2.1. Applications

We provide some applications of our main results. The proofs are provided in Section 5. Our first result gives the limit theory for an unbiased estimator of the distribution function of the volume of a typical cell in a weighted Poisson–Voronoi tessellation.

Our next result gives the limit theory for an unbiased estimator of the $(d-1)$ -dimensional Hausdorff measure $\mathcal{H}^{d-1}$ of the boundary of a typical cell in a weighted Poisson–Voronoi tessellation.

There are naturally other applications of the general theorems. By choosing h appropriately, one could for example use the general results to deduce the limit theory for an unbiased estimator of the distribution function of either the surface area, inradius, or circumradius of a typical cell in a weighted Poisson–Voronoi tessellation.

4.1. Preliminary lemmas

In this section we omit in the notation the dependence on the weight $\rho$ that defines the tessellation. For simplicity, we write

\begin{equation*}H_{\lambda}(\eta \cap \hat{W}_\lambda ) \coloneqq H_{\lambda}^{\rho}(\eta \cap \hat{W}_\lambda ), \qquad H_{\lambda}(\eta ) \coloneqq H_{\lambda}^{\rho}(\eta),\end{equation*}

as well as

\begin{equation*}\hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda ) \coloneqq \hat{H}_{\lambda}^{\rho}(\eta \cap \hat{W}_\lambda ), \qquad \hat{H}_{\lambda}(\eta ) \coloneqq \hat{H}_{\lambda}^{\rho}(\eta ).\end{equation*}

Let us start with some useful first-order results.

Lemma 4.1. Under the assumptions of Theorem 2.2(ii), we have

\begin{equation*}\lim_{\lambda \to \infty} \lambda\,\mathbb{E} \left| H_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) \right| = 0.\end{equation*}

Proof. We denote by $\hat{\mathbb{Q}}$ the product of the Lebesgue measure on $\mathbb{R}^d$ and $\mathbb{Q}_\mathbb{M}$ . By the refined Campbell theorem and stationarity,

\begin{align*}&\mathbb{E} \left| H_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) \right| \\&\leq \mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \frac{ |h(C(\hat{x},\eta))|}{\textrm{Vol}(W_\lambda \ominus C(\hat{x},\eta))}\,{\bf 1}\{C(\hat{x},\eta) \subseteq W_\lambda\}\,{\bf 1}\bigg\{\textrm{Vol}(W_\lambda \ominus C(\hat{x}, \eta) ) < \frac{\lambda} {2} \bigg\} \\&= \int_{\hat{W}_\lambda} \mathbb{E}^{\hat{x}} \left( \frac{ |h(C(\hat{x},\eta))|}{\textrm{Vol}(W_\lambda \ominus C(\hat{x},\eta))}\,{\bf 1}\{C(\hat{x},\eta) \subseteq W_\lambda\}\,{\bf 1}\bigg\{\textrm{Vol}(W_\lambda \ominus C(\hat{x}, \eta) ) < \frac{\lambda} {2} \bigg\}\right) \,\hat{\mathbb{Q}}(\textup{d} \hat{x}) \\&= \int_{W_\lambda}\int_\mathbb{M} \mathbb{E}^{({\bf 0},m)} \Biggl( \frac{ |h(C(({\bf 0},m),\eta))|}{\textrm{Vol}(W_\lambda \ominus C(({\bf 0},m),\eta))}\,{\bf 1}\{x \in W_\lambda \ominus C(({\bf 0},m),\eta)\} \\&\quad \times {\bf 1}\bigg\{\textrm{Vol}(W_\lambda \ominus C(({\bf 0},m), \eta) ) < \frac{\lambda} {2} \bigg\}\Biggr)\,\mathbb{Q}_\mathbb{M}(\textup{d} m)\,\textup{d} x.\end{align*}

Changing the order of integration we get

(4.1) \begin{align}\mathbb{E} \left| H_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) \right| &\leq \int_{\mathbb{M}} \!\! \mathbb{E}^{{\bf 0}_m} \bigg( |h(C({\bf 0}_m,\eta))|{\bf 1}\bigg\{ \textrm{Vol}( W_\lambda \ominus C({\bf 0}_m,\eta)) < \frac{\lambda} {2} \bigg\} \nonumber\\&\quad\times \int_{W_\lambda}\frac{{\bf 1}\{ x \in W_\lambda \ominus C({\bf 0}_m,\eta) \}}{\textrm{Vol}(W_\lambda \ominus C({\bf 0}_m,\eta))} \,\textup{d} x\bigg)\,\mathbb{Q}_\mathbb{M}(\textup{d} m),\end{align}

where ${\bf 0}_m \coloneqq ({\bf 0},m)$ . The inner integral over $W_\lambda$ is bounded by one, showing that for all $p \in (1, \infty)$ we have

\begin{align*}& \mathbb{E} \left| H_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) \right|\\&\quad\leq \int_{\mathbb{M}} \mathbb{E}^{{\bf 0}_m} \left( |h(C({\bf 0}_m,\eta))|\,{\bf 1}\bigg\{ \textrm{Vol}( W_\lambda \ominus C({\bf 0}_m,\eta)) < \frac{\lambda} {2} \bigg\}\right)\,\mathbb{Q}_\mathbb{M}(\textup{d} m) \\ &\quad\leq \int_{\mathbb{M}} \left(\mathbb{E}^{{\bf 0}_m} |h(C({\bf 0}_m,\eta))|^{p}\right)^{\frac{1}{p}} \, \mathbb{P}^{{\bf 0}_m} \left( \textrm{Vol}( W_\lambda \ominus C({\bf 0}_m,\eta)) < \frac{\lambda} {2} \right)^{\frac{p-1}{p}} \,\mathbb{Q}_\mathbb{M}(\textup{d} m).\end{align*}

The random variable D at (3.1) satisfies $C(\hat{\bf 0},\eta) \subseteq B_D({\bf 0})$ almost surely. Thus,

\begin{equation*}\mathbb{P}^{\hat{\bf 0}} \left( \textrm{Vol}( W_\lambda \ominus C(\hat{\bf 0},\eta) ) < \frac{\lambda} {2} \right) \leq \mathbb{P}^{\hat{\bf 0}} \left( \textrm{Vol}( W_\lambda \ominus B_D({\bf 0}) ) < \frac{\lambda} {2} \right).\end{equation*}

The volume of the erosion on the right-hand side equals $(\lambda^{1/d}-2D)_+ ^d$ . By conditioning on $Y \coloneqq {\bf 1}\{\lambda^{1/d}\geq 2D\}$ , we obtain

\begin{align*}\mathbb{P}^{\hat{\bf 0}} \left((\lambda^{1/d}-2D)_+ ^d < \frac{\lambda} {2} \right)&= \mathbb{P}^{\hat{\bf 0}} \left((\lambda^{1/d}-2D)_+ ^d < \frac{\lambda} {2} \, \Big| \, Y=1\right)\, \mathbb{P}^{\hat{\bf 0}}(Y=1) \\[4pt] &\quad + \mathbb{P}^{\hat{\bf 0}} \left((\lambda^{1/d}-2D)_+ ^d < \frac{\lambda} {2} \, \Big| \, Y=0\right)\, \mathbb{P}^{\hat{\bf 0}}(Y=0)\\[4pt] & \leq \mathbb{P}^{\hat{\bf 0}} \left((\lambda^{1/d}-2D) ^d < \frac{\lambda} {2} \right) + \mathbb{P}^{\hat{\bf 0}} (\lambda^{1/d} < 2D)\\[4pt] & \leq 2 \mathbb{P}^{\hat{\bf 0}} (D > e(\lambda)),\end{align*}

where $e(\lambda) \coloneqq (\lambda^{1/d} - (\lambda/2)^{1/d})/2$ . Finally, recalling that D has exponentially decaying tails as at (2.3), we obtain

\begin{equation*}\mathbb{P}^{\hat{\bf 0}} \left( \textrm{Vol}( W_\lambda \ominus C(\hat{\bf 0},\eta) ) < \frac{\lambda} {2} \right) \le 2 \, c_\textrm{diam} \exp\left(-\frac{1}{c_\textrm{diam}}\, e(\lambda)^d \right).\end{equation*}

Using this bound, we have

\begin{align*}& \lambda\,\mathbb{E} \left| H_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) \right|\\&\quad \leq \lambda\int_{\mathbb{M}} ( \mathbb{E}^{{\bf 0}_m} |h(C({\bf 0}_m,\eta))|^{p})^{\frac{1}{p}} \left( 2\, c_\textrm{diam} \exp\left(- \frac{1}{c_\textrm{diam}}\, e(\lambda)^d\right)\right)^{\frac{p-1}{p}}\,\mathbb{Q}_\mathbb{M}(\textup{d} m).\end{align*}

Now $\xi$ satisfies the p-moment condition (2.2) for $p \in (1, \infty)$ and thus the first factor is bounded by a constant uniformly over all $m \in \mathbb{M}.$ Lemma 4.1 follows.

Lemma 4.2. Under the assumptions of Theorem 2.2(i), we have

\begin{equation*}\lim_{\lambda \to \infty} \lambda\,\mathbb{E} \left| H_{\lambda}(\eta) - \hat{H}_{\lambda}(\eta) \right| = 0.\end{equation*}

Proof. We follow the proof of Lemma 4.1. In (4.1), we integrate over $\mathbb{R}^d$ instead of over $W_\lambda$ , yielding a value of one for the inner integral. Now follow the proof of Lemma 4.1 verbatim.

Lemma 4.3. Under the assumptions of Theorem 2.2(i), we have

\begin{equation*}\lim_{\lambda \to \infty} \mathbb{E} \left| \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) - \hat{H}_{\lambda}(\eta) \right| = 0.\end{equation*}

Proof. Write

(4.2) \begin{align*} \hat{\nu}_\lambda(\hat{x}, \eta) \coloneqq & \frac{ h(C( \hat{x}, \eta)) \, {\bf 1}\{ C(\hat{x}, \eta) \subseteq W_\lambda \}} { \textrm{Vol}( W_\lambda \ominus C(\hat{x}, \eta) ) }\\[5pt] & \times {\bf 1}\bigg\{ \textrm{Vol}( W_\lambda \ominus C( \hat{x}, \eta) ) \geq \frac{\lambda} {2} \bigg\}\,{\bf 1}\{D_{\hat{x}} \geq d(x, W_\lambda)\},\end{align*}

where $D_{\hat{x}}$ is the radius of the ball centered at x and containing $C(\hat{x}, \eta)$ , and where $D_{\hat{x}}$ is equal in distribution to D, with D as in (3.1). Here, $d(x, W_\lambda)$ denotes the Euclidean distance between x and $W_\lambda$ . We observe that

\begin{equation*}\mathbb{E} \left| \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) - \hat{H}_{\lambda}(\eta) \right| \leq \mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda ^c} \left| \hat{\nu}_\lambda(\hat{x}, \eta) \right|.\end{equation*}

From now on, we use the notation c to denote a universal positive constant whose value may change from line to line. By the Hölder inequality, the p-moment condition on $\xi$ , and the assumption that $C(\hat{x}, \eta)$ has an exponentially decaying tail, we have $\mathbb{E} |\hat{\nu}_\lambda(\hat{x}, \eta)| \leq (c/\lambda) \exp\left( - \frac{1}{c} d(x, W_\lambda)^d \right)$ . Thus,

\begin{equation*}\mathbb{E} \left| \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) - \hat{H}_{\lambda}(\eta) \right| \leq \frac{c}{\lambda} \int_{W_\lambda^c} \exp\left( - \frac{1}{c} \, d(x, W_\lambda)^d \right)\,\textup{d} x.\end{equation*}

Let $W_{\lambda,\varepsilon}$ be the set of points in $W_\lambda^c$ at distance $\varepsilon$ from $W_\lambda$ . The co-area formula implies

\begin{equation*}\mathbb{E} \left| \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) - \hat{H}_{\lambda}(\eta) \right| \leq \frac{c}{\lambda} \int_0^{\infty} \int_{W_{\lambda,\varepsilon}} \exp\left( - \frac{1}{c} \, \varepsilon^d \right)\, \mathcal{H}^{d-1}(\textup{d} y)\,\textup{d} \varepsilon.\end{equation*}

Since $\mathcal{H}^{d-1} ( W_{\lambda,\varepsilon}) \leq c\,(\lambda^{1/d}(1 + \varepsilon) )^{d-1}$ , we get

\begin{equation*}\hspace*{110pt}\mathbb{E} \left| \hat{H}_{\lambda}(\eta\cap \hat{W}_\lambda) - \hat{H}_{\lambda}(\eta) \right| = O(\lambda^{-1/d}). \hspace*{110pt}\square\end{equation*}

4.2. Proof of Theorem 2.2

1. (i) The asymptotic unbiasedness of $H_{\lambda}(\eta \cap \hat{W}_\lambda)$ , $\hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )$ , and $\hat{H}_{\lambda}(\eta)$ is a consequence of Lemmas 4.1, 4.2, and 4.3. For example, concerning $H_{\lambda}(\eta \cap \hat{W}_\lambda)$ , one may write

   \begin{align*}& |\mathbb{E} H_{\lambda}(\eta \cap \hat{W}_\lambda )-\mathbb{E}^0 h(K_{\bf 0}(\eta) )| \leq \mathbb{E} |H_{\lambda}(\eta \cap \hat{W}_\lambda ) - H_{\lambda}(\eta)| \\&\quad \leq \left( \mathbb{E} |H_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )| + \mathbb{E} |\hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda ) - \hat{H}_{\lambda}(\eta)| + \mathbb{E} |\hat{H}_{\lambda}(\eta) - H_{\lambda}(\eta)| \right),\end{align*}
   which in view of Lemmas 4.1, 4.2, and 4.3 goes to zero as $\lambda \to \infty$ . This gives the asymptotic unbiasedness of $H_{\lambda}(\eta \cap \hat{W}_\lambda )$ . One may similarly show the asymptotic unbiasedness for $\hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )$ and $\hat{H}_{\lambda}(\eta)$ .

2. (ii) To show consistency, we introduce $T_\lambda(\eta \cap \hat{W}_\lambda)= \lambda^{-1} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \xi(\hat{x}, \eta)$ . By assumption, $\xi$ stabilizes and satisfies the p-moment condition for $p \in (1, \infty)$ . Thus, using Theorem 2.1 of [Reference Penrose and Yukich15], we get that $T_\lambda(\eta \cap \hat{W}_\lambda)$ is a consistent estimator of $\mathbb{E}^0 h(K_{\bf 0}(\eta) ).$ To prove the consistency of the estimators in Theorem 2.2(iii), it is enough to show for one of them that it has the same $L_1$ limit as $T_\lambda(\eta \cap \hat{W}_\lambda)$ . We choose $ \hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )$ and write

   \begin{align*}&\mathbb{E} \left| \hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )-T_\lambda(\eta \cap \hat{W}_\lambda)\right| \\[4pt] &\quad = \mathbb{E} \left|\lambda^{-1} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \xi(\hat{x}, \eta) \left( \frac{ \lambda\,{\bf 1}\{ C(\hat{x}, \eta) \subseteq W_\lambda \}\,{\bf 1}\{ \textrm{Vol}( W_\lambda \ominus C(\hat{x}, \eta) ) \geq \frac{\lambda}{2} \} } { \textrm{Vol}( W_\lambda \ominus C(\hat{x}, \eta) ) } -1 \right) \right| \\[4pt] &\quad \leq \lambda^{-1} \mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} |\xi(\hat{x}, \eta)| \left| \frac{ \lambda\,{\bf 1}\{ C(\hat{x}, \eta) \subseteq W_\lambda \}\,{\bf 1}\{ \textrm{Vol}( W_\lambda \ominus C(\hat{x}, \eta) ) \geq \frac{\lambda}{2}\} } { \textrm{Vol}( W_\lambda \ominus C(\hat{x}, \eta) ) } -1 \right| \\[4pt] &\quad \leq \int_{W_\lambda} \lambda^{-1} \mathbb{E}^0 \left( |h(K_{\bf 0}(\eta) )| \left| \frac{ \lambda\,{\bf 1}\{ x + K_{\bf 0}(\eta) \subseteq W_\lambda \}\,{\bf 1}\{ \textrm{Vol}( W_\lambda \ominus K_{\bf 0}(\eta) ) \geq \frac{\lambda}{2} \}} {\textrm{Vol}( W_\lambda \ominus K_{\bf 0}(\eta) ) } -1 \right| \right)\,\textup{d} x\\[4pt] &\quad = \int_{[\!- \frac{1} {2}, \frac{1} {2 } ]^d} \mathbb{E}^0 \big( |h(K_{\bf 0}(\eta))| Y_\lambda(u)\big)\,\textup{d} u,\end{align*}
   where we substituted $\lambda^{1/d}u$ for x in the last equality and the defined random variables
   \begin{equation*}Y_{\lambda} (u) \coloneqq \left| \frac{ \lambda\,{\bf 1}\{ \lambda^{1/d} u + K_{\bf 0}(\eta) \subseteq W_\lambda \}\,{\bf 1}\{ \textrm{Vol}( W_\lambda \ominus K_{\bf 0}(\eta) ) \geq \frac{\lambda}{2} \}} { \textrm{Vol}( W_\lambda\ominus K_{\bf 0}(\eta) )} -1 \right|.\end{equation*}

We show that $Y_{\lambda} (u)$ converges to zero in $\mathbb{P}^0$ probability for any $u \in (-1/2, 1/2)^d$ . Write the inclusion $K_{\bf 0}(\eta) \subseteq B_D({\bf 0})$ , where D has exponentially decaying tails by assumption. We conclude that both $\lambda/\textrm{Vol}(W_\lambda \ominus K_{\bf 0}(\eta))$ and ${\bf 1}\{ \textrm{Vol}( W_\lambda\ominus K_{\bf 0}(\eta) ) \geq \lambda/2 \}$ tend to one in $\mathbb{P}^0$ probability. To prove the convergence of $Y_{\lambda}(u)$ to zero in $\mathbb{P}^0$ probability, it remains to show that ${\bf 1}\{ \lambda^{1/d} u + K_{\bf 0}(\eta) \subseteq W_\lambda \}$ converges to one in $\mathbb{P}^0$ probability. Equivalently, we show that the $\mathbb{P}^0$ probability of the event $\{\lambda^{1/d} u + K_{\bf 0}(\eta) \subseteq W_\lambda \}$ goes to 1. Let $u \in (-1/2, 1/2)^d$ be fixed. Then

\begin{align*}&\mathbb{P}^0(\lambda^{1/d}u \in W_\lambda \ominus K_{\bf 0}(\eta) ) \geq \mathbb{P}^0(\lambda^{1/d}u \in W_\lambda \ominus B_D({\bf 0})) \\[2pt] &\quad = \mathbb{P}^0 \left(u \in \left[\!-\frac{1}{2} + \frac{D}{\lambda^{1/d}}, \frac{1}{2} - \frac{D}{\lambda^{1/d}} \right]^d \right)\\[2pt] &\quad = \mathbb{P}^0 \left(u \in \left[\!-\frac{1}{2} + \frac{D}{\lambda^{1/d}}, \frac{1}{2} - \frac{D}{\lambda^{1/d}} \right]^d \, \Big| \, D\leq \log{\lambda}\right) \mathbb{P}^0(D\leq \log{\lambda})\\[2pt] &\qquad + \mathbb{P}^0 \left(u \in \left[\!-\frac{1}{2} + \frac{D}{\lambda^{1/d}}, \frac{1}{2} - \frac{D}{\lambda^{1/d}} \right]^d \, \Big| \, D > \log{\lambda}\right) \mathbb{P}^0(D > \log{\lambda})\displaybreak \\ &\quad \geq {\bf 1} \left\{u \in \left[\!-\frac{1}{2} + \frac{\log{\lambda}}{\lambda^{1/d}}, \frac{1}{2} - \frac{\log{\lambda}}{\lambda^{1/d}} \right]^d \right\} \mathbb{P}^0(D\leq \log{\lambda}) \\ &\qquad + \mathbb{P}^0 \left(u \in \left[\!-\frac{1}{2} + \frac{D}{\lambda^{1/d}}, \frac{1}{2} - \frac{D}{\lambda^{1/d}} \right]^d \, \Big| \, D > \log{\lambda}\right) \mathbb{P}^0(D > \log{\lambda}).\end{align*}

Again, D has exponentially decaying tails, so the lower bound converges to $\mathbb{P}^0 (u \in (-1/2, 1/2)^d) = 1$ , showing that $Y_\lambda(u)$ goes to zero in probability as $\lambda \to \infty$ . We proved that the $Y_{\lambda} (u)$ converge to zero in $\mathbb{P}^0$ probability, but they are also uniformly bounded by one, hence it follows from the moment condition on $\xi$ that $h(K_{\bf 0}(\eta))Y_\lambda(u)$ goes to zero in $L^1$ . Finally, by the dominated convergence theorem, we get

\begin{equation*}\lim_{\lambda \to \infty}\mathbb{E} \left| \hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )-T_\lambda(\eta \cap \hat{W}_\lambda)\right| = 0.\end{equation*}

Thus, $ \hat{H}_{\lambda}(\eta \cap \hat{W}_\lambda )$ converges to $\mathbb{E}^0 h(K_{\bf 0}(\eta))$ in $L^1$ and also in probability. The consistency of the remaining estimators in Theorem 2.2 follows from Lemmas 4.1, 4.2, and 4.3. This completes the proof of Theorem 2.2.

4.3. Proof of Theorem 2.3(i)

We prove the variance asymptotics (2.6). The proof is split into two lemmas (Lemmas 4.5 and 4.6). We first show an auxiliary result used in the proofs of both lemmas. Then we prove the variance asymptotics for $\hat{H}_\lambda(\eta \cap \hat{W}_\lambda)$ . This is easier, since, after scaling by $\lambda$ , the scores are bounded by $2 |\xi(\hat{x},\eta)|$ and thus, by assumption, satisfy a p-moment condition for some $p \in (2, \infty)$ . Finally, we conclude the proof by showing that the asymptotic variance of $\hat{H}_\lambda(\eta)$ is the same as the asymptotic variance of $\hat{H}_\lambda(\eta \cap \hat{W}_\lambda)$ .

Lemma 4.4. Let $\varphi\,{:}\, \hat{\mathbb{R}^d} \times {\bf N} \to \mathbb{R}$ be an exponentially stabilizing function with respect to $\eta$ that satisfies the p-moment condition for some $p \in (2, \infty)$ . Then there exists a constant $c \in (0, \infty)$ such that, for all $\hat{x}, \hat{y} \in \hat{\mathbb{R}^d}$ ,

(4.3) \begin{align} &|\mathbb{E} \varphi(\hat{x}, \eta \cup \{\hat{y}\})\varphi(\hat{y}, \eta \cup \{\hat{x}\}) - \mathbb{E} \varphi(\hat{x}, \eta)\,\mathbb{E}\varphi(\hat{y}, \eta)| \nonumber \\&\qquad\leq {c}\,\left( \sup_{\hat{x}, \hat{y} \in \hat{\mathbb{R}^d} } \mathbb{E} | \varphi(\hat{x}, \eta \cup \{ \hat{y} \} )|^{p}\right)^{\frac{2}{p}}\exp\left(-\frac{1}{c}\, \|x - y\|^{ \alpha } \right),\end{align}

where $\varphi(\hat{x},\eta) \coloneqq \varphi(\hat{x},\eta \cup \{\hat{x}\})$ if $\hat{x} \not\in \eta$ .

Proof. We follow the proof of Lemma 5.2 in [Reference Baryshnikov and Yukich2] and show that the constant $A_{1,1}$ there involves the moment $\smash{(\mathbb{E} | \varphi(\hat{x}, \eta \cup \{\hat{y}\})|^p)^{\frac{2}{p}}}$ . Put $R \coloneqq \max (R_{\hat{x}}, R_{\hat{y}})$ , where $R_{\hat{x}}, R_{\hat{y}}$ are the radii of stabilization as in Proposition 3.2 for $\hat{x}$ and $\hat{y}$ , respectively. Furthermore, put $r \coloneqq \|x-y\|/3$ and define the event $E \coloneqq \{ R \leq r \}$ . Hölder’s inequality gives

(4.4) \begin{align} & | \mathbb{E} \varphi(\hat{x}, \eta \cup \{\hat{y}\} ) \varphi(\hat{y}, \eta \cup \{\hat{x}\} ) - \mathbb{E} \varphi(\hat{x}, \eta \cup \{\hat{y}\} ) \varphi(\hat{y}, \eta \cup \{\hat{x}\} ){\bf 1}\{E\} | \nonumber \\&\qquad \leq c \left(\sup_{ \hat{x}, \hat{y} \in \hat{\mathbb{R}^d} } \mathbb{E} | \varphi(\hat{x}, \eta \cup \{\hat{y}\} )|^p\right)^{\frac{2}{p}} \, \mathbb{P}(E^c)^{\frac{p-2}{p}}.\end{align}

Notice that

\begin{align*}& \mathbb{E} \varphi(\hat{x}, \eta \cup \{\hat{y}\} ) \varphi(\hat{y}, \eta \cup \{\hat{x}\} ){\bf 1}\{E\} \\&\quad = \mathbb{E} \varphi(\hat{x}, (\eta \cup \{\hat{y}\}) \cap \hat{B}_{r}(\hat{x}) ) \varphi(\hat{y}, (\eta \cup \{\hat{x}\}) \cap \hat{B}_{r}(\hat{x}) ){\bf 1}\{E\} \\&\quad =\mathbb{E} \varphi(\hat{x}, (\eta \cup \{\hat{y}\}) \cap \hat{B}_{r}(\hat{x}) ) \varphi(\hat{y}, (\eta \cup \{\hat{x}\}) \cap \hat{B}_{r}(\hat{x}) ) (1 - {\bf 1}\{E^c\}).\end{align*}

A second application of Hölder’s inequality gives

(4.5) \begin{align} & | \mathbb{E} \varphi(\hat{x}, \eta \cup \{\hat{y}\} ) \varphi(\hat{y}, \eta \cup \{\hat{x}\} ){\bf 1}\{E\}- \mathbb{E} \varphi(\hat{x}, (\eta \cup \{\hat{y}\}) \cap \hat{B}_{r}(\hat{x}) ) \varphi(\hat{y}, (\eta \cup \{\hat{x}\}) \cap \hat{B}_{r}(\hat{y}) ) |\nonumber \\&\quad \leq c \left(\sup_{ \hat{x}, \hat{y} \in \hat{\mathbb{R}^d} } \mathbb{E} | \varphi(\hat{x}, \eta \cup \{\hat{y}\} )|^p\right)^{\frac{2}{p}} \, \mathbb{P}(E^c)^{\frac{p-2}{p}}.\end{align}

Thus, combining (4.4) and (4.5) and using the independence of $\varphi(\hat{x}, (\eta \cup \{\hat{y}\}) \cap \hat{B}_{r}(\hat{x}) )$ and $\varphi(\hat{y}, (\eta \cup \{\hat{x}\}) \cap \hat{B}_{r}(\hat{y}) )$ we have

(4.6) \begin{align} & | \mathbb{E} \varphi(\hat{x}, \eta \cup \{\hat{y}\} ) \varphi(\hat{y}, \eta \cup \{\hat{x}\} )- \mathbb{E} \varphi(\hat{x}, (\eta \cup \{\hat{y}\}) \cap \hat{B}_{r}(\hat{x}) ) \mathbb{E} \varphi(\hat{y}, (\eta \cup \{\hat{x}\}) \cap \hat{B}_{r}(\hat{y}) ) | \nonumber \\&\qquad \leq c \left(\sup_{ \hat{x}, \hat{y} \in \hat{\mathbb{R}^d} } \mathbb{E} | \varphi(\hat{x}, \eta \cup \{\hat{y}\} )|^p\right)^{\frac{2}{p}} \, \mathbb{P}(E^c)^{\frac{p-2}{p}}.\end{align}

Likewise, we may show that

(4.7) \begin{align} & | \mathbb{E} \varphi(\hat{x}, \eta ) \mathbb{E} \varphi(\hat{y}, \eta ) - \mathbb{E} \varphi(\hat{x}, \eta \cap \hat{B}_{r}(\hat{x}) ) \mathbb{E} \varphi(\hat{y}, \eta \cap \hat{B}_{r}(\hat{y}) ) | \nonumber \\&\qquad \leq c \left(\sup_{ \hat{x}, \hat{y} \in \hat{\mathbb{R}^d} } \mathbb{E} | \varphi(\hat{x}, \eta \cup \{\hat{y}\} )|^p\right)^{\frac{2}{p}} \, \mathbb{P}(E^c)^{\frac{p-2}{p}}.\end{align}

Combining (4.6) and (4.7), and using that $\mathbb{P}(E^c)$ decreases exponentially in $\|x - y\|^{\alpha}$ , we thus obtain (4.3).

Lemma 4.5. If $\xi$ is exponentially stabilizing with respect to $\eta$ then

\begin{equation*}\lim_{\lambda \to \infty} \lambda \textrm{Var} \hat{H}_\lambda(\eta \cap \hat{W}_\lambda) = \sigma^2(\xi),\end{equation*}

where $\sigma^2(\xi)$ is as in (2.5).

Proof. Put, for all $\hat{x} \in \hat{\mathbb{R}^d}$ and any marked point process $\mathcal{P}$ ,

\begin{equation*}\zeta_\lambda(\hat{x}, \mathcal{P}) \coloneqq \frac{ \lambda\,\xi( \hat{x}, \mathcal{P}) } { \textrm{Vol}( W_\lambda \ominus C(\hat{x}, \mathcal{P}) ) }\, {\bf 1}\bigg\{ \textrm{Vol}( W_\lambda \ominus C( \hat{x}, \mathcal{P}) ) \geq \frac{\lambda} {2} \bigg\}\end{equation*}

and

\begin{equation*}\nu_\lambda(\hat{x}, \mathcal{P}) \coloneqq \zeta_\lambda(\hat{x},\mathcal{P})\, {\bf 1}\{ C(\hat{x}, \mathcal{P}) \subseteq W_\lambda \}.\end{equation*}

Note that $\zeta_\lambda$ is translation invariant whereas $\nu_\lambda$ is not translation invariant. Then, $\lambda\,\hat{H}_\lambda(\eta \cap \hat{W}_\lambda) = \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \nu_\lambda(\hat{x},\eta)$ .

Recall that $\hat{\mathbb{Q}}$ is the product measure of Lebesgue measure on $\mathbb{R}^d$ and $\mathbb{Q}_{\mathbb{M}}$ . By the Slivnyak–Mecke theorem [Reference Schneider and Weil16, Corollary 3.2.3] we have

\begin{align*} \lambda \textrm{Var} \hat{H}_\lambda(\eta \cap \hat{W}_\lambda) & = \lambda^{-1} \mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \nu_\lambda^2(\hat{x},\eta) \\ & \quad + \lambda^{-1} \mathbb{E} \sum_{\hat{x}, \hat{y} \in \eta \cap \hat{W}_\lambda; \hat{x} \neq \hat{y}} \nu_\lambda (\hat{x},\eta)\nu_\lambda (\hat{y},\eta)- \lambda^{-1} \left(\mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \nu_\lambda(\hat{x},\eta)\right)^2 \\ & = \lambda^{-1} \int_{\hat{W}_\lambda} \mathbb{E} \nu_\lambda^2(\hat{x}, \eta)\,\hat{\mathbb{Q}}(\textup{d} \hat{x}) \\[4pt] & \quad + \lambda^{-1} \int_{\hat{W}_\lambda} \int_{\hat{W}_\lambda} \left[ \mathbb{E} \nu_\lambda(\hat{x}, \eta \cup \{\hat{y}\} ) \nu_\lambda(\hat{y}, \eta \cup \{\hat{x}\} )\right.\\[4pt] & \quad \left. -\ \mathbb{E} \nu_\lambda(\hat{x}, \eta)\,\mathbb{E} \nu_\lambda (\hat{y}, \eta)\right]\,\hat{\mathbb{Q}}(\textup{d} \hat{y})\,\hat{\mathbb{Q}}(\textup{d} \hat{x}) \\ & \eqqcolon\ I_1(\lambda) + I_2(\lambda).\end{align*}

Here we use the convention that $\nu_\lambda(\hat{x},\mathcal{P}) \coloneqq \nu_\lambda(\hat{x},\mathcal{P} \cup \{\hat{x}\})$ if $\hat{x} \not\in \mathcal{P}$ .

Using stationarity and the transformation $u \coloneqq \lambda^{1/d}x$ , we rewrite $I_1(\lambda)$ as

\begin{equation*}I_1(\lambda) = \lambda^{-1} \int_{W_\lambda}\int_{\mathbb{M}} \mathbb{E} Z_\lambda^2({\bf 0}_m,\eta,x)\,\mathbb{Q}_\mathbb{M}(\textup{d} m)\,\textup{d} x = \int_{W_1} \mathbb{E} Z_\lambda^2({\bf 0}_M,\eta,\lambda^{1/d}u)\,\textup{d} u,\end{equation*}

where $Z_\lambda((z,m_z),\mathcal{P},x) \coloneqq \zeta_\lambda((z,m_z), \mathcal{P})\,{\bf 1}\{C((z,m_z),\mathcal{P}) \subseteq W_\lambda-x\}$ . Similarly, by translation invariance of $\zeta_\lambda$ , we have

\begin{align*}I_2(\lambda) &= \lambda^{-1} \int_{W_\lambda} \int_{W_\lambda-x} \int_{\mathbb{M}} \int_{\mathbb{M}} [ \mathbb{E} Z_\lambda({\bf 0}_{m_1}, \eta \cup \{z_{m_2}\}, x)\,Z_\lambda(z_{m_2}, \eta \cup \{{\bf 0}_{m_1}\}, x) \\[4pt] &\quad - \mathbb{E} Z_\lambda({\bf 0}_{m_1}, \eta, x)\,\mathbb{E} Z_\lambda(z_{m_2}, \eta, x)]\,\mathbb{Q}_\mathbb{M}(\textup{d} m_1)\,\mathbb{Q}_\mathbb{M}(\textup{d} m_2)\,\textup{d} z\,\textup{d} x \\[4pt] &= \int_{W_1} \int_{W_\lambda-\lambda^{1/d}u} [\mathbb{E} Z_\lambda({\bf 0}_M,\eta \cup \{z_M\},\lambda^{1/d}u)\, Z_\lambda(z_M,\eta \cup \{{\bf 0}_M\},\lambda^{1/d}u) \\[4pt] &\quad - \mathbb{E} Z_\lambda({\bf 0}_M,\eta,\lambda^{1/d}u)\,\mathbb{E} Z_\lambda(z_M,\eta,\lambda^{1/d}u)]\,\textup{d} z\,\textup{d} u,\end{align*}

where ${\bf 0}_{m_1} \coloneqq ({\bf 0},m_1)$ , $z_{m_2} \coloneqq (z,m_2)$ , ${\bf 0}_M \coloneqq ({\bf 0},M_{\bf 0})$ , $z_M \coloneqq (z,M_z)$ , and $M_{\bf 0}$ , $M_z$ are random marks distributed according to $\mathbb{Q}_\mathbb{M}$ .

Since $|\zeta_\lambda(\hat{x},\eta)| \leq 2|\xi(\hat{x},\eta)|$ , $\zeta_\lambda$ satisfies a p-moment condition, $p \in (2, \infty)$ . Recall that $\textrm{Vol}(W_\lambda \ominus C(\hat{x},\eta))/\lambda$ tends in probability to 1, and notice that $W_\lambda - \lambda^{1/d}u$ for $u \in (-1/2,1/2)^d$ increases to $\mathbb{R}^d$ as $\lambda \to \infty$ . Thus, as $\lambda \to \infty$ , we have, for any $\hat{\bf 0} \coloneqq ({\bf 0},m_{\bf 0})$ , $\hat{z} \coloneqq (z,m_z) \in \hat{\mathbb{R}^d}$ , and $u \in (-1/2,1/2)^d$ ,

(4.8) \begin{align}\mathbb{E} Z_\lambda(\hat{\bf 0},\eta,\lambda^{1/d}u) &\to \mathbb{E} \xi(\hat{\bf 0},\eta), \end{align}

(4.9) \begin{align}\mathbb{E} Z_\lambda^2(\hat{\bf 0}, \eta,\lambda^{1/d}u) &\to \mathbb{E} \xi^2(\hat{\bf 0}, \eta), \end{align}

(4.10) \begin{align}\mathbb{E} Z_\lambda(\hat{\bf 0}, \eta \cup \{ \hat{z}\}, \lambda^{1/d}u)Z_\lambda(\hat{z},\eta \cup \{ \hat{\bf 0} \} ,\lambda^{1/d}u) &\to \mathbb{E}\xi(\hat{\bf 0}, \eta \cup \{ \hat{z}\}) \xi(\hat{z}, \eta \cup \{\hat{\bf 0} \}). \end{align}

These ingredients are enough to establish variance asymptotics for $\hat{H}_\lambda(\eta \cap \hat{W}_\lambda)$ . Indeed, $I_1(\lambda)$ converges to $\mathbb{E} \xi^2({\bf 0}_M, \eta)$ by (4.9). Concerning $I_2(\lambda)$ , for each $u \in (-1/2,1/2)^d$ we have

\begin{align*}& \lim_{\lambda \to \infty} \int_{W_\lambda-\lambda^{1/d}u}[\mathbb{E} Z_\lambda({\bf 0}_M,\eta \cup \{z_M\},\lambda^{1/d}u) Z_\lambda(z_M,\eta \cup \{{\bf 0}_M\},\lambda^{1/d}u) \\[4pt] &\quad - \mathbb{E} Z_\lambda({\bf 0}_M,\eta,\lambda^{1/d}u)\,\mathbb{E} Z_\lambda(z_M,\eta,\lambda^{1/d}u)]\,\textup{d} z\\[4pt] &= \int_{\mathbb{R}^d} [ \mathbb{E} \xi({\bf 0}_M,\eta \cup \{z_M\}) \xi(z_M,\eta \cup \{{\bf 0}_M\}) - \mathbb{E} \xi({\bf 0}_M,\eta) \mathbb{E} \xi(z_M,\eta)]\,\textup{d} z.\end{align*}

Here we use that for any $x \in \mathbb{R}^d$ , the function $Z_\lambda(\cdot, \cdot, x)\,{:}\, \hat{\mathbb{R}^d} \times {\bf N} \to \mathbb{R}$ is exponentially stabilizing with respect to $\eta$ and satisfies the p-moment condition for some $p \in (2, \infty)$ . Thus, from Lemma 4.4, the integrand is dominated by an exponentially decaying function of $\|z\|^{ \alpha } $ . Applying the dominated convergence theorem, together with (4.8) and (4.10), we obtain the desired variance asymptotics since $\textrm{Vol}(W_1) = 1$ .

The next lemma completes the proof of Theorem 2.3(i).

Lemma 4.6. If $\xi$ is exponentially stabilizing with respect to $\eta$ then

\begin{equation*}\lim_{\lambda \to \infty} \lambda \textrm{Var} \hat{H}_\lambda(\eta) = \lim_{\lambda \to \infty} \lambda \textrm{Var} \hat{H}_\lambda(\eta\cap \hat{W}_\lambda ) = \sigma^2(\xi).\end{equation*}

Proof. Write

\begin{equation*}\lambda\,\hat{H}_\lambda(\eta) = \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \nu_\lambda(\hat{x},\eta) + \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \nu_\lambda(\hat{x},\eta).\end{equation*}

Now,

\begin{align*}\lambda \textrm{Var} \hat{H}_\lambda(\eta) = & \lambda^{-1} \textrm{Var} \left( \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \nu_\lambda(\hat{x},\eta)\right) + \lambda^{-1} \textrm{Var} \left( \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \nu_\lambda(\hat{x},\eta) \right) \\ & + 2 \lambda^{-1} \textrm{Cov} \left( \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda} \nu_\lambda(\hat{x},\eta), \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \nu_\lambda(\hat{x},\eta) \right).\end{align*}

It suffices to show that $\textrm{Var} \big( \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \nu_\lambda(\hat{x},\eta) \big) = O(\lambda^{(d-1)/d})$ , for then the Cauchy–Schwarz inequality shows that the covariance term in the above expression is negligible compared to $\lambda$ .

We show that $\textrm{Var} \big( \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \nu_\lambda(\hat{x},\eta) \big) = O(\lambda^{(d-1)/d})$ as follows. Note that $\hat{H}_\lambda(\eta) = \sum_{\hat{x} \in \eta} \hat{\nu}_\lambda(\hat{x}, \eta),$ where $\hat{\nu}_\lambda(\hat{x}, \eta)$ is as in (4.2). By the Slivnyak–Mecke theorem we have

\begin{align*}& \lambda \textrm{Var} \left( \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \nu_\lambda(\hat{x},\eta) \right) = \lambda^{-1} \mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \hat{\nu}_\lambda^2(\hat{x},\eta) \\ &\qquad + \lambda^{-1} \mathbb{E} \sum_{\hat{x}, \hat{y} \in \eta \cap \hat{W}_\lambda^c; \hat{x} \neq \hat{y}} \hat{\nu}_\lambda (\hat{x},\eta)\hat{\nu}_\lambda (\hat{y},\eta) - \lambda^{-1} \left(\mathbb{E} \sum_{\hat{x} \in \eta \cap \hat{W}_\lambda^c} \hat{\nu}_\lambda(\hat{x},\eta)\right)^2 \displaybreak\\ &\quad = \lambda^{-1} \int_{\hat{W}_\lambda^c} \mathbb{E} \hat{\nu}_\lambda^2(\hat{x}, \eta)\,\hat{\mathbb{Q}}(\textup{d} \hat{x}) \\ &\qquad + \lambda^{-1} \int_{\hat{W}_\lambda^c} \int_{\hat{W}_\lambda^c} [ \mathbb{E} \hat{\nu}_\lambda(\hat{x}, \eta \cup \{\hat{y}\} ) \hat{\nu}_\lambda(\hat{y}, \eta \cup \{\hat{x}\} ) - \mathbb{E} \hat{\nu}_\lambda(\hat{x}, \eta)\, \mathbb{E} \hat{\nu}_\lambda (\hat{y}, \eta)]\,\hat{\mathbb{Q}}(\textup{d} \hat{x})\,\hat{\mathbb{Q}}(\textup{d} \hat{y})\\ &\quad \eqqcolon I_1^*(\lambda) + I_2^*(\lambda).\end{align*}

By the Hölder inequality, the moment condition on $\xi$ , and the assumed exponential decay of the tail of the diameter of $C(\hat{x}, \eta)$ , we have $\mathbb{E} \hat{\nu}_\lambda(\hat{x}, \eta)^p \leq c \exp\left( - \frac{1}{c} \, d(x, W_\lambda)^d \right)$ for some positive constant c. Then, similarly to Lemma 4.3, we may use the co-area formula to obtain $I_1^*(\lambda) = O(\lambda^{-1/d}).$

To bound $I_2^*(\lambda)$ we appeal to Lemma 4.3. Notice that $|\hat{\nu}_\lambda(\hat{x},\eta)| \leq 2|\xi(\hat{x},\eta)|$ . Since $\hat{\nu}_\lambda, \lambda \geq 1$ are exponentially stabilizing with respect to $\eta$ and satisfy the p-moment condition for $p \in (2, \infty)$ , then, by Lemma 4.4,

\begin{align*}& | \mathbb{E} \hat{\nu}_\lambda(\hat{x}, \eta \cup \{\hat{y}\} ) \hat{\nu}_\lambda(\hat{y}, \eta \cup \{\hat{x}\} ) - \mathbb{E} \hat{\nu}_\lambda(\hat{x}, \eta)\, \mathbb{E} \hat{\nu}_\lambda (\hat{y}, \eta)|\\&\qquad \leq c\,\left(\sup_{ \hat{x}, \hat{y} \in \hat{\mathbb{R}^d} } \mathbb{E} | \hat{\nu}_\lambda(\hat{x}, \eta \cup \{\hat{y}\} )|^p\right)^{\frac{2}{p}} \exp\left(-\frac{1}{c} \, \|x - y\|^\alpha \right).\end{align*}

Using this estimate we compute

\begin{align*}I_2^*(\lambda) &\leq \lambda^{-1} \int_{\hat{W}_\lambda^c} \int_{W_\lambda^c} c \,(\mathbb{E} | \hat{\nu}_\lambda(\hat{x}, \eta)|^{p})^{\frac{2}{p}} \exp\left(-\frac{1}{c}\, \|x - y\|^\alpha \right)\,\textup{d} y\,\hat{\mathbb{Q}}(\textup{d} \hat{x}) \\[3pt] &\leq c\, \lambda^{-1} \int_{\hat{W}_\lambda^c} ( \mathbb{E} | \hat{\nu}_\lambda(\hat{x}, \eta)|^{p})^{\frac{2}{p}} \int_{\mathbb{R}^d} \exp\left(-\frac{1}{c} \, \|x - y\|^\alpha\right)\,\textup{d} y\,\hat{\mathbb{Q}}(\textup{d} \hat{x}) \\[3pt] &\leq c\, \lambda^{-1} \int_{W_\lambda^c} \exp\left( -\frac{1}{c} \, d(x, W_\lambda)^d \right) \,\textup{d} x \int_{\mathbb{R}^d} \exp\left(-\frac{1}{c}\, \|y\|^\alpha\right)\,\textup{d} y.\end{align*}

Since $\int_{\mathbb{R}^d} \exp(- \| y\|^\alpha/c )\,\textup{d} y < \infty$ , we obtain

\begin{equation*}I_2^*(\lambda) \leq c\, \lambda^{-1} \int_{W_\lambda^c} \exp\left( -\frac{1}{c} \, d(x, W_\lambda)^d \right)\,\textup{d} x.\end{equation*}

Arguing as we did for $I_1^*(\lambda)$ we obtain $I_2^*(\lambda) = O(\lambda^{-1/d}).$

4.4. Proof of Theorem 2.3(ii)

Now we prove the central limit theorems for $H_\lambda(\eta \cap \hat{W}_\lambda)$ and $H_\lambda(\eta)$ . Let us first introduce some notation. Define, for any stationary marked point process $\mathcal{P}$ on $\hat{\mathbb{R}^d}$ ,

\begin{align*}\xi_\lambda (\hat{x}, \mathcal{P}) &\coloneqq \frac{\lambda\,\xi( \lambda^{1/d} \hat{x}, \lambda^{1/d} \mathcal{P}) } { \textrm{Vol}( W_\lambda \ominus C( \lambda^{1/d} \hat{x}, \lambda^{1/d} \mathcal{P}) ) } \, {\bf 1}\{ C(\lambda^{1/d} \hat{x}, \lambda^{1/d}\mathcal{P}) \subseteq W_\lambda \},\\\hat{\xi}_\lambda(\hat{x}, \mathcal{P}) & \coloneqq \xi_\lambda(\hat{x}, \mathcal{P})\,{\bf 1}\bigg\{ \textrm{Vol}( W_\lambda \ominus C( \lambda^{1/d} \hat{x}, \lambda^{1/d} \mathcal{P}) ) \geq \frac{\lambda} {2} \bigg\},\end{align*}

where $\lambda^{1/d}\hat{x} \coloneqq (\lambda^{1/d} x,m_x)$ and $\lambda^{1/d}\mathcal{P} \coloneqq \{\lambda^{1/d}\hat{x}\,{:}\, \hat{x} \in \mathcal{P}\}$ .

Put

\begin{align*}S_\lambda(\eta_\lambda \cap \hat{W}_1) \coloneqq \sum_{\hat{x} \in \eta_\lambda \cap \hat{W}_1} \xi_\lambda(\hat{x}, \eta_\lambda), & \qquad \hat{S}_\lambda(\eta_\lambda \cap \hat{W}_1) \coloneqq \sum_{\hat{x} \in \eta_\lambda \cap \hat{W}_1} \hat{\xi}_\lambda(\hat{x}, \eta_\lambda),\end{align*}

as well as

\begin{align*} S_\lambda(\eta_\lambda) \coloneqq \sum_{\hat{x} \in \eta_\lambda} \xi_\lambda(\hat{x}, \eta_\lambda), & \qquad \hat{S}_\lambda (\eta_\lambda) \coloneqq \sum_{\hat{x} \in \eta_\lambda} \hat{\xi}_\lambda(\hat{x}, \eta_\lambda).\end{align*}

Notice that

\begin{equation*}S_\lambda(\eta_\lambda \cap \hat{W}_1) \stackrel{\textrm{D}}{=} \lambda\,H_\lambda(\eta \cap \hat{W}_\lambda), \qquad S_\lambda(\eta_\lambda) \stackrel{\textrm{D}}{=} \lambda\,H_\lambda(\eta) \end{equation*}

and

\begin{equation*}\hat{S}_\lambda(\eta_\lambda \cap \hat{W}_1) \stackrel{\textrm{D}}{=} \lambda\,\hat{H}_\lambda(\eta \cap \hat{W}_\lambda), \qquad \hat{S}_\lambda(\eta_\lambda) \stackrel{\textrm{D}}{=} \lambda\,\hat{H}_\lambda(\eta)\end{equation*}

due to the distributional identity $\lambda^{1/d} \eta_\lambda \stackrel{\textrm{D}}{=} \eta_1$ . The reason for expressing the statistic $\lambda\,H_\lambda(\eta \cap \hat{W}_\lambda)$ in terms of the scores $\xi_\lambda(\hat{x}, \eta_\lambda)$ is that it puts us in a better position to apply the normal approximation results of [Reference Lachièze-Rey, Schulte and Yukich6] to the sums $S_\lambda(\eta_\lambda \cap \hat{W}_1)$ .

In particular, we appeal to Theorem 2.3 of [Reference Lachièze-Rey, Schulte and Yukich6], with s replaced by $\lambda$ there, to establish a central limit theorem for $\hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1)$ . Indeed, in that paper we may put $\mathbb{X}$ to be $\mathbb{R}^d$ , we let $\mathbb{Q}$ be Lebesgue measure on $\mathbb{R}^d$ so that $\eta_\lambda $ has intensity measure $\lambda \mathbb{Q}$ , and we put $K = W_1$ . We may write $\hat{S}_\lambda(\eta_\lambda \cap \hat{W}_1) = \sum_{\hat{x} \in \eta_\lambda \cap \hat{W}_1 } \hat{\xi}_\lambda(\hat{x}, \eta_\lambda)\,{\bf 1}\{x \in W_1\}.$ Note that $\hat{\xi}_\lambda(\hat{x}, \eta_\lambda){\bf 1}\{x \in W_1\}$ , $\hat{x} \in \hat{\mathbb{X}}$ , are exponentially stabilizing with respect to the input $\eta_\lambda$ , they satisfy the p-moment condition for some $p \in (4, \infty)$ , they vanish for $x \in W_1^c$ , and they (trivially) decay exponentially fast with respect to the distance to K. (Here, the notion of decaying exponentially fast with respect to the distance to K is defined at (2.8) of [Reference Lachièze-Rey, Schulte and Yukich6]; since the distance to K is zero for $x \in K$ this condition is trivially satisfied.) This makes $I_{K, \lambda} = \Theta(\lambda)$ , where $I_{K, \lambda}$ is defined in (2.10) of [Reference Lachièze-Rey, Schulte and Yukich6]. Thus, all conditions of Theorem 2.3 of [Reference Lachièze-Rey, Schulte and Yukich6] are fulfilled and we deduce a central limit theorem for $\hat{S}_\lambda(\eta_\lambda \cap \hat{W}_1)$ , and hence for $\hat{H}_\lambda(\eta \cap \hat{W}_\lambda)$ .

We may also apply Theorem 2.3 of [Reference Lachièze-Rey, Schulte and Yukich6] to show a central limit theorem for $\hat{S}_\lambda(\eta_\lambda)$ . For $x \in W_1^c$ we find the radius $D_x$ such that $C(\lambda^{1/d}\hat{x},\lambda^{1/d}\eta_\lambda) \subseteq B_{D_x}(\lambda^{1/d}x)$ . Then the score $\hat{\xi}_\lambda(\hat{x}, \eta_\lambda)$ vanishes if $D_x > d(\lambda^{1/d}x,W_\lambda)$ . As in Section 3, $D_x$ has exponentially decaying tails and thus $\hat{\xi}_\lambda$ decays exponentially fast with respect to the distance to K.

Let $d_\textrm{K}(X, Y)$ denote the Kolmogorov distance between random variables X and Y. Applying Theorem 2.3 of [Reference Lachièze-Rey, Schulte and Yukich6] we obtain

\begin{equation*} d_\textrm{K} \left( \frac{ \hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1) - \mathbb{E} \hat{S}_\lambda(\eta_\lambda \cap \hat{W}_1) } { \sqrt{ \textrm{Var} \hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1) }}, N(0,1) \right) \leq \frac{ c } { \sqrt{ \textrm{Var} \hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1) }\,}\end{equation*}

and

\begin{equation*} d_\textrm{K} \left( \frac{ \hat{S}_\lambda (\eta_\lambda) - \mathbb{E} \hat{S}_\lambda(\eta_\lambda) } { \sqrt{ \textrm{Var} \hat{S}_\lambda(\eta_\lambda) }}, N(0,1) \right) \leq \frac{ c } { \sqrt{ \textrm{Var} \hat{S}_\lambda (\eta_\lambda) }\,}.\end{equation*}

Combining this with (2.6) and using $\textrm{Var} \hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1) \geq c\, \lambda$ , we obtain, as $\lambda \to \infty$ ,

\begin{equation*} \frac{ \hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1) - \mathbb{E} \hat{S}_\lambda (\eta_\lambda \cap \hat{W}_1)} { \sqrt{ \lambda}} \stackrel{\textrm{D}}{\longrightarrow} N(0, \sigma^2(\xi))\end{equation*}

and

\begin{equation*}\frac{ \hat{S}_\lambda (\eta_\lambda) - \mathbb{E} \hat{S}_\lambda (\eta_\lambda)} { \sqrt{ \lambda}} \stackrel{\textrm{D}}{\longrightarrow} N(0, \sigma^2(\xi)).\end{equation*}

To show that

(4.11) \begin{equation} \frac{ {S}_\lambda (\eta_\lambda \cap \hat{W}_1) - \mathbb{E} {S}_\lambda(\eta_\lambda \cap \hat{W}_1) } { \sqrt{ \lambda}} \stackrel{\textrm{D}}{\longrightarrow} N(0, \sigma^2(\xi)) \end{equation}

as $\lambda \to \infty$ , it suffices to show that $\lim_{\lambda \to \infty} \mathbb{E}| S_\lambda (\eta_\lambda \cap \hat{W}_1)- \hat{S}_\lambda(\eta_\lambda \cap \hat{W}_1)| = 0$ . Since $\mathbb{E}| S_\lambda (\eta_\lambda \cap \hat{W}_1) - \hat{S}_\lambda (\eta_\lambda\cap \hat{W}_1)| = \lambda\,\mathbb{E} |H _\lambda (\eta_\lambda \cap \hat{W}_\lambda) - \hat{H} _\lambda (\eta_\lambda \cap \hat{W}_\lambda)|$ , we may use Lemma 4.1 to prove (4.11). Likewise, to obtain the central limit theorem for $ {S}_\lambda (\eta_\lambda)$ , it suffices to show that $\lim_{\lambda \to \infty} \mathbb{E}| S_\lambda (\eta_\lambda)- \hat{S}_\lambda(\eta_\lambda)| = 0$ , which is a consequence of Lemma 4.2. Hence, we deduce from the central limit theorem for $\hat{S}_\lambda(\eta_\lambda)$ that, as $\lambda \to \infty$ ,

\begin{equation*} \frac{ {S}_\lambda(\eta_\lambda) - \mathbb{E} {S}_\lambda(\eta_\lambda) } { \sqrt{ \lambda}} \stackrel{\textrm{D}}{=} \sqrt{\lambda} \left( H_\lambda(\eta) - \mathbb{E}^0 h(K_{\bf 0}(\eta)) \right) \stackrel{\textrm{D}}{\longrightarrow} N(0, \sigma^2(\xi)).\end{equation*}

This completes the proof of Theorem 2.3(ii).

5.1. Proof of Theorem 2.4

1. (i) The assertion of unbiasedness follows from Theorem 2.1.

2. (ii) To prove asymptotic normality, we write

   \begin{equation*}h(C^{\rho_i}(\hat{x}, \eta)) \coloneqq {\bf 1}\{\textrm{Vol} (C^{\rho_i}(\hat{x}, \eta)) \leq t \} \eqqcolon \varphi^{\rho_i}(\hat{x}, \eta).\end{equation*}
   To deduce (2.7) from Theorem 2.3(ii) we need only verify the p-moment condition for $p \in (4, \infty)$ and the positivity of $\sigma^2(\varphi^{\rho_i})$ . The moment condition holds for all $p \ \in [1, \infty) $ since $\varphi$ is bounded by 1. To verify the positivity of $\sigma^2(\varphi^{\rho_i})$ , we recall Remark 2.1. More precisely, we may use Theorem 2.1 of [Reference Penrose and Yukich14] and show that there is an almost surely finite random variable S and a non-degenerate random variable $\Delta^{\rho_i}(\infty)$ such that, for all finite $\mathcal{A} \subseteq \hat{B}_S({\bf 0})^c$ , we have
   \begin{align*}\Delta^{\rho_i}(\infty) = & \sum_{ \hat{x} \in (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M \} }{\bf 1}\{\textrm{Vol} (C^{\rho_i}(\hat{x}, (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M\} ))\leq t \} \\ & - \sum_{ \hat{x} \in (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} } {\bf 1}\{\textrm{Vol} (C^{\rho_i}(\hat{x}, (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} )) \leq t \}.\end{align*}
   We first explain the argument for the Voronoi case and then indicate how to extend it to treat Laguerre and Johnson–Mehl tessellations.

Let $t \in (0,\infty)$ be arbitrary but fixed. Let N be the smallest integer of even parity that is larger than $4\sqrt{d}$ . The choice of this value will be explained later in the proof. For $L>0$ we consider a collection of $N^d$ cubes $Q_{L,1},\ldots, Q_{L,N^d}$ centered around $x_i$ , $i = 1,\ldots, N^d$ , such that

1. (i) $Q_{L,i}$ has side length $\frac{L} {N}$ , and

2. (ii) $\cup \{ Q_{L,i}, i=1, \ldots, N^d\} = \big[\!- \frac{L} {2} , \frac{L} {2} \big]^d$ .

Put $\varepsilon_L \coloneqq L/100N$ and $\hat{Q}_{L,i} \coloneqq Q_{L,i} \times \mathbb{M}$ . Define the event

\begin{equation*}E_{L, N} \coloneqq \left\{ |\eta \cap \hat{Q}_{L,i} \cap \hat{B}_{\varepsilon_L}(x_i)|=1,\, |\eta \cap \hat{Q}_{L,i} \cap \hat{B}_{\varepsilon_L}^c(x_i)|=0 \text{ for all }i=1,\ldots,N^d\right\}.\end{equation*}

Elementary properties of the Poisson point process show that $\mathbb{P}(E_{L,N}) > 0$ for all L and N.

On $E_{L,N}$ the faces of the tessellation restricted to $\big[\frac{-L} {2}, \frac{L} {2}\big]^d$ nearly coincide with the union of the boundaries of $Q_{L,i}, i=1, \ldots, N^d$ , and the cell generated by $\hat{x} \in \eta \cap \big[\frac{-L} {2} + \frac{L} {N}, \frac{L} {2} - \frac{L} {N}\big]^d$ is determined only by $\eta \cap(\cup \{Q_{L,j}, j \in I(\hat{x})\})$ , where $j \in I(\hat{x})$ if and only if $\hat{x} \in \hat{Q}_{L,j}$ or $\hat{Q}_{L,j} \cap \hat{Q}_{L,i}\neq \emptyset$ for i such that $\hat{x} \in \hat{Q}_{L,i}$ . Thus, inserting a point at the origin will not affect the cells far from the origin. More precisely, the cells around the points outside ${\hat R}_{L,N} \coloneqq \big[\!-\frac{2L} {N}, \frac{2L} {N} \big]^d \times \mathbb{M}$ are not affected by inserting a point at the origin. For $S_L \coloneqq L/2$ we have ${\hat R}_{L,N} \subseteq {\hat B}_{S_L}({\bf 0})$ due to our choice of the value N. Therefore,

\begin{equation*}C^{\rho_1}(\hat{x},(\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M\}) =C^{\rho_1}(\hat{x},(\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A})\end{equation*}

for any finite $\mathcal{A} \subseteq {\hat B}_{S_L}({\bf 0})^c$ and $\hat{x} \in (\eta \cap ({\hat B}_{S_L}({\bf 0}) \setminus {\hat R}_{L,N})) \cup \mathcal{A}$ . Consequently, on $E_{L,N}$ ,

\begin{align*}\Delta^{\rho_1}(\infty) = & \sum_{ \hat{x} \in (\eta \cap {\hat R}_{L,N}) \cup \{{\bf 0}_M \} }{\bf 1}\{\textrm{Vol} (C^{\rho_1}(\hat{x}, (\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M\} ))\leq t \} \\ & - \sum_{ \hat{x} \in \eta \cap {\hat R}_{L,N} } {\bf 1}\{\textrm{Vol} (C^{\rho_1}(\hat{x}, (\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A} )) \leq t \}.\end{align*}

Figure 1 illustrates the difference appearing in $\Delta^{\rho_1}(\infty)$ on $E_{L,N}$ for $d=2$ . The cells generated by the points outside the square $\big[\!-\frac{2L} {N}, \frac{2L} {N}\big]^2$ are identical for both point configurations, whereas the cells generated by the points inside the square may differ.

On the event $E_{L,N}$ , the cell generated by $\hat{x} \in (\eta \cap {\hat R}_{L,N}) \cup \{{\bf 0}_M \}$ is contained in $\cup \{Q_{L,j}, j \in I(\hat{x})\}$ , and thus

\begin{equation*}\sup_{\hat{x} \in (\eta \cap {\hat R}_{L,N}) \cup \{{\bf 0}_M\}} \textrm{Vol}(C^{\rho_1}(\hat{x},(\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A})) \leq \left(\frac{3L} {N}\right)^d.\end{equation*}

If $L \in (0, N t^{1/d}/3)$ , then all cell volumes in ${\hat R}_{L,N}$ are at most t; thus, $\Delta^{\rho_1}(\infty) = 1$ on the event $E_{L_1,N}$ with $L_1 \coloneqq \frac16 N t^{1/d}$ . Similarly,

\begin{equation*}\inf_{\hat{x} \in (\eta \cap {\hat R}_{L,N}) \cup \{{\bf 0}_M\}} \textrm{Vol}(C^{\rho_1}(\hat{x}, (\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M\})) \geq\left(\frac{L}{3N}\right)^d.\end{equation*}

If $L \in (3N t^{1/d}, \infty)$ , then all the cell volumes in ${\hat R}_{L,N}$ exceed t and thus $\Delta^{\rho_1}(\infty) = 0$ on the event $E_{L_2,N}$ with $L_2 \coloneqq 6 N t^{1/d}$ . Taking $S \coloneqq S_{L_1}{\bf 1}\{E_{L_1,N}\} + S_{L_2}{\bf 1}\{E_{L_2,N}\}$ , we have found two disjoint events $E_{L_1,N}$ and $E_{L_2,N}$ , each having positive probability, such that $\Delta^{\rho_1} (\infty)$ takes different values on these events, and thus it is non-degenerate. Hence, $\sigma^2(\varphi^{\rho_1})>0$ and we can apply Theorem 2.3(ii).

Figure 1: Voronoi tessellations in $[\!-\frac{L} {2}, \frac{L} {2}]^2$ generated by $(\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A}$ (left) and $(\eta \cap \hat{B}_{S_L}({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M\}$ (right). The ball $B_{S_L}({\bf 0})$ encloses the square $[\!-\frac{2L} {N}, \frac{2L} {N}]^2$ , where N here has a value of 10.

To prove the positivity of $\sigma^2(\varphi^{\rho_2})$ and $\sigma^2(\varphi^{\rho_3})$ we shall consider a subset of $E_{L,N}$ . Assume there exists a parameter $\mu^* \in [0, \mu]$ and a small interval $I_{\alpha}(\mu^*) \subseteq [0, \mu]$ for some $\alpha \geq 0$ such that $\mathbb{Q}_{\mathbb{M}}(I_{\alpha}(\mu^*))>0$ . Define $\hat{E}_{L,N}$ to be the intersection of $E_{L,N}$ and the event $F_{L,N, \alpha}$ that the Poisson points in $[\!-L/2,L/2]^d$ have marks in $I_{\alpha}(\mu^*)$ . If $\alpha$ is small enough, then the Laguerre and Johnson–Mehl cells nearly coincide with the Voronoi cells on the event $\hat{E}_{L,N}$ . Consideration of the events $\hat{E}_{L_1,N}$ and $\hat{E}_{L_2,N}$ shows that $\Delta^{\rho_2} (\infty)$ and $\Delta^{\rho_3} (\infty)$ are non-degenerate, implying that $\sigma^2(\varphi^{\rho_2})>0$ and $\sigma^2(\varphi^{\rho_3}) > 0$ . Thus, Theorem 2.4 holds for the Laguerre and Johnson–Mehl tessellations.

Remark 5.1. In the same way, one can establish that Theorem 2.4 holds for any h taking the form

\begin{equation*}h(K) = \textbf{1} \{g(K) \leq t\} \quad \text{or} \quad h(K) = \textbf{1} \{g(K) > t\}\end{equation*}

for $t \in (0, \infty)$ fixed and $g\,{:}\,({\bf F}^d, \mathcal{B}({\bf F}^d)) \rightarrow (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ a homogeneous function of order q, i.e. $g(\alpha K) = \alpha^q g(K)$ for all $K \in \textbf{F}^d$ and $\alpha \in (0, \infty).$ Examples of the function g include (a) $g(K) \coloneqq \mathcal{H}^{d-1}(\partial K)$ , (b) $g(K) \coloneqq \text{diam}(K)$ , (c) $g(K) \coloneqq \text{radius}$ of the circumscribed ball of K, and (d) $g(K) \coloneqq \text{radius}$ of the circumscribed ball of K.

5.2. Proof of Theorem 2.5

The unbiasedness is again a consequence of Theorem 2.1. To prove the asymptotic normality, we need to check the p-moment condition for

\begin{equation*}\xi^{\rho_i}(\hat{x}, \eta) \coloneqq \mathcal{H}^{d-1}(\partial C^{\rho_i} (\hat{x}, \eta)){\bf 1}\{C^{\rho_i}(\hat{x},\eta) \text{ is bounded}\}\end{equation*}

and the positivity of $\sigma^2(\xi^{\rho_i})$ , $i=1, 2, 3$ .

First, we verify the moment condition with $p=5$ . Given any $\hat{x},\hat{y} \in \hat{\mathbb{R}^d}$ , we assert that $\mathbb{E}^{\hat{x},\hat{y}} \mathcal{H}^{d-1}(\partial C^{\rho_i}(\hat{x},\eta))^5= \mathbb{E} \mathcal{H}^{d-1}(\partial C^{\rho_i}(\hat{x},\eta \cup \{\hat{y}\}))^5 \leq c < \infty$ for some constant c that does not depend on $\hat{x}$ and $\hat{y}$ . From Proposition 3.2 there is a random variable $R_{\hat{x}}$ such that

\begin{equation*}C^{\rho_i}(\hat{x},\eta \cup \{\hat{y}\}) = \bigcap_{\hat{z} \in (\eta \cup \{\hat{y}\} \setminus \{\hat{x}\}) \cap \hat{B}_{R_{\hat{x}}}(x)} \mathbb{H}_{\hat{z}}(\hat{x}).\end{equation*}

As in Proposition 3.1 we find $D_{\hat{x}}$ such that $C^{\rho_i}(\hat{x},\eta \cup \{\hat{y}\}) \subseteq B_{D_{\hat{x}}}(\hat{x})$ . Then

\begin{align*}\mathcal{H}^{d-1}(\partial C^{\rho_i}(\hat{x},\eta \cup \{\hat{y}\})) &\leq \sum_{\hat{z} \in (\eta \cup \{\hat{y}\} \setminus \{\hat{x}\}) \cap \hat{B}_{R_{\hat{x}}}(x)}\mathcal{H}^{d-1}(\partial \mathbb{H}_{\hat{z}}(\hat{x}) \cap B_{D_{\hat{x}}}(\hat{x})) \\&\leq c_{i,d} D_{\hat{x}}^{d-1} \eta(\hat{B}_{R_{\hat{x}}}(x))\end{align*}

for some constant $c_{i,d}$ that depends only on i and d. Using the Cauchy–Schwarz inequality we get

\begin{equation*}\mathbb{E} \mathcal{H}^{d-1}(\partial C^{\rho_i}(\hat{x},\eta \cup \{\hat{y}\}))^5 \leq c_{i,d}^5 (\mathbb{E} D_{\hat{x}}^{10(d-1)})^{1/2} (\mathbb{E} \eta(\hat{B}_{R_{\hat{x}}}(x))^{10})^{1/2}.\end{equation*}

By the property of the Poisson distribution we have

\begin{equation*}\mathbb{E} \eta(\hat{B}_{R_{\hat{x}}}(x))^{10} = \mathbb{E} (\mathbb{E} (\eta(\hat{B}_{R_{\hat{x}}}(x))^{10} \mid R_{\hat{x}})) = \mathbb{E} P(\textrm{Vol}(B_{R_{\hat{x}}}(x))),\end{equation*}

where $P(\!\cdot\!)$ is a polynomial of degree 10. Both $D_{\hat{x}}$ and $R_{\hat{x}}$ have exponentially decaying tails, and the decay does not depend on $\hat{x}$ . Therefore, $(\mathbb{E} D_{\hat{x}}^{10(d-1)})^{1/2} (\mathbb{E} \eta(\hat{B}_{R_{\hat{x}}}(x))^{10})^{1/2}$ is bounded and the moment condition is satisfied with $p=5$ .

The positivity of the asymptotic variance can be shown similarly to the proof of Theorem 2.4. We will show it only for the Voronoi case, as the Laguerre and Johnson–Mehl tessellations can be treated similarly. We will again find a random variable S and a $\Delta^{\rho_1}(\infty)$ such that, for all finite $\mathcal{A} \subseteq \hat{B}_S({\bf 0})^c$ , we have

\begin{align*}\Delta^{\rho_1}(\infty) = & \sum_{ \hat{x} \in (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M \} } \xi^{\rho_1} (\hat{x}, (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} \cup \{{\bf 0}_M\} ) \\[3pt] & - \sum_{ \hat{x} \in (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} } \xi^{\rho_1} (\hat{x}, (\eta \cap \hat{B}_S({\bf 0})) \cup \mathcal{A} )\end{align*}

and, moreover, $\Delta^{\rho_1}(\infty)$ assumes different values on two events having positive probability and is thus non-degenerate. By Theorem 2.1 of [Reference Penrose and Yukich14], this is enough to show the positivity of $\sigma^2(\xi^{\rho_1})$ .

Let $L>0$ and let $N \in \mathbb{N}$ have odd parity. Abusing notation, we construct a collection of $N^d$ cubes $Q_{L,1}, \ldots, Q_{L,N^d}$ centered around $x_i \in \mathbb{R}^d$ , $i= 1, \ldots, N^d$ , such that

1. (i) $Q_{L,i}$ has side length $ \frac{L} {N} $ , and

2. (ii) $\cup \{ Q_{L,i}, i=1, \ldots, N^d\} = \big[\!-\frac{L} {2}, \frac{L} {2}\big]^d$ .

There is a unique index $i_0 \in \{1, \ldots, N^d\}$ such that $x_{i_0} = {\bf 0}$ . We define $\varepsilon_L, \hat{Q}_{L,i}$ and the event $E_{L,N}$ as in the proof of Theorem 2.4. Note that, under $E_{L,N}$ ,

\begin{equation*}\inf_{(x, m_x) \in \eta \cap \hat{Q}_{L,i_0} } \|x\| \leq \varepsilon_L.\end{equation*}

Hence, on the event $E_{L,N}$ , the insertion of the origin into the point configuration creates a new face of the tessellation whose surface area is bounded below by $c_{\min} (L/N)^{d-1}$ and bounded above by $c_{\max} (L/N)^{d-1}$ . Thus,

\begin{equation*}c_{\min} \left(\frac{L}{N}\right)^{d-1} + O\left(\varepsilon_L\left(\frac{L}{N}\right)^{d-2}\right)\leq \Delta^{\rho_1}(\infty) \leq c_{\max} \left(\frac{L}{N}\right)^{d-1} - O\left(\varepsilon_L\left(\frac{L}{N}\right)^{d-2}\right),\end{equation*}

where $O\big(\varepsilon_L\left(\frac{L}{N}\right)^{d-2}\big)$ is the change in the combined surface areas of the already existing faces after inserting the origin. Events $E_{L_1,N}, E_{L_2, N}$ , $L_1 < L_2$ , both occur with positive probability for any $L_1, L_2$ . Similarly to the proof of Theorem 2.4, we can find N, S, $L_1$ , and $L_2$ ( $L_2-L_1$ large enough) such that the value of $\Delta^{\rho_1}(\infty)$ differs on each event. Thus, $\sigma^2(\xi^{\rho_1})$ is strictly positive.

To show that $\sigma^2(\xi^{\rho_2})$ and $\sigma^2(\xi^{\rho_3})$ are strictly positive we argue as follows. The Laguerre and Johnson–Mehl tessellations are close to the Voronoi tessellation on the event $F_{L,N, \alpha}$ , for $\alpha$ small. Arguing as we did in the proof of Theorem 2.4, and considering the event $\hat{E}_{L,N}$ given in the proof of that theorem, we may conclude that $\sigma^2(\xi^{\rho_2})>0$ and $\sigma^2(\xi^{\rho_3}) > 0$ .