal X_{i_\ell}, r_n(t_2)) \text{ is connected} \}\, b_{j_\ell, r_n(t_\ell)} (\mathcal X_{i_\ell})\, \alpha_{r_n(t_2)}({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) \bigg]{.}\]

By virtue of this bound we have

(6.18) \begin{align}n^{-1} |A_{2,n}| &\leq 3 \sum_{i_1 = k+2}^{\infty} \sum_{i_2 = k+2}^{\infty} \sum_{j_1>0} \sum_{j_2>0} j_1\, j_2\, \dfrac{n^{i_1 + i_2 - 1}}{i_1! i_2!} \notag \\[3pt] &\quad\, \times {\mathbb{E}} \bigg[ \prod_{\ell=1}^2 {{\bf 1}} \{ \skew4\check{C} (\mathcal X_{i_\ell}, r_n(t_2)) \text{ is connected} \}\, b_{j_\ell, r_n(t_\ell)} (\mathcal X_{i_\ell})\, \alpha_{r_n(t_2)}({\mathcal X_{i_1}}, {\mathcal X_{i_2}}) \bigg] \notag \\[3pt] &\leq 3 \sum_{i_1 = k+2}^{\infty} \sum_{i_2 = k+2}^{\infty} \binom{i_1}{k+1} \binom{i_2}{k+1} \, \dfrac{n^{i_1 + i_2 - 1}}{i_1! i_2!} \notag \\[3pt] &\quad\,\times {\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_{i_\ell}, r_n(t_2)) \text{ is connected for } \ell=1,2,\mathcal{B}({\mathcal X_{i_1}};\, r_n(t_2)) \cap \mathcal{B}({\mathcal X_{i_2}};\, r_n(t_2)) \neq \emptyset ).\end{align}

We claim here that

(6.19) \begin{align}&{\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_{i_\ell}, r_n(t_2)) \text{ is connected for } \ell=1,2, \ \mathcal{B}({\mathcal X_{i_1}};\, r_n(t_2)) \cap \mathcal{B}({\mathcal X_{i_2}};\, r_n(t_2)) \neq \emptyset ) \notag\\[3pt] &\quad \leq 2^d i_1^{i_1-1} i_2^{i_2-1} ( r_n(t_2)^d \|\, f \|_\infty \theta_d )^{i_1+i_2-1}.\end{align}

To see this, by the change of variables as in (6.10), we have

\begin{align*}&{\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_{i_\ell}, r_n(t_2)) \text{ is connected for } \ell=1,2, \ \mathcal{B}({\mathcal X_{i_1}};\, r_n(t_2)) \cap \mathcal{B}({\mathcal X_{i_2}};\, r_n(t_2)) \neq \emptyset ) \\*&\quad \leq ( r_n(t_2)^d \|\, f \|_\infty )^{i_1+i_2-1} \int_{{\mathbb R}^{d(i_1+i_2-1)}} {{\bf 1}} \{ {\skew4\check{C}} (\{0,y_1,\ldots,y_{i_1-1} \}, 1) \text{ is connected} \} \\&\quad\quad\, \times {{\bf 1}} \{ {\skew4\check{C}} (\{y_{i_1},\ldots,y_{i_1+i_2-1} \}, 1) \text{ is connected} \} \\*&\quad\quad\, \times {{\bf 1}} \{ \mathcal{B}(\{ 0,y_1,\ldots,y_{i_1-1} \};\, 1) \cap \mathcal{B}(\{ y_{i_1},\ldots,y_{i_1+i_2-1} \};\, 1) \neq \emptyset \} \text{d}{\bf y}.\end{align*}

Note that there are $i_1^{i_1-2}$ spanning trees on the set of points $\{ 0,y_1,\ldots,y_{i_1-1} \}$ with unit connectivity radius, and there are $i_2^{i_2-2}$ spanning trees on $\{ y_{i_1}, \ldots, y_{i_1 + i_2 -1} \}$ with unit connectivity radius as well. In addition there are $i_1 \times i_2$ possible ways of picking one vertex from $\{ 0,y_1,\ldots,y_{i_1-1} \}$ and another from $\{ y_{i_1}, \ldots, y_{i_1+i_2-1} \}$ , and connecting the two chosen vertices with connectivity radius 2. Therefore, the expression above is eventually bounded by

\begin{align*}&( r_n(t_2)^d \|\, f \|_\infty )^{i_1+i_2-1} i_1^{i_1-2} i_2^{i_2-2} \theta_d^{i_1 + i_2 - 2} (i_1 i_2 2^d \theta_d) = 2^d i_1^{i_1-1} i_2^{i_2-1} ( r_n(t_2)^d \|\, f \|_\infty \theta_d )^{i_1+i_2-1}.\end{align*}

Now we have

\[n^{-1} |A_{2,n}| \leq \dfrac{3\cdot 2^d}{( (k+1)! )^2 t_2^d \|\,f \|_\infty \theta_d}\, \bigg\{ \sum_{i=k+2}^\infty \dfrac{i^{i-1}}{(i-k-1)!}\, ( t_2^d \|\, f \|_\infty \theta_d )^i \bigg\}^2.\]

The constraint $t_2 \lt (e\|\, f \|_\infty \theta_d)^{-1/d}$ , together with the ratio test, guarantees that the last term converges. Hence the proof is completed.

Proof of Theorem 4.1. We begin by proving the corresponding result for the truncated Betti number in 6.1 for every $M \in {\mathbb N}$ , that is,

\[n^{-1/2} ( \beta_{k,n}^{(M)}(t_i) -{\mathbb{E}} [\beta_{k,n}^{(M)}(t_i)], \, i=1,\ldots,m ) \Rightarrow ( \mathcal H_k^{(M)}(t_i) \, i=1,\ldots,m ),\]

where $\mathcal H_k^{(M)}$ is the ‘truncated’ limiting centered Gaussian process given by

\[\mathcal H_k^{(M)}(t) = \sum_{i=k+2}^M \sum_{j>0} j \mathcal H_k^{(i,j)}(t).\]

We now restrict ourselves to the case in which the corresponding leftmost points belong to a fixed bounded set A. By the Cramér–Wold device [Reference Durrett13, page 176], we need todemonstrate a univariate central limit theorem for $\sum_{i=1}^m a_i \beta_{k,n,A}^{(M)}(t_i)$ , where $a_i \in {\mathbb R}$ , $m \geq 1$ . The asymptotic variance of $\sum_{i=1}^m a_i \beta_{k,n,A}^{(M)}(t_i)$ scaled by $n^{-1/2}$ can be derived from Proposition 6.1(i):

(6.20) \begin{align}{\textrm{Var}}\biggl(n^{-1/2} \sum_{i=1}^m a_i \beta_{k,n,A}^{(M)}(t_i) \biggr) &= \sum_{i=1}^m \sum_{j=1}^m a_i a_j n^{-1} {\textrm{Cov}}(\beta_{k,n,A}^{(M)}(t_i), \beta_{k,n,A}^{(M)}(t_j)) \notag \\[3pt] &\to \sum_{i=1}^m \sum_{j=1}^m a_i a_j \Phi_{k,A}^{(M)}(t_i,t_j), \quad n\to\infty.\end{align}

Our proof exploits Stein’s normal approximation method for weakly dependent random variables, as in Theorem 2.4 in [Reference Penrose26]. We assume the limit in (6.20) is positive, as otherwise our proof is trivial. Recall that in the statement of Theorem 4.1 we have set $0 \lt t_1 \lt t_2 \lt \cdots \lt t_m$ , and let $(Q_{j,n}, j \in \mathbb{N})$ be an enumeration of almost disjoint closed cubes (i.e. their interiors are disjoint) of side length $r_n(t_m)$ , such that $\cup_{j \in {\mathbb N}} Q_{j,n} = {\mathbb{R}}^d$ . Further,

\[V_n \,:\!= \{j \in \mathbb{N}\,\colon Q_{j,n} \cap A \neq \emptyset\},\]

and

\[\xi_{j, n} \,:\!= \sum_{i=1}^m a_i \beta_{k,n,A\cap Q_{j,n}}^{(M)}(t_i),\]

so that

\[\sum_{i=1}^m a_i \beta_{k,n,A}^{(M)}(t_i) = \sum_{j \in V_n} \xi_{j, n} .\]

We now turn $V_n$ into the vertex set of a dependency graph (see Section 2.1 in [Reference Penrose26] for the definition) by declaring that for $j, j' \in V_n$ , $j \sim {j^{\prime}}$ if and only if

\[\inf\{ |x-y|\,\colon x \in Q_{j,n}, y \in Q_{{j^{\prime}}, n} \} \leq 2Mr_n(t_m).\]

It is easy to show that this provides us with the required independence properties, that is, for any vertex set $I_1, I_2 \subset V_n$ with no edges connecting vertices in $I_1$ and those in $I_2$ , we have $(\xi_{j,n}, \, j \in I_1)$ and $(\xi_{j,n}, \, j \in I_2)$ are independent. Note moreover that the degree of $(V_n, \sim)$ is uniformly bounded regardless of n. Since A is a bounded set, we have $| V_n| = {\textrm{O}} (s_n^{-d})$ . Let $Y_{j,n}$ denote the number of points of $\mathcal{P}_n$ belonging to

\[\text{Tube}(Q_{j,n}, Mr_n(t_m)) \,:\!= \Big\{ x \in {\mathbb R}^d\,\colon \inf_{y \in Q_{j,n}} |x-y| \leq Mr_n(t_m) \Big\}.\]

Then we have

\begin{align*}|\xi_{j,n}| &\leq \sum_{i=1}^m |a_i| \beta_{k,n, A\cap Q_{j,n}}^{(M)} (t_i) \\*&\leq \sum_{i=1}^m |a_i| \beta_k ( \skew4\check{C} ( \mathcal{P}_n \cap \text{Tube} ( Q_{j,n}, Mr_n(t_m) ), r_n(t_i) ) ) \\*&\leq \sum_{i=1}^m |a_i| \binom{Y_{j,n}}{k+1}.\end{align*}

By definition, $Y_{j,n}$ is Poisson-distributed with parameter

\[\lambda_{j,n} \,:\!= n\int_{\text{Tube}(Q_{j,n}, Mr_n(t_m))} f(z) \, d{z},\]

which itself yields an upper bound of the form

(6.21) \begin{equation}\lambda_{j,n} \leq n \|\, f \|_{\infty} m ( \text{Tube}(Q_{j,n}, Mr_n(t_m))) \,:\!= c.\end{equation}

This implies that $Y_{j,n}$ is stochastically dominated by a Poisson random variable, which we call Y, with parameter c. The assumption $ns_n^d=1$ ensures that c does not depend on n, and for the rest of the proof, let $C^*$ denote a generic positive constant which is independent of n but may vary between lines.

For $\alpha \in \mathbb{N}$ we obtain

(6.22) \begin{equation}{\mathbb{E}} [|\xi_{j,n}|^\alpha] \leq \bigg( \sum_{i=1}^m |a_i| \bigg)^\alpha \, {\mathbb{E}} \bigg[ \binom{Y_{j,n}}{k+1}^\alpha \bigg]\leq \bigg( \sum_{i=1}^m |a_i| \bigg)^\alpha \, {\mathbb{E}} \bigg[ \binom{Y}{k+1}^\alpha \bigg] =C^* {.}\end{equation}

Letting

\[\xi^{\prime}_{j,n} \,:\!= \dfrac{\xi_{j,n} - {\mathbb{E}}[\xi_{j,n}]}{\sqrt{{\textrm{Var}}(\sum_{i=1}^m a_i \beta_{k,n,A}^{(M)}(t_i))}},\]

it is clear that $(V_n, \sim)$ still constitutes a dependency graph for the $({\xi}^{\prime}_{j,n}, \, j \in \mathbb{N})$ because independence is not affected by affine transformations. Let Z be a standard normal random variable. It then follows from Stein’s normal approximation method (i.e. Theorem 2.4 from [Reference Penrose26]) that for all $x\in {\mathbb R}$ ,

\begin{align*}\bigg| {\mathbb{P}}\bigg( \sum_{j \in V_n} \xi'_{\!\!j,n} \leq x \bigg) - {\mathbb{P}}(Z \leq x) \bigg|& \leq C^*\big( \sqrt{s_n^{-d} \, {\mathbb{E}} [ |\xi'_{\!\!j,n}|^3 ]} + \sqrt{s_n^{-d} \, {\mathbb{E}} [ |\xi'_{\!\!j,n}|^4 ]} \big) \\[3pt] &\leq C^*\big( \sqrt{s_n^{-d}n^{-3/2} \, {\mathbb{E}} [ |\xi_{j,n}-{\mathbb{E}}[\xi_{j,n}]|^3 ]}\\[3pt] &\quad + \sqrt{s_n^{-d} n^{-2} \, {\mathbb{E}} [ |\xi_{j,n} -{\mathbb{E}}[ \xi_{j,n}]|^4 ]} \big),\end{align*}

where we have applied (6.20) for the second inequality.

Now we have by (6.22) that ${\mathbb{E}}[|\xi_{j,n} - {\mathbb{E}}[\xi_{j,n}]|^p ] \leq C^*$ for $p=3, 4$ , so that

\[s_n^{-d} n^{-p/2} \, {\mathbb{E}} [ |\xi_{j,n} - {\mathbb{E}} [\xi_{j,n}]|^p ] \leq C^* n^{1-p/2} \to 0, \quad n\to\infty.\]

From the argument thus far we conclude that

\[\sum_{j \in V_n} \xi'_{\!\!j,n} \Rightarrow Z,\]

which in turn implies

\[n^{-1/2} ( \beta_{k,n,A}^{(M)}(t_i) - {\mathbb{E}} [ \beta_{k,n,A}^{(M)}(t_i) ], \, i=1,\ldots,m ) \Rightarrow \mathcal N ( 0, (\Phi_{k,A}^{(M)}(t_i, t_j))_{i,j=1}^m )\]

for all bounded sets A. The case when A is unbounded can be established by standard approximation arguments nearly identical to those in [Reference Kahle and Meckes21] and [Reference Penrose26], so we omit the details and conclude that, as $n\to\infty$ ,

\begin{equation*}n^{-1/2} ( \beta_{k,n}^{(M)}(t_i) - {\mathbb{E}} [ \beta_{k,n}^{(M)}(t_i) ], \, i=1,\ldots,m ) \Rightarrow \mathcal N ( 0, (\Phi_{k,{\mathbb R}^d}^{(M)}(t_i,t_j))_{i,j=1}^m ).\end{equation*}

This is equivalent to

\[n^{-1/2} ( \beta_{k,n}^{(M)}(t_i) - {\mathbb{E}} [ \beta_{k,n}^{(M)}(t_i) ], \, i=1,\ldots,m ) \Rightarrow ( \mathcal H_k^{(M)}(t_i), \, i=1,\ldots,m ),\]

as $n\to\infty$ . Further, as $M\to\infty$ ,

\[( \mathcal H_k^{(M)} (t_i), \, i=1,\ldots,m ) \Rightarrow ( \mathcal H_k (t_i), \, i=1,\ldots,m ),\]

since $\Phi_{k,{\mathbb R}^d}^{(M)}(t_i, t_j) \to \Phi_{k,{\mathbb R}^d} (t_i, t_j)$ as $M\to\infty$ . According to Theorem 3.2 in [Reference Billingsley3] it suffices to show that, for every $t>0$ and $\epsilon>0$ ,

(6.23) \begin{equation}\lim_{M\to\infty}\limsup_{n\to\infty} {\mathbb{P}} ( | \beta_{k,n}(t) - \beta_{k,n}^{(M)}(t) -{\mathbb{E}} [\beta_{k,n}(t) - \beta_{k,n}^{(M)}(t)] | \gt \epsilon n^{1/2} ) = 0.\end{equation}

By Chebyshev’s inequality, the probability in (6.23) is bounded by

\[\dfrac{1}{\epsilon^2 n} \text{Var} ( \beta_{k,n}(t) - \beta_{k,n}^{(M)}(t) ),\]

which itself converges to

(6.24) \begin{equation}\dfrac{1}{\epsilon^2}\, \sum_{i_1=M+1}^\infty \sum_{i_2=M+1}^\infty \sum_{j_1 \gt 0} \sum_{j_2 \gt 0} j_1\, j_2\, \bigg( \dfrac{\eta_{k, \mathbb R^d}^{(i_1, j_1, j_2)}(t_1, t_2) \delta_{i_1, i_2}}{i_1!}+ \dfrac{\nu_{k, \mathbb R^d}^{(i_1, i_2, j_1, j_2)}(t_1, t_2)}{i_1! i_2!}\bigg), \quad n\to\infty.\end{equation}

Since $\Phi_{k,{\mathbb R}^d}(t,t)$ is a finite constant, (6.24) goes to 0 as $M\to\infty$ .

Proof of Corollary 4.1. Theorem 4.6 in [Reference Yogeshwaran, Subag and Adler34] verified that

\[\lim_{n\to\infty} n^{-1} ( \beta_{k,n}(t) - {\mathbb{E}} [\beta_{k,n}(t)] ) = 0\]

almost surely. Combining this with Proposition 6.1(ii) proves the claim.

6.2. Central limit theorem in the sparse regime

As with the critical regime case, the key results for proving a central limit theorem are those on asymptotic moments that can be seen in the proposition below. As discussed in Section 3, the probabilistic features of these moments are asymptotically determined by $S_{k,n}(t)$ . Many functions and objects in Section 6.1 will be carried over for use in this section.

Proposition 6.2. Let f be an almost everywhere bounded and continuous density function. If $ns_n^d \to 0$ and $A \subset \mathbb{R}^d$ is open with $m(\partial A) = 0$ , then for $t \gt 0$ we have

\[\rho_n^{-1} \, {\mathbb{E}} [\beta_{k,n,A}(t)] \to \mu_{k,A}(t,t), \quad n\to\infty,\]

and for $t_1, t_2 \gt 0$ ,

\begin{equation*}\rho_n^{-1} {\textrm{Cov}}(\beta_{k,n,A}(t_1), \beta_{k,n,A}(t_2)) \to \mu_{k,A}(t_1,t_2), \quad n\to\infty,\end{equation*}

where

\[\mu_{k,A} (t_1, t_2) \,:\!= \dfrac{1}{(k+2)!} \int_{A} f(x)^{k+2} \ \text{d}{x} \int_{{\mathbb R}^{d(k+1)}} h_{t_1}(0, {\bf y})h_{t_2}(0, {\bf y})\, \text{d}{{\bf y}}.\]

Proof. We only discuss the covariance result in the case $A={\mathbb R}^d$ . Throughout the proof we assume $0 \lt t_1 \leq t_2$ . We first derive the same expression as in 6.11,

\[\text{Cov} (\beta_{k,n}(t_1), \beta_{k,n}(t_2)) = A_{1,n} + A_{2,n},\]

where $A_{1,n}$ and $A_{2,n}$ are given in (6.12), (6.13) respectively. Observing that $g_{r_n(t)}^{(k+2,j)}({\mathcal{X}_{i}}, {\mathcal{X}_{i}} \cup \mathcal{P}_n) = 0$ for all $j \geq 2$ and any $t \gt 0$ , we can split $A_{1,n}$ into two parts, $A_{1,n} = D_{1,n} + D_{2,n}$ , where

\begin{align*}D_{1, n} & \,:\!= \dfrac{n^{k+2}}{(k+2)!}\, {\mathbb{E}} [ g_{r_n(t_1)} ({\mathcal{X}_{{k+2}}}, {\mathcal{X}_{{k+2}}} \cup \mathcal{P}_n)\, g_{r_n(t_2)} ({\mathcal{X}_{{k+2}}}, {\mathcal{X}_{{k+2}}} \cup \mathcal{P}_n) ],\\*D_{2,n} & \,:\!= A_{1,n} - D_{1,n} = \sum_{i=k+3}^\infty \sum_{j_1>0} \sum_{j_2>0} j_1\,j_2\dfrac{n^i}{i!}\, {\mathbb{E}} \big[ g_{r_n(t_1)}^{(i, j_1)} (\mathcal X_i, \mathcal X_i \cup \mathcal{P}_n) g_{r_n(t_2)}^{(i, j_2)} (\mathcal X_i, \mathcal X_i \cup \mathcal{P}_n) \big].\end{align*}

Based on this decomposition, we claim that

(6.25) \begin{equation}\rho_n^{-1} D_{1,n} \to \mu_{k,{\mathbb R}^d}(t_1,t_2), \quad n\to\infty,\end{equation}

and $\rho_n^{-1} D_{2,n}$ and $\rho_n^{-1} A_{2,n}$ both converge to 0 as $n\to\infty$ . An important implication of these convergence results is that

\[\rho_n^{-1} \text{Cov} ( S_{k,n}(t_1), S_{k,n}(t_2) ) \to \mu_{k,{\mathbb R}^d}(t_1,t_2), \quad n\to\infty{,}\]

namely, the covariance of $\beta_{k,n}(t)$ asymptotically coincides with that of $S_{k,n}(t)$ .

By what should now be a familiar argument and the customary change of variable, we see that

(6.26) \begin{align}\rho_n^{-1} D_{1,n} &= \dfrac{\rho_n^{-1} n^{k+2}}{(k+2)!} \, {\mathbb{E}} [ h_{r_n(t_1)} ({\mathcal{X}_{{k+2}}}) h_{r_n(t_2)}({\mathcal{X}_{{k+2}}}) {\mathbb{E}} [J_{k+2, r_n(t_2)} ({\mathcal{X}_{{k+2}}}, {\mathcal{X}_{{k+2}}} \cup \mathcal{P}_n) \mid {\mathcal{X}_{{k+2}}} ] ]\nonumber \\[3pt] &= \dfrac{\rho_n^{-1} n^{k+2}}{(k+2)!} \int_{\mathbb{R}^{d(k+2)}} h_{r_n(t_1)}({{\bf x}})h_{r_n(t_2)}({{\bf x}}) \exp({-}nI_{r_n(t_2)}({{\bf x}})) \prod_{j=1}^{k+2} f(x_j)\, \text{d}{{{\bf x}}} \nonumber\\[3pt] &= \dfrac{1}{(k+2)!} \int_{\mathbb{R}^{d(k+1)}} \int_{{\mathbb R}^d} h_{t_1}(0,{\bf y}) h_{t_2}(0,{\bf y}) \exp ({-}nI_{r_n(t_2)}(x, x+s_n{\bf y})) \nonumber\\[3pt] &\quad\, \times f(x)\prod_{j=1}^{k+1} f(x+s_ny_j) \text{d}{x}\, \text{d}{{\bf y}}.\end{align}

By the continuity of f it holds that

\[\prod_{j=1}^{k+1} f(x+s_ny_j) \to f(x)^{k+1}\quad \text{a.e.\ as $n\to \infty$.}\]

Moreover, the exponential term converges to 1 because we see that

\[nI_{r_n(t_2)}(x, x+s_n {\bf y}) \leq ns_n^d \|\, f \|_\infty m ( \mathcal{B}( \{ 0,{\bf y} \};\, t_2 ) ) \to 0, \quad n\to\infty.\]

Thus (6.25) follows from the dominated convergence theorem.

Next let us turn to the asymptotics of $\rho_n^{-1}D_{2,n}$ . Proceeding as in (6.17), while applying (6.7) and (6.9), we have

\begin{align*}\rho_n^{-1} D_{2,n} &\leq \sum_{i=k+3}^\infty \sum_{j_1>0} \sum_{j_2>0}j_1\, j_2\, \dfrac{\rho_n^{-1} n^{i}}{i!} \, {\mathbb{E}}\bigg[ {{\bf 1}} \{ {\skew4\check{C}} (\mathcal{X}_i, r_n(t_1)) \text{ is connected} \} \prod_{\ell=1}^{2} b_{j_{\ell}, r_n(t_{\ell})}(\mathcal{X}_i) \bigg] \\[3pt]&\leq \sum_{i = k+3}^{\infty} \binom{i}{k+1}^2 \dfrac{\rho_n^{-1}n^{i}}{i!}\, {\mathbb{P}}( {\skew4\check{C}} (\mathcal X_{i}, r_n(t_1)) \text{ is connected} ) \\[3pt]&\leq \dfrac{( t_1^d \|\, f \|_\infty \theta_d )^{k+1}}{((k+1)!)^2} \sum_{i=k+3}^\infty b_{i,n},\end{align*}

where

\[b_{i,n} \,:\!= \dfrac{i! i^{i-2}}{( (i-k-1)! )^2}\, ( nr_n(t_1)^d \|\, f \|_\infty \theta_d )^{i-(k+2)}.\]

Obviously $b_{i,n} \to 0$ , $n\to\infty$ for all $i \geq k+3$ . Since $ns_n^d \to 0$ , it is easy to find a summable upper bound $c_i \geq b_{i,n}$ for sufficiently large n. Now the dominated convergence theorem for sums concludes $\rho_n^{-1} D_{2,n} \to 0$ as $n\to\infty$ .

For the evaluation of $n^{-1}|A_{2,n}|$ , we apply (6.19) to the right-hand side of (6.18). Slightly changing the description of the resulting bound, we obtain

\begin{align*}\rho_n^{-1} |A_{2,n}| &\leq 3 \cdot 2^d\, \dfrac{( t_2^d \|\, f \|_\infty \theta_d )^{k+1}}{( (k+1)!)^2} \\[3pt] &\quad\, \times\sum_{i_1=k+2}^\infty \sum_{i_2=k+2}^\infty \dfrac{i_1^{i_1-1} i_2^{i_2-1}}{(i_1-k-1)! (i_2-k-1)!}\, ( nr_n(t_2)^d \|\, f \|_\infty \theta_d )^{i_1+i_2 - (k+2)}.\end{align*}

Since $ns_n^d \to 0$ as $n\to\infty$ , it follows from the dominated convergence theorem for sums that $\rho_n^{-1} A_{2,n} \to 0$ , $n\to\infty$ , as desired.

Proof of Theorem 3.1. We first establish the central limit theorem for $S_{k,n}(t)$ by proceeding in an almost identical fashion to Theorem 4.1. As in that proof, we require the leftmost point of each subset ${\mathcal{Y}} \subset \mathcal{P}_n$ to lie in an (open) bounded set $A \subset {\mathbb{R}}^d$ , with $m(\partial A) = 0$ . Let $V_n, Q_{j,n}$ be defined as in the proof of Theorem 4.1 and recall once more that we assume $0 \lt t_1 \lt t_2 \lt \cdots \lt t_m$ . In this case, however, we let $V_n$ be the vertex set of a dependency graph by letting $j \sim j^{\prime}$ if and only if

\[\inf\{ |x-y|\,\colon x \in Q_{j,n}, y \in Q_{{j^{\prime}}, n} \} \leq 2(k+2)r_n(t_m).\]

We modify $\xi_{j,n}$ to be defined as

\[\xi_{j,n} \,:\!= \sum_{i=1}^m a_i \sum_{{\mathcal{Y}} \subset \mathcal{P}_n} g_{r_n(t_i), A \cap Q_{j,n}}({\mathcal{Y}}, \mathcal{P}_n)\]

so that

\[\sum_{i=1}^m a_i S_{k,n,A}(t_i) = \sum_{j \in V_n} \xi_{j,n}.\]

Furthermore, $Y_{j,n}$ denotes the number of points of $\mathcal{P}_n$ in $\text{Tube} (Q_{j,n}, (k+2)r_n(t_m))$ . Then,

\[|\xi_{j,n}| \leq \sum_{i=1}^m |a_i| \binom{Y_{j,n}}{k+2}.\]

It is easy to demonstrate that the Poisson parameter of $Y_{j,n}$ is bounded by $cns_n^d$ for some constant $c>0$ ; see (6.21). Letting $C^*$ be a general positive constant as in the proof of Theorem 4.1, we obtain for $\alpha \in {\mathbb N}$

\[{\mathbb{E}} [| \xi_{j,n} |^\alpha] \leq C^* (ns_n^d)^{k+2}.\]

This in turn implies

\[ {\mathbb{E}} [ |\xi_{j,n} - {\mathbb{E}} [\xi_{j,n}]|^p ] \leq C^* (ns_n^d)^{k+2} \]

for $p=3,4$ . Let

\[{\xi}^{\prime}_{j,n} \,:\!= \dfrac{\xi_{j,n} - {\mathbb{E}}[\xi_{j,n}]}{\sqrt{{\textrm{Var}} (\sum_{i=1}^m a_i S_{k,n,A} (t_i))}}\]

and $Z \sim \mathcal N(0,1)$ . As in the critical regime case, Stein’s normal approximation method gives

\begin{align*}& \bigg| {\mathbb{P}}\bigg( \sum_{j \in V_n} \xi'_{\!\!j,n} \leq x \bigg) - {\mathbb{P}}(Z \leq x) \bigg| \\[3pt] &\quad \leq C^* \big( \sqrt{s_n^{-d}\rho_n^{-3/2} \, {\mathbb{E}} [ |\xi_{j,n}-{\mathbb{E}}[\xi_{j,n}]|^3 ]} + \sqrt{s_n^{-d} \rho_n^{-2}\, {\mathbb{E}} [ |\xi_{j,n} -{\mathbb{E}}[ \xi_{j,n}]|^4 ]} \big),\end{align*}

The right-hand side vanishes as $n\to\infty$ , since for $p=3,4$ ,

\[s_n^{-d}\rho_n^{-p/2} \, {\mathbb{E}} [ |\xi_{j,n}-{\mathbb{E}}[\xi_{j,n}]|^p ] \leq C^* \rho_n^{1-p/2} \to 0, \quad n\to\infty.\]

Thus we have obtained

(6.27) \begin{equation}\rho_n^{-1/2} ( S_{k,n}(t_i) - {\mathbb{E}} [ S_{k,n}(t_i) ], \, i=1,\ldots,m ) \Rightarrow \mathcal N ( 0, (\mu_{k,{\mathbb R}^d}(t_i,t_j))_{i,j=1}^m ).\end{equation}

The limiting covariance matrix above coincides with the covariance functions of the process $\mathcal G_k$ , that is,

\[{\mathbb{E}} [ \mathcal G_k(t_i) \mathcal G_k(t_j) ] = C_{f,k} \int_{{\mathbb R}^{d(k+1)}} h_{t_i} (0,{\bf y}) h_{t_j}(0,{\bf y}) \text{d}{\bf y} = \mu_{k,{\mathbb R}^d}(t_i,t_j), \quad i,j = 1,\ldots,m.\]

Therefore (6.27) is equivalent to

\[\rho_n^{-1/2} ( S_{k,n}(t_i) - {\mathbb{E}} [ S_{k,n}(t_i) ], \, i=1,\ldots,m ) \Rightarrow ( \mathcal G_k(t_i), \, i=1,\ldots,m ).\]

Now we can finish the entire proof, provided that, for every $t>0$ ,

\[\rho_n^{-1/2} ( \beta_{k,n}(t) - {\mathbb{E}} [ \beta_{k,n}(t) ] )- \rho_n^{-1/2} ( S_{k,n}(t) - {\mathbb{E}} [ S_{k,n}(t) ] ) \stackrel{p}{\to} 0, \quad n\to\infty.\]

This can be proved immediately by Chebyshev’s inequality. That is, for every $\epsilon>0$ ,

\[{\mathbb{P}} ( \rho_n^{-1/2} | R_{k,n}(t) - {\mathbb{E}} [R_{k,n}(t)] | \gt \epsilon ) \leq \dfrac{1}{\epsilon^2 \rho_n}\, {\textrm{Var}} ( R_{k,n}(t) ) \to 0,\]

where the convergence is a direct consequence of $\rho_n^{-1}D_{2,n} \to 0$ and $\rho_n^{-1}A_{2,n} \to 0$ , which were verified in the proof of Proposition 6.2.

6.3. Poisson limit theorem in the sparse regime

Proof of Theorem 5.1. We begin by defining

\[H_{k,n}(t) \,:\!= \sum_{{\mathcal{Y}} \subset \mathcal{P}_n} h_{r_n(t)}({\mathcal{Y}}),\]

and show that

(6.28) \begin{equation}( H_{k,n}(t_i), \, i=1,\ldots,m ) \Rightarrow ( \mathcal V_k(t_i), \, i=1,\ldots,m ).\end{equation}

Subsequently we shall verify that, for every $t>0,$

(6.29) \begin{align}H_{k,n}(t) - S_{k,n}(t) \stackrel{p}{\to} 0,\\[-30pt]\nonumber\end{align}

(6.30) \begin{align} \beta_{k,n}(t) - S_{k,n}(t) &\stackrel{p}{\to} 0.\end{align}

Then the proof of 5.1 will be complete.

Part 1. For the proof of 6.28, it is sufficient to show that, for any $a_1, a_2, \ldots, a_m \gt 0$ , $m \geq 1$ ,

\begin{equation*}\sum_{i=1}^{m} a_i H_{k,n}(t_i) \Rightarrow \sum_{i=1}^{m} a_i \mathcal{V}_k(t_i).\end{equation*}

We may use positive constants because of the fact that the Laplace transform characterizes a random vector with values in ${\mathbb{R}}_+^m$ . We proceed by using Theorem 3.1 from [Reference Decreusefond, Schulte and Thäle12]. First let $(\Omega, \mathcal F, {\mathbb{P}})$ denote a generic probability space on which all objects are defined. Let $\mathbf N ({\mathbb R}_+)$ be the set of finite counting measures on ${\mathbb R}_+$ . We equip $\mathbf N ({\mathbb R}_+)$ with the vague topology; see e.g. [Reference Resnick28] for more information on the vague topology. Let us define a point process $\xi_n\,\colon \Omega \to \mathbf{N}({\mathbb{R}}_+)$ by

\[\xi_n(\cdot) \,:\!= \sum_{{\mathcal{Y}} \subset \mathcal{P}_n}{{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{r_n(t_i)} ({\mathcal{Y}}) \gt0 \bigg\}\, \delta_{\sum_{i=1}^m a_i h_{r_n(t_i)}({\mathcal{Y}})}(\cdot),\]

where $\delta$ is a Dirac measure.

Further, let $\zeta\,\colon \Omega \to \mathbf N({\mathbb R}_+)$ denote a Poisson random measure with mean measure $C_{f,k} \tau_k$ , where

\[\tau_k(A) \,:\!= m_k \bigg\{ {\bf y} \in {\mathbb R}^{d(k+1)}\,\colon \sum_{i=1}^m a_i h_{t_i} (0,{\bf y}) \in A \setminus \{ 0 \} \bigg\}, \quad A \subset {\mathbb R}_+.\]

The rest of Part 1 is devoted to showing that

(6.31) \begin{equation}\xi_n \Rightarrow \zeta \quad \text{in } \mathbf N({\mathbb R}_+).\end{equation}

According to Theorem 3.1 in [Reference Decreusefond, Schulte and Thäle12], the following two conditions suffice for (6.31). Let $\mathbf L_n(\cdot) \,:\!= {\mathbb{E}} [\xi_n(\cdot)]$ and $\mathbf M (\cdot) \,:\!= {\mathbb{E}}[\zeta (\cdot)] = C_{f,k}\tau_k(\cdot)$ . The first requirement for (6.31) is the convergence in terms of the total variation distance:

(6.32) \begin{equation}d_{\text{TV}} (\mathbf L_n, \mathbf M) \,:\!= \sup_{A \in \mathcal{B}({\mathbb R}_+)} | \mathbf L_n(A) - \mathbf M(A) | \to 0, \quad n \to \infty,\end{equation}

where $\mathcal{B}({\mathbb R}_+)$ is the Borel $\sigma$ -field over ${\mathbb R}_+$ . In addition, the second requirement for (6.31) is

(6.33) \begin{align}v_n& \,:\!= \max_{1 \leq \ell \leq k+1} \int_{{\mathbb R}^{d\ell}} \!\bigg(\! \int_{{\mathbb R}^{d(k+2-\ell)}} {{\bf 1}} \bigg\{\! \sum_{i=1}^m a_i h_{r_n(t_i)} (x_1,\ldots,x_{k+2}) \gt 0 \bigg\} \lambda^{k+2-\ell} ( \text{d}\, (x_{\ell+1}, \ldots, x_{k+2}))\! \bigg)^2 \notag\\[3pt] &\quad\, \times\lambda^\ell ( \text{d}\, (x_1,\ldots,x_\ell)) \to 0\end{align}

as $n\to\infty$ , where $\lambda^m = \lambda \otimes \cdots \otimes \lambda$ is a product measure on ${\mathbb R}^m$ with $\lambda(\cdot) = n\int_\cdot f(z)\, \text{d}z$ .

Let us now return to (6.32) and present its proof here. As usual, we have assumed $0 \lt t_1 \lt t_2 \lt \cdots \lt t_m$ . Then, for any $A \in \mathcal{B}({\mathbb R}_+)$ we have from Palm theory, the change of variables $x_1=x$ , $x_i=x+s_n y_{i-1}$ for $i=2,\ldots,k+2$ , and $\rho_n = 1$ that

\begin{align*}\mathbf L_n(A) &= \dfrac{n^{k+2}}{(k+2)!}\, \int_{{\mathbb R}^{d(k+2)}} {{\bf 1}} \bigg\{ \, \sum_{i=1}^m a_i h_{r_n(t_i)}({{\bf x}}) \in A \setminus \{ 0 \} \bigg\} \prod_{j=1}^{k+2}f(x_j) \text{d}{{\bf x}} \\*&= \dfrac{1}{(k+2)!}\, \int_{{\mathbb R}^{d(k+2)}} {{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{t_i}(0, {\bf y}) \in A \setminus \{ 0 \} \bigg\} f(x) \prod_{j=1}^{k+1} f(x+s_n y_j)\, \text{d}x\, \text{d}{\bf y}.\end{align*}

Therefore,

\begin{align*}| \mathbf L_n(A) - \mathbf M(A) | &\leq \dfrac{1}{(k+2)!}\, \int_{{\mathbb R}^{d(k+2)}} {{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{t_i}(0, {\bf y}) \in A \setminus \{ 0 \} \bigg\} \\*& \quad\, \times f(x)\, \bigg| \prod_{j=1}^{k+1} f(x+s_n y_j) -f(x)^{k+1} \bigg| \text{d}x\, \text{d}{\bf y}.\end{align*}

If the indicator function above is equal to 1, then $h_{t_i}(0,{\bf y}) = 1$ for at least one i, which means that the distance of each component in y from the origin must be less than $t_m$ . Otherwise one cannot form a required empty $(k+1)$ -simplex. Hence we have

\begin{align*}| \mathbf L_n(A) - \mathbf M(A) | &\leq \dfrac{1}{(k+2)!}\, \int_{{\mathbb R}^{d(k+2)}} \prod_{i=1}^{k+2} {{\bf 1}} \{ |y_i| \leq t_m \}f(x)\, \bigg| \prod_{j=1}^{k+1} f(x+s_n y_j) -f(x)^{k+1} \bigg| \text{d}x\, \text{d}{\bf y}.\end{align*}

We have by continuity of f that

\[\bigg| \prod_{j=1}^{k+1} f(x+s_n y_j) -f(x)^{k+1} \bigg|\]

converges to 0 a.e. as $n\to\infty$ and is bounded by $2\|\,f \|_{\infty}^{k+1} \lt \infty$ . So the dominated convergence theorem applies to get $| \mathbf L_n(A) - \mathbf M(A) | \to 0$ as $n\to\infty$ . Since this convergence holds uniformly for all $A\in \mathcal{B}({\mathbb R}_+)$ , we have now established (6.32).

Next we turn to proving (6.33). First we can immediately see that

\begin{align*}v_n &= \max_{1 \leq \ell \leq k+1} n^{2k+4-\ell} \int_{{\mathbb R}^{d(2k+4-\ell)}} {{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{r_n(t_i)} (x_1,\ldots,x_{k+2}) \gt 0 \bigg\} \\[3pt] &\quad\, \times {{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{r_n(t_i)} (x_1,\ldots,x_\ell, x_{k+3}, \ldots, x_{2k+4-\ell}) \gt 0 \bigg\} \prod_{j=1}^{2k+4-\ell} f(x_j)\, \text{d} {{\bf x}}.\end{align*}

Making a change of variables with $x_1 = x$ and $x_i = x + s_ny_{i-1}$ for $i=2,\ldots,2k+4-\ell$ , while using $f(x+s_ny_{i-1}) \leq \|\, f\|_\infty$ , we obtain

\begin{align*}v_n &\leq \|\, f \|_\infty^{2k+3-\ell} \max_{1 \leq \ell \leq k+1} n^{2k+4-\ell}s_n^{d(2k+3-\ell)} \int_{{\mathbb R}^{d(2k+3-\ell)}} {{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{t_i} (0,y_1,\ldots,y_{k+1}) \gt 0 \bigg\} \\[3pt] &\quad\, \times {{\bf 1}} \bigg\{ \sum_{i=1}^m a_i h_{t_i} (0,y_1, \ldots,y_{\ell-1}, y_{k+2}, \ldots, y_{2k+3-\ell}) \gt 0 \bigg\}\text{d}{\bf y}.\end{align*}

Obviously the above integral is finite, and

\[\max_{1 \leq \ell \leq k+1} n^{2k+4-\ell}s_n^{d(2k+3-\ell)} = \max_{1 \leq \ell \leq k+1} (ns_n^d)^{k+2-\ell} \to 0, \quad n\to\infty,\]

by the assumption $\rho_n=1$ . So $v_n \to 0$ follows and (6.33) is obtained.

Part 2. Define the map $\widehat T\,\colon \mathbf N ({\mathbb R}_+) \to {\mathbb R}_+$ by $\widehat T (\!\sum_n \delta_{x_n}) = \sum_n x_n$ . This map is continuous because it is defined on the space of finite counting measures. Applying the continuous mapping theorem to (6.31) gives $\widehat T(\xi_n) \Rightarrow \widehat T (\zeta)$ . Equivalently, we have

\[\sum_{i=1}^m a_i H_{k,n}(t_i)\Rightarrow \sum_{i=1}^m a_i \mathcal V_k (t_i).\]

To see such equivalence, note that $\widehat T(\xi_n) = \sum_{i=1}^m a_i H_{k,n}(t_i)$ , so it now suffices to show that $\widehat T(\zeta)$ is equal in distribution to $\sum_{i=1}^m a_i \mathcal V_k(t_i)$ . To this end let us represent $\zeta$ as

\[\zeta \stackrel{d}{=} \sum_{i=1}^{M_n} \delta_{Y_i},\]

where $Y_1,Y_2,\ldots$ are i.i.d. with common distribution $\tau_k(\cdot)/\tau_k ({\mathbb R}_+)$ and $M_n$ is Poisson-distributed with parameter $C_{f,k}\tau_k ({\mathbb R}_+)$ . Further, $(Y_i)$ and $M_n$ are independent. On one hand, it follows from the Laplace functional of a Poisson random measure (see Theorem 5.1 in [Reference Resnick29]) that, for every $\lambda \gt0$ ,

\begin{align*}{\mathbb{E}} \bigg[\! \exp \bigg({-}\lambda \sum_{i=1}^m a_i \mathcal V_k(t_i) \bigg) \bigg] &= {\mathbb{E}} \bigg[\! \exp\bigg({-}\int_{{\mathbb R}^{d(k+1)}} \lambda \sum_{i=1}^m a_i h_{t_i} (0,{\bf y}) M_k(\text{d}{\bf y}) \bigg) \bigg] \notag \\[3pt] &= \exp \bigg({-} C_{f,k} \int_{{\mathbb R}^{d(k+1)}} ( 1-e^{-\lambda \sum_{i=1}^m a_i h_{t_i} (0,{\bf y})} ) \text{d}{\bf y} \bigg).\end{align*}

On the other hand it is straightforward to compute that

\begin{align*}{\mathbb{E}} [\! \exp({-}\lambda \widehat T (\zeta) ) ] &= {\mathbb{E}} \bigg[\! \exp\bigg({-}\lambda \sum_{i=1}^{M_n} Y_i \bigg) \bigg] \\[3pt] &= \exp ({-}C_{f,k}\tau_k({\mathbb R}_+) (1-{\mathbb{E}}[e^{-\lambda Y_1}]) ) \\[3pt] &= \exp \bigg({-} C_{f,k} \int_{{\mathbb R}^{d(k+1)}} ( 1-e^{-\lambda \sum_{i=1}^m a_i h_{t_i} (0,{\bf y})} ) \text{d}{\bf y} \bigg),\end{align*}

implying $\widehat T(\zeta) \stackrel{d}{=} \sum_{i=1}^m a_i \mathcal V_k(t_i)$ as required.

Part 3. It remains to show (6.29) and (6.30). As for (6.29), we know from (6.25) with $\rho_n = 1$ and $t_1 = t_2=t$ , that

\[{\mathbb{E}} [S_{k,n}(t)] \to \mu_{k,{\mathbb R}^d}(t,t), \quad n\to\infty.\]

Since the exponential term in (6.26) converges to 1 without affecting the value of the limit, it must be that the ${\mathbb{E}} [H_{k,n}(t)]$ and ${\mathbb{E}}[S_{k,n}(t)]$ have the same limit. That is,

\[{\mathbb{E}} [H_{k,n}(t)] \to \mu_{k,{\mathbb R}^d}(t,t), \quad n\to\infty,\]

and thus the Markov inequality gives (6.29).

Finally we turn our attention to (6.30). By Markov’s inequality, it suffices to show that ${\mathbb{E}} [R_{k,n}(t)] \to 0$ as $n\to\infty$ . Mimicking the derivation of (6.8) with $\rho_n = 1$ , we obtain

\[{\mathbb{E}}[R_{k,n}(t)] \leq \sum_{i=k+3}^\infty \binom{i}{k+1} \dfrac{n^i}{i!}\, {\mathbb{P}} ( {\skew4\check{C}} (\mathcal X_i, r_n(t)) \text{ is connected} ).\]

Recalling the bound in (6.9), we have

\begin{align*}{\mathbb{E}} [R_{k,n}(t)] &\leq \dfrac{( t^d \|\, f \|_\infty \theta_d )^{k+1}}{(k+1)!}\, \sum_{i=k+3}^\infty \dfrac{i^{i-2}}{(i-k-1)!}\, ( nr_n(t)^d \|\, f \|_\infty \theta_d )^{i-(k+2)} \to 0\end{align*}

as $n\to\infty$ .