2.1 High-intensity regime

We look at the high-intensity regime where $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to \infty$ . First, we define the space of signed measures $\mathcal{M}^{\gamma_1,\gamma_2}$ where the theorem of convergence holds.

Definition 2.1. Let $\mathcal{M}^{\gamma_1,\gamma_2}$ be the subset of $\mathcal{M}_2$ with the following property. For each $\mu \in \mathcal{M}^{\gamma_1,\gamma_2}$ , there exist constants $C>0$ and $\alpha_i$ with $\gamma_i < \alpha_i \leq 2$ for $i=1,2$ such that, for all $u \in \mathbb{R}_+^2$ ,

(2.1) $$ \begin{equation} \int_{\mathbb{R}^2} \mu (B(x,u)) ^2 {\rm d} x \leq C \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ). \label{eqn2} \end{equation} $$

In order to proceed quickly to the results, we postpone a discussion of the space $\mathcal{M}^{\gamma_1,\gamma_2}$ to Section 3.1. Here, we only note that this subspace is closed under translation and dilation; cf. Section 2.6.

The limiting random field is given by a centred Gaussian linear random field.

Theorem 2.1. Let $\gamma_i \in (1,2)$ for $i=1,2, \lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to \infty$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \frac{\skew5\widetilde{J}_\rho(\cdot)}{\sqrt{ \lambda_\rho \rho^{\gamma_1+ \gamma_2}}\,} \xrightarrow{\mathcal{M}^{\gamma_1,\gamma_2}} Z(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $(Z(\mu))_\mu$ is the centred Gaussian linear random field with covariance function

(2.2) $$ \begin{equation} C_Z(\mu,\nu)= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \mu(B(x,u))\nu(B(x,u)) \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} x {\rm d} u. \label{eqn3} \end{equation} $$

2.2 Intermediate-intensity regime

In the intermediate intensity regime where $ \lambda_\rho \rho^{\gamma_1+\gamma_2} \to a \in (0,\infty)$ , the space of signed measures is identical with that in the high-intensity regime. The limiting random field consists of compensated Poisson integrals.

Theorem 2.2. Let $\gamma_i \in (1,2)$ for $i=1,2, \lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to 1$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \skew5\widetilde{J}_\rho(\cdot) \xrightarrow{\mathcal{M}^{\gamma_1,\gamma_2}} J_I(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $ (J_I(\mu))_\mu$ is the linear random field of compensated Poisson integrals

$$ \begin{equation*} J_I(\mu)\,:\!= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \mu(B(x,u)) \widetilde{N}_I(\mathrm{d} x,\mathrm{d} u), \end{equation*} $$

where the intensity measure is given by

$$ \begin{equation*} n_I(\mathrm{d} x,\mathrm{d} u)= \mathrm{d} x \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u_1 {\rm d} u_2. \end{equation*} $$

2.3 Low-intensity regime

The low-intensity regime is defined by $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to 0$ with $\gamma_1 < \gamma_2$ , which is divided once more into three different subregimes. In these subregimes, we additionally have to assume that the density function of the length of a box for small values is bounded, i.e. we assume that there is some $c_{f_1}>0$ such that the inequality

(2.3) $$ \begin{equation} f_1(u_1) \leq \frac{c_{f_1}}{u_1^{\gamma_1+1}} \label{eqn4} \end{equation} $$

holds for all $u_1 \in \mathbb{R}_+$ . This technical assumption ensures the existence of a suitable majorant for $f_1$ in the proofs below. From now on, we treat the three subregimes separately.

2.3.1 Gaussian-lines scaling regime

We define the space of signed measures $\mathcal{M}_{L}$ for the Gaussian-lines scaling regime where $\lambda_\rho \rho^{\gamma_1+\eta} \to a \in (0,\infty)$ for some $\eta \in (0,\gamma_2)$ .

Definition 2.2. Let $\mathcal{M}_{L}$ be the subset of $\mathcal{M}_2$ where

• each $\mu \in \mathcal{M}_{L}$ has a density function $f_\mu$ , i.e. $\mu(\mathrm{d} x) = f_\mu(x) {\rm d} x$ ;

• for each $\mu \in \mathcal{M}_{L},$ the density function $f_\mu$ is bounded and decays at least exponentially fast, i.e. there exist constants $C_\mu>0$ and $c_\mu>0$ such that, for all $x \in \mathbb{R}^2$ ,

  (2.4) $$ \begin{equation} \vert \kern2ptf_\mu(x) \vert \leq C_\mu {\rm e}^{-c_\mu(\vert x_1 \vert + \vert x_2 \vert)}; \label{eqn5} \end{equation} $$

• for each $\mu \in \mathcal{M}_{L},$ the pointwise convergence

  (2.5) $$ \begin{equation} \frac{1}{{\varepsilon}} \int_{B( x, \binom{u_1}{{\varepsilon}})} f_\mu(y) \mathrm{d} y \to \int_{[x_1-{u_1}/{2},x_1+{u_1}/{2}]} f_\mu(y_1,x_2) {\rm d} y_1 \label{eqn6} \end{equation} $$
  as ${\varepsilon} \to 0$ holds for all $(x,u_1) \in \mathbb{R}^2 \times \mathbb{R}_+$ .

We note that this subspace is closed under translation and dilation; cf. Section 2.6. For a discussion about the properties of this space and the relation to $\mathcal{M}^{\gamma_1,\gamma_2}$ , we refer the reader to Section 3.1.

In the Gaussian-lines scaling regime, we require a further condition on the ‘lighter’ tail index $\gamma_2$ , namely $\gamma_2>2$ . Consequently, the width of a box has a finite variance. The limiting random field is a centred Gaussian linear random field.

Theorem 2.3. Let $\gamma_1 \in (1,2)$ , $\gamma_2>2, \lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1+\eta} \to 1$ for some $\eta \in (0,\gamma_2)$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \frac{\skew5\widetilde{J}_\rho(\cdot) }{{\rho}^{1-\eta / 2}} \xrightarrow{\mathcal{M}_{L}} Y(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $(Y(\mu))_\mu$ is the centred Gaussian linear random field with covariance function

(2.6) $$ \begin{equation} C_Y(\mu,\nu) = \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \int_{[x_1-{u_1}/{2},x_1+{u_1}/{2}]^2} f_\mu(\,y_1,x_2)\ f_\nu(\,y_2,x_2) {\rm d} y \frac{u_2^2\ f_2(u_2)}{u_1^{\gamma_1+1}} {\rm d} x {\rm d} u. \label{eqn7} \end{equation} $$

2.3.2 Poisson-lines scaling regime

In the Poisson-lines scaling regime where $ \lambda_\rho \rho^{\gamma_1} \to a \in (0,\infty)$ , we provide the theorem of convergence to a random field consisting of compensated Poisson integrals. The corresponding space of signed measures coincides with that from the Gaussian-lines scaling regime.

Theorem 2.4. Let $\gamma_1 \in (1,2)$ , $\gamma_1 < \gamma_2, \lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1} \to 1$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \frac{\skew5\widetilde{J}_\rho(\cdot) }{\rho} \xrightarrow{\mathcal{M}_{L}} J_L(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $(J_L(\mu))_\mu$ is the linear random field of compensated Poisson integrals

(2.7) $$ \begin{equation} J_L(\mu)\,:\!= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \bigg( u_2 \int_{[x_1-{u_1}/{2},x_1+{u_1}/{2}]} f_\mu(y_1,x_2) {\rm d} y_1 \bigg) \widetilde{N}_L(\mathrm{d} x,\mathrm{d} u), \label{eqn8} \end{equation} $$

where the intensity measure is given by

(2.8) $$ \begin{equation} n_L(\mathrm{d} x,\mathrm{d} u)= \mathrm{d} x \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 f_2(u_2) {\rm d} u_2. \label{eqn9} \end{equation} $$

In the Poisson-lines scaling regime, we have $ \lambda_\rho \rho^{\gamma_1} \to a \in (0,\infty)$ and $ \lambda_\rho \rho^{\gamma_2} \to 0$ as $\rho \to 0$ . This indicates a different behaviour for the length and the width of the boxes. For a graphical representation, we ran simulations of the Poisson point processes for some small $\rho$ and appropriate $\lambda_\rho$ . We generated random Poisson points, where we chose Pareto distributions for the length and the width of the boxes. Then, we plotted the boxes that are filled with black colour, i.e. black rectangles. Two samples of the random boxes model in the Poisson-lines scaling regime are given in Figure 1. Besides points, we spot horizontal lines in the sample on the left-hand side. In the sample on the right-hand side, each box is just additionally randomly rotated around the center point of itself (cf. Section 2.7 below for the definition of this modified model).

Figure 1

Poisson-lines scaling regime.

2.3.3 Points scaling regime

In the points scaling regime where $\lambda_\rho \rho^{\gamma_1} \to 0$ , we investigate the scaling behaviour of $\,\skew5\widetilde{J}_\rho$ on the space of signed measures $\mathcal{M}_{P}$ which is given as follows.

Definition 2.3. Let $\mathcal{M}_{P}$ be the subset of $\mathcal{M}_2$ where

• each signed measure $\mu \in \mathcal{M}_{P}$ has a continuous density function $f_\mu$ , i.e. $\mu(\mathrm{d} x) = f_\mu(x) {\rm d} x$ ;

• for each $\mu \in \mathcal{M}_{P},$ the density function $f_\mu$ is bounded and decays at least exponentially fast, i.e. there exist constants $C_\mu>0$ and $c_\mu>0$ such that, for all $x \in \mathbb{R}^2$ ,

  $$ \begin{equation*} \vert \kern2ptf_\mu(x) \vert \leq C_\mu {\rm e}^{-c_\mu(\vert x_1 \vert + \vert x_2 \vert)}. \end{equation*} $$

We note that this subspace is closed under translation and dilation; cf. Section 2.6. A further discussion of the space is given in Section 3.1.

The limiting random field consists of integrals with respect to an $\alpha$ -stable random measure. For $\alpha \in (1,2)$ , we denote by $\Lambda_\alpha$ the independently scattered $\alpha$ -stable random measure with unit skewness and Lebesgue control measure (cf., e.g. [Reference Samorodnitsky and Taqqu27]). We define the random linear functional

(2.9) $$ \begin{equation} S_{\gamma_1}(\mu)\,:\!= \int_{\mathbb{R}^2} f_\mu(x) \Lambda_{\gamma_1}(\mathrm{d} x), \qquad \mu \in \mathcal{M}_{P}, \label{eqn10} \end{equation} $$

by its characteristic function at 1, i.e.

$$ \begin{equation*} \mathbb{E} ({\rm e}^{{\rm i} S_{\gamma_1}(\mu) })= \exp \bigg( -\sigma_\mu^{\gamma_1} \bigg(1-{\rm i} \beta_\mu \tan \bigg( \frac{\pi \gamma_1}{2} \bigg) \bigg) \bigg), \end{equation*} $$

where (excluding the trivial case $\Vert \kern2ptf_\mu \Vert_{\gamma_1} = 0$ )

(2.10) $$ \begin{equation} \sigma_\mu=\Vert \kern2ptf_\mu \Vert_{\gamma_1} , \qquad \beta_\mu= \Vert \kern2ptf_\mu \Vert_{\gamma_1}^{-\gamma_1} (\Vert {f_\mu}_+ \Vert_{\gamma_1}^{\gamma_1} - \Vert {f_\mu}_- \Vert_{\gamma_1}^{\gamma_1} ) \label{eqn11} \end{equation} $$

and ${f_\mu}_+\,:\!= \max({\kern2ptf_\mu},0 )$ , ${f_\mu}_-\,:\!= - \min({\kern2ptf_\mu},0)$ .

Theorem 2.5. Let $\gamma_1 \in (1,2)$ , $\gamma_1 < \gamma_2$ , $\lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1} \to 0$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \frac{\skew5\widetilde{J}_\rho(\cdot) }{c_{\gamma_1,\gamma_2} \lambda_\rho^{{1}/{\gamma_1}} \rho^2} \xrightarrow{\mathcal{M}_{P}} S_{\gamma_1}(\cdot) \end{equation*} $$

as $\rho \to 0$ , where the linear random field of functionals $(S_{\gamma_1}(\mu))_\mu$ and the constant $c_{\gamma_1,\gamma_2}$ are defined in (2.9) and (4.8) below, respectively.

2.4 The finite-variance case

Finally, we want to investigate the scaling behaviour in the case where the area of a box has a finite variance. We assume that the length and the width of the boxes have finite second moments instead of heavy tails. Similar to above, let F be a probability measure on $\mathbb{R}_+^2$ given by

$$ \begin{equation*}\label{eq:Fdu_finite_variance} F(\mathrm{d} u)=f_1(u_1)f_2(u_2) {\rm d} u_1 {\rm d} u_2. \end{equation*} $$

Furthermore, we define, for $i=1,2$ ,

(2.11) $$ \begin{equation} v_i\,:\!= \int_{\mathbb{R}_+} u_i^2 f_i(u_i) {\rm d} u_i < \infty. \label{eqn12} \end{equation} $$

The following result shows that the centred and renormalised random field on the space $\mathcal{M}_{P}$ converges to a centred Gaussian linear random field. We emphasise that there does not exist a diversity of regimes to distinguish in the finite-variance case, which is also the much simpler case. Nevertheless, the proof of this result can be viewed as a ‘prototype proof’ for all other regimes.

Moreover, we conjecture that the following theorem holds for a larger space of signed measures than $\mathcal{M}_{P}$ (but our proof cannot be adapted to this larger space). Actually, the limiting random field is well defined for $f_\mu \in L^1(\mathbb{R}^2) \cap L^2(\mathbb{R}^2)$ .

Theorem 2.6. Let $\lambda_\rho \to \infty$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \frac{\skew5\widetilde{J}_\rho(\cdot)}{\rho^2 \sqrt{ \lambda_\rho v_1 v_2}\,} \xrightarrow{\mathcal{M}_{P}} X(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $v_i$ is defined in (2.11) for $i=1,2$ and where $( X(\mu))_\mu$ is the centred Gaussian linear random field with covariance function

(2.12) $$ \begin{equation} C_X(\mu,\nu)= \int_{\mathbb{R}^2} f_\mu(x) f_\nu(x) {\rm d} x. \label{eqn13} \end{equation} $$

2.5 Comparison of the different regimes

First, we make a remark regarding the comparison of the different regimes among each other. In each low intensity subregime, the different parameters $\gamma_1$ and $\gamma_2$ contribute in different ways to the limit. This contrasts the limits in the high- and intermediate-intensity regimes, where both parameters $\gamma_1$ and $\gamma_2$ are present in a homogeneous way in each limit. For example, in the points scaling regime the ‘heavier’ tail index $\gamma_1$ for the length of a box appears primarily in the limit, i.e. the ‘lighter’ tail index $\gamma_2$ only enters into a constant (see Theorem 2.5). More precisely, the limit $S_{\gamma_1}(\mu)$ is a $\gamma_1$ -stable random variable and the constant $c_{\gamma_1,\gamma_2}$ given in (4.8) below is the only quantity depending on the tail index $\gamma_2$ .

Second, we want to say a few words about the comparison to previous work. Two limiting random fields in this paper have already arisen in identical form. The centred Gaussian linear random field with covariance function given in (2.12) coincides with the corresponding one in the finite-variance case of the random grain model where the size of a grain is given by a single distribution (cf. Theorem 1 of [Reference Kaj, Leskelä, Norros and Schmidt18]). Moreover, the limiting random field consisting of integrals with respect to a stable random measure in the points scaling regime has also appeared there (cf. Equation (13) of [Reference Kaj, Leskelä, Norros and Schmidt18]). The index of stability is given by the index of the regularly varying tail of the volume of a grain there and by the ‘heavier’ tail index $\gamma_1$ for the length of a box in our random boxes model. All other limiting random fields seem to be new.

2.6 Statistical properties

In the following paragraphs, we give some statistical properties of the different scaling limits $Z, J_I, Y, J_L$ , $S_{\gamma_1},$ and X on their respective spaces of signed measures. With regard to these properties, we note that these subspaces are closed under translation and dilation. We will omit the proofs of all these facts because they can be verified easily.

Covariance. The covariance functions of the Gaussian random fields Z, Y, and X are given in (2.2), (2.6), and (2.12), respectively. The covariance function of $J_I$ in the intermediate-intensity regime is exactly the same as in the high-intensity regime (see (2.2)), but the limit $J_I$ is not a Gaussian random field. In the points scaling regime, the scaling limit $S_{\gamma_1}(\mu)$ is $\gamma_1$ -stable and thus does not have a finite variance. We distinguish two cases in the Poisson-lines scaling regime. If $\gamma_2 < 2$ , the compensated Poisson integral $J_L(\mu)$ does not have a finite variance. In contrast, if we assume that $\gamma_2 > 2$ , i.e. the width of a box has a finite variance, the scaling limit $J_L(\mu)$ has a finite variance as well and the covariance function coincides with that in the Gaussian-lines scaling regime (see (2.6)).

Translation invariance. Let $s \in \mathbb{R}^2$ . We define the translation of a signed measure $\tau_s \mu$ by $\tau_s \mu (A)\,:\!= \mu(A-s)$ for any Borel set A. We call a random field W on $\mathcal{M}_W$ translation invariant (cf. Definition 3.1 of [Reference Biermé, Estrade and Kaj4]) if we have

$$ \begin{equation*} (W(\tau_s \mu))_{\mu \in \mathcal{M}_W} = (W(\mu))_{\mu \in \mathcal{M}_W} \end{equation*} $$

in finite-dimensional distributions for all $s \in \mathbb{R}^2$ ( $\mathcal{M}_W$ has to be closed under translations $\tau_s$ ). All limiting random fields Z, $J_I$ , Y, $J_L$ , $S_{\gamma_1},$ and X are translation invariant on the respective spaces of signed measures.

Dilation. For all $a >0,$ the dilation of a signed measure $\mu_a$ is given by $\mu_a (A)\,:\!= \mu(a^{-1}A)$ for any Borel set A. We call a random field W on $\mathcal{M}_W$ self-similar with index H (cf. Definition 3.3 of [Reference Biermé, Estrade and Kaj4]) if we have

$$ \begin{equation*} (W(\mu_a))_{\mu \in \mathcal{M}_W} = (a^H W(\mu))_{\mu \in \mathcal{M}_W} \end{equation*} $$

in finite-dimensional distributions for all $a >0$ ( $\mathcal{M}_W$ has to be closed under dilations $\mu_a$ ).

The limiting (Gaussian) random fields Z, Y, and X are self-similar with indices $H = (2-\gamma_1 - \gamma_2)/{2}$ , $H=-\gamma_1/{2},$ and $H=-1$ , respectively. In the points scaling regime, the limit $S_{\gamma_1}$ is self-similar with index $H=2 / \gamma_1 - 2$ . We emphasise that H is negative in these cases. If the reader expects H to be positive, a reason may be found in the way of defining the dilation of a signed measure which, however, is common in the literature. We can also verify that the random field $J_I$ in the intermediate intensity regime is not self-similar (cf. [18, p.537]).

We call a random field W with $\mathbb{E} W=0$ on $\mathcal{M}_W$ (which has to be again closed under dilation) aggregate-similar (cf. Definition 3.5 of [Reference Biermé, Estrade and Kaj4] and [Reference Kaj16]) if there is a positive sequence $(a_m)_{m\geq 1}$ such that we have

$$ \begin{equation*} (W(\mu_{a_m}))_{\mu \in \mathcal{M}_W} = \bigg(\sum_{k=1}^m W^k(\mu)\bigg)_{\mu \in \mathcal{M}_W} \end{equation*} $$

in finite-dimensional distributions for all $m \geq 1$ , where $(W^k)_{k \geq 1}$ are independent and identically distributed (i.i.d.) copies of W.

We find that the random fields Z, Y, X, $J_I,$ and $S_{\gamma_1}$ are aggregate-similar, where we have $a_m=\smash{m^{1/(2-\gamma_1-\gamma_2)}}$ , $a_m=\smash{m^{-1/\gamma_1}}$ , $a_m=m^{-1/2}$ , $a_m=\smash{m^{1/(2-\gamma_1-\gamma_2)}},$ and $a_m=\smash{m^{1/(2-2\gamma_1)}}$ , respectively. Regarding the dilation in the Poisson-lines scaling regime, we mention that the scaling limit $J_L$ does not fulfil aggregate-similarity. In fact, it fulfils the following property, which is some kind of aggregate-similarity with dilation:

$$ \begin{equation*} (J_L(\mu_{a_m}))_{\mu \in \mathcal{M}_L} = \bigg(\sum_{k=1}^m J_{L,a_m}^k(\mu)\bigg)_{\mu \in \mathcal{M}_L}. \end{equation*} $$

Here $a_m=\smash{m^{1/(4-\gamma_1)}}$ and $\smash{(J_{L,a_m}^k)_{k \geq 1}}$ are i.i.d. copies of $J_{L,a_m}$ . The modification $J_{L,a_m}$ here is also given by (2.7), but $f_2(\cdot)$ in (2.8) has to be replaced by $\smash{a_m^{-1}f_2(a_m \cdot)}$ , i.e. the measure for the width is dilated simultaneously.

2.7 Extension to randomly rotated boxes

A modification of the random boxes model consists in additionally endowing the rectangles with independent and uniformly distributed orientations. A similar model was considered in Section 3.3 of [Reference Kaj, Leskelä, Norros and Schmidt18]. We introduce the Haar measure $\mathrm{d} \theta$ on the group of rotations ${\rm SO}(2)$ in $\mathbb{R}^2$ and consider the Poisson random measure $N^\circ_\rho$ on $\mathbb{R}^2 \times \mathbb{R}_+^2 \times {\rm SO}(2)$ with intensity measure given by

$$ \begin{equation*} n^\circ_\rho(\mathrm{d} x, \mathrm{d} u,\mathrm{d} \theta) = \lambda_\rho {\rm d} x F_\rho(\mathrm{d} u) {\rm d} \theta. \end{equation*} $$

Then the centred Poisson integral

$$ \begin{equation*} \skew5\widetilde{J}^\circ_\rho(\mu)\,:\!= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2 \times {\rm SO}(2)} \mu(B_{\theta}(x,u)) \widetilde{N}^\circ_\rho(\mathrm{d} x,\mathrm{d} u, \mathrm{d} \theta) \end{equation*} $$

is the object of interest, where $B_{\theta}(0,u)\,:\!= \theta B(0,u)$ denotes the rectangle B(0,u) rotated by $\theta$ and $B_{\theta}(x,u)$ for $x \neq 0$ is defined by

$$ \begin{equation*} B_{\theta}(x,u)\,:\!= x + B_{\theta}(0,u). \end{equation*} $$

In order to obtain results for this modified random boxes model, we have to adapt the spaces of signed measures slightly. In the following, we treat the high- and intermediate-intensity regimes as an example.

Definition 2.4 Let $\smash{\mathcal{M}_\circ^{\gamma_1,\gamma_2}}$ be the subset of $\mathcal{M}_2$ where, for each $\smash{\mu \in \mathcal{M}_\circ^{\gamma_1,\gamma_2}}$ , there exist constants $C>0$ and $\alpha_i$ with $\gamma_i < \alpha_i \leq 2$ for $i=1,2$ such that, for all $u \in \smash{\mathbb{R}_+^2}$ ,

$$ \begin{equation*} \int_{\mathbb{R}^2\times {\rm SO}(2)} \mu (B_{\theta}(x,u)) ^2 {\rm d} x {\rm d} \theta \leq C \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ). \end{equation*} $$

Using this new subspace of signed measures, we get analogous theorems of convergence. For the high-intensity regime, the limiting random field is again Gaussian.

Theorem 2.7. Let $\gamma_i \in (1,2)$ for $i=1,2$ , $\lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to \infty$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \frac{\skew5\widetilde{J}^\circ_\rho(\cdot)}{\sqrt{ \lambda_\rho \rho^{\gamma_1+\gamma_2}}\,} \xrightarrow{\mathcal{M}_\circ^{\gamma_1,\gamma_2}} Z^\circ(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $(Z^\circ(\mu))_\mu$ is the centred Gaussian linear random field with covariance function

$$ \begin{equation*} C_{Z^\circ}(\mu,\nu)= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2\times {\rm SO}(2)} \mu(B_{\theta}(x,u))\nu(B_{\theta}(x,u)) \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} x {\rm d} u {\rm d} \theta. \end{equation*} $$

In the intermediate-intensity regime, the limiting field is again a compensated Poisson random field.

Theorem 2.8. Let $\gamma_i \in (1,2)$ for $i=1,2$ , $\lambda_\rho \to \infty,$ and $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to 1$ as $\rho \to 0$ . Then, we have

$$ \begin{equation*} \skew5\widetilde{J}^\circ_\rho(\cdot) \xrightarrow{\mathcal{M}_\circ^{\gamma_1,\gamma_2}} J^\circ_I(\cdot) \end{equation*} $$

as $\rho \to 0$ , where $\smash{ (J^\circ_I(\mu))_\mu}$ is the linear random field of compensated Poisson integrals

$$ \begin{equation*} J^\circ_I(\mu)\,:\!= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2 \times {\rm SO}(2)} \mu(B_{\theta}(x,u)) \widetilde{N}_I(\mathrm{d} x,\mathrm{d} u,\mathrm{d} \theta), \end{equation*} $$

where the intensity measure is given by

$$ \begin{equation*} n_I(\mathrm{d} x,\mathrm{d} u,\mathrm{d} \theta)= \mathrm{d} x \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u_1 {\rm d} u_2 {\rm d} \theta. \end{equation*} $$

Since the probability measure $\mathrm{d} \theta$ on the group ${\rm SO}(2)$ is not affected by the scaling of the edge lengths as $\rho \to 0$ , we obtain for each regime the same type of limiting random field as in the model without rotations. Moreover, we can proceed in the proofs as in Theorems 2.1 and 2.2. We just have to use the new subspace of signed measures in order to obtain the analogous limiting random fields for this modified random boxes model. To keep the exposition comprehensible, we will not enter into more details for the proofs in this paper and omit the low-intensity regime, where the modified subspaces are more complicated and technical.

In addition, further extensions of our random boxes model are feasible. For example, it is possible to allow nonnegative $\sigma$ -finite measures F instead of restricting ourselves to probability measures or to consider boxes (hyper-rectangles) in $\mathbb{R}^d$ with $d \geq 3$ .

3.1 Spaces of signed measures

We investigate the spaces of signed measures where the theorems of convergence in the high-, intermediate-, and low-intensity regimes hold, respectively. The following proposition, which we can prove easily, ensures the linearity of these subspaces.

Remark 3.1. The linear space $\mathcal{M}^{\gamma_1,\gamma_2}$ , where the theorems of convergence in the high- and intermediate-intensity regimes hold, is not yet the technically largest possible. We are able to weaken the condition in (2.1) as follows. For each $\mu \in \mathcal{M}^{\gamma_1,\gamma_2}$ , there exist some constants $C>0$ , $\underline{\alpha}_i,$ and $\overline{\alpha}_i$ with $0<\underline{\alpha}_i< \gamma_i < \overline{\alpha}_i \leq 2$ for $i=1,2$ such that the inequality

$$ \begin{equation*} \int_{\mathbb{R}^2} \mu (B(x,u)) ^2 {\rm d} x \leq C \min(u_1^{\underline{\alpha}_1},u_1^{\overline{\alpha}_1} ) \min(u_2^{\underline{\alpha}_2},u_2^{\overline{\alpha}_2} ) \end{equation*} $$

holds for all $u \in \smash{\mathbb{R}_+^2}$ .

Next, we touch on the comparison of these spaces of signed measures for $\gamma_i \in (1,2)$ for $i=1,2$ . We observe that the space $\mathcal{M}^{\gamma_1,\gamma_2}$ contains measures which do not have to have a density. Therefore, there exist some $\mu \in \mathcal{M}^{\gamma_1,\gamma_2}$ , but $\mu \notin \mathcal{M}_{k}$ for $k \in \{L,P\}$ . Conversely, we obtain the following result.

Proposition 3.2. Let $\gamma_i \in (1,2)$ for $i=1,2$ . We have $\mathcal{M}_{k} \subseteq \mathcal{M}^{\gamma_1,\gamma_2}$ for $k \in \{L,P\}$ .

Sketch of the proof. Note that the density function of a signed measure in $\mathcal{M}_{k}$ for $k \in \{L,P\}$ satisfies

$$ \begin{equation*} \vert \kern2ptf_\mu(x) \vert \leq C_\mu {\rm e}^{-c_\mu(\vert x_1 \vert + \vert x_2 \vert)} \end{equation*} $$

for all $x \in \mathbb{R}^2$ for some $C_\mu>0$ and $c_\mu>0$ . We can compute that

$$ \begin{equation*} \int_{\mathbb{R}} \bigg( \int_{[x_1-{u_1}/{2}, x_1+{u_1}/{2}]} {\rm e}^{-c_\mu \vert y_1 \vert} {\rm d} y_1 \bigg)^2 \mathrm{d} x_1 \leq c \min(u_1,u_1^{2}) \end{equation*} $$

by a case distinction for $0 \leq u_1 \leq 1$ and $u_1 > 1$ . Using the product form, the validity of inequality (2.1) follows.

In order to obtain signed measures which are contained in the spaces $\mathcal{M}_{k}$ for $k \in \{L,P\}$ , we can stick closely to the definitions of these spaces. For example, we can consider the measure $\mu$ given by $\mu(\mathrm{d} x)\,:\!= {\rm e}^{-\vert x_1\vert}{\rm e}^{-\vert x_2\vert}{\rm d} x$ .

3.2 Existence of the random fields

We deal with the existence of the random field $\,\smash{\skew5\widetilde{J}}$ of interest and all the limiting random fields in the different scaling regimes. Using Lemma 12.13 of [Reference Kallenberg19], we can verify that the random fields J and $\,\smash{\skew5\widetilde{J}}$ exist because we have

$$ \begin{equation*} \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \vert \mu(B(x,u)) \vert n(\mathrm{d} x,\mathrm{d} u) \leq \lambda \Vert \mu \Vert \int_{\mathbb{R}_+^2} u_1 u_2 F(\mathrm{d} u)< \infty. \end{equation*} $$

Furthermore, by standard facts on Poisson integrals and Fubini’s theorem, the expected value of $J(\mu)$ is finite and given by

$$ \begin{equation*} \mathbb{E} J(\mu) = \lambda \mu (\mathbb{R}^2) \int_{\mathbb{R}_+} u_1 f_1(u_1) {\rm d} u_1 \int_{\mathbb{R}_+} u_2 f_2(u_2) {\rm d} u_2. \end{equation*} $$

Using the function $\Psi$ defined in (3.1), the characteristic function of $\,\skew5\widetilde{J}(\mu)$ is given by

$$ \begin{equation*} \mathbb{E} ({\rm e}^{{\rm i}\skew5\widetilde{J}(\mu)}) = \exp \bigg( \int_{\mathbb{R}_+^2} \int_{\mathbb{R}^2} \Psi(\mu(B(x,u))) \lambda {\rm d} x F(\mathrm{d} u) \bigg). \end{equation*} $$

Lemma 3.1 We have, for all $\mu \in \mathcal{M}^{\gamma_1,\gamma_2}$ ,

$$ \begin{equation*} \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \mu(B(x,u))^2 \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} x {\rm d} u < \infty. \end{equation*} $$

Proof. This follows directly from Definition 2.1 of the space $\mathcal{M}^{\gamma_1,\gamma_2}$ by using the estimate in (0.2) for the function $\varphi$ defined by

(3.2) $$ \begin{equation} \varphi(u)\,:\!= \int_{\mathbb{R}^2} \mu(B(x,u))^2 {\rm d} x \label{eqn15} \end{equation} $$

for $u \in \mathbb{R}_+^2$ .

In the following, we briefly note that all the limiting random fields obtained in Theorems 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6 are well defined.

• Using Lemma 3.1, we can easily check that the right-hand side of (2.2) is a symmetric, positive-semidefinite function such that there is a centred Gaussian linear random field Z with covariance function given by (2.2).

• The existence of $J_I$ follows from Lemma 12.13 of [Reference Kallenberg19] and Lemma 3.1.

• The proof of Theorem 2.3 shows that $\sigma^2$ given in (4.17) is finite. Hence, it can serve to construct the covariance function of a centred Gaussian linear random field Y.

• The existence of the compensated Poisson integral $J_L(\mu)$ for $\mu \in \mathcal{M}_{L}$ given in (2.7) can be verified by Lemma 12.13 of [Reference Kallenberg19]. We just have to show that

  $$ \begin{equation*} \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \min ( \vert g(x,u) \vert, g(x,u) ^2 )\frac{1}{u_1^{\gamma_1+1}} f_2 ( u_2 ) {\rm d} x {\rm d} u < \infty, \end{equation*} $$
  where
  $$ \begin{equation*} g(x,u)\,:\!= u_2 \int_{[x_1-{u_1}/{2}, x_1+{u_1}/{2}]} f_\mu(y_1,x_2){\rm d} y_1. \end{equation*} $$

This can be seen by a case distinction. Let us start with the following general consideration. There is an ${\varepsilon} >0$ such that

(3.3) $$ \begin{equation} \min ( \vert v \vert , v^2 ) \leq \min ( \vert v \vert^{\gamma_1-{\varepsilon}} , \vert v \vert^{\gamma_1+{\varepsilon}} ) \label{eqn16} \end{equation} $$

with $1<\gamma_1 - {\varepsilon} $ and $\gamma_1 + {\varepsilon} < \min ( \gamma_2, 2 )$ . Furthermore, we observe that

(3.4) $$ \begin{align} & \min ( \vert ab \vert^{\gamma_1-{\varepsilon}} , \vert ab \vert^{\gamma_1+{\varepsilon}} )\nonumber \\ &\qquad\leq \min ( \vert a \vert^{\gamma_1-{\varepsilon}} (\vert b \vert^{\gamma_1-{\varepsilon}}+\vert b \vert^{\gamma_1+{\varepsilon}}) , \vert a \vert^{\gamma_1+{\varepsilon}} (\vert b \vert^{\gamma_1+{\varepsilon}}+\vert b \vert^{\gamma_1-{\varepsilon}}) )\label{eqn17}\end{align} $$

$$ \begin{align} &\qquad= \min ( \vert a \vert^{\gamma_1-{\varepsilon}} , \vert a \vert^{\gamma_1+{\varepsilon}} ) (\vert b \vert^{\gamma_1-{\varepsilon}}+\vert b \vert^{\gamma_1+{\varepsilon}}).\nonumber \end{align} $$

We use (3.3), (3.4), and assumption (2.4) from Definition 2.2 to obtain

(3.5) $$ \begin{align} &\min ( \vert g(x,u) \vert, g(x,u) ^2 ) \nonumber \\ &\qquad\leq \min ( (C_\mu g_1(x_1,u_1) {\rm e}^{-c_\mu \vert x_2 \vert} )^{\gamma_1-{\varepsilon}} , (C_\mu g_1(x_1,u_1) {\rm e}^{-c_\mu \vert x_2 \vert})^{\gamma_1+{\varepsilon}} )\nonumber \\ &\qquad \times (\vert u_2 \vert^{\gamma_1-{\varepsilon}}+\vert u_2 \vert^{\gamma_1+{\varepsilon}}), \label{eqn18}\end{align} $$

where

(3.6) $$ \begin{equation} g_1(x_1,u_1)\,:\!= \int_{[x_1-{u_1}/{2}, x_1+{u_1}/{2}]} {\rm e}^{-c_\mu \vert y_1 \vert } {\rm d} y_1. \label{eqn19} \end{equation} $$

Since

$$ \begin{equation*} \int_{\mathbb{R}_+} (\vert u_2 \vert^{\gamma_1-{\varepsilon}}+\vert u_2 \vert^{\gamma_1+{\varepsilon}}) f_2 ( u_2 ) {\rm d} u_2 < \infty \end{equation*} $$

due to $\gamma_1 + {\varepsilon} < \gamma_2$ and the asymptotic behaviour of $f_2$ , and since

$$ \begin{equation*} \int_{\mathbb{R}} {\rm e}^{-c_\mu \vert x_2 \vert({\gamma_1 \pm {\varepsilon}})} {\rm d} x_2 < \infty, \end{equation*} $$

it remains to show that

(3.7) $$ \begin{equation} \int_{\mathbb{R} \times \mathbb{R}_+} \min ( g_1(x_1,u_1)^{\gamma_1-{\varepsilon}} , g_1(x_1,u_1) ^{\gamma_1+{\varepsilon}} ) \frac{1}{u_1^{\gamma_1+1}} {\rm d} x_1 {\rm d} u_1 \label{eqn20} \end{equation} $$

is finite. For $0 \leq u_1 \leq 1$ , we obtain

$$ \begin{align*} \int_{\mathbb{R} } g_1(x_1,u_1) ^{\gamma_1+{\varepsilon}} {\rm d} x_1 &= \int_{\mathbb{R} } \bigg( \int_{[x_1-{u_1}/{2}, x_1+{u_1}/{2}]} {\rm e}^{-c_\mu \vert y_1 \vert } \mathrm{d} y_1 \bigg)^{\gamma_1+{\varepsilon}} \mathrm{d} x_1 \\ &\leq 2 \int_{\mathbb{R}_+ } \bigg( \int_{[x_1-{u_1}/{2}, x_1+{u_1}/{2}]} {\rm e}^{-c_\mu y_1} \mathrm{d} y_1 \bigg)^{\gamma_1+{\varepsilon}} \mathrm{d} x_1 \\ &\leq 2 \int_{\mathbb{R}_+ } \bigg( \int_{[x_1-{u_1}/{2}, x_1+{u_1}/{2}]} {\rm e}^{-c_\mu ( x_1-{u_1}/{2}) } \mathrm{d} y_1 \bigg)^{\gamma_1+{\varepsilon}} \mathrm{d} x_1 \\ &= 2 {\rm e}^{c_\mu {u_1}({\gamma_1+{\varepsilon}})/{2}} u_1^{\gamma_1+{\varepsilon}}\int_{\mathbb{R}_+ } {\rm e}^{-c_\mu ({\gamma_1+{\varepsilon}}) x_1}{\rm d} x_1 \\ &\leq C u_1^{\gamma_1+{\varepsilon}}. \end{align*} $$

In the case of $u_1 \geq 1$ , we observe that

$$ \begin{align*} & \int_{\mathbb{R} } g_1(x_1,u_1) ^{\gamma_1-{\varepsilon}} {\rm d} x_1 \\ &\qquad\leq \int_{[-{u_1}/{2}, {u_1}/{2}]} \bigg( \int_{\mathbb{R}} {\rm e}^{-c_\mu \vert y_1 \vert} {\rm d} y_1 \bigg)^{\gamma_1-{\varepsilon}} \mathrm{d} x_1 + 2 \int_{({u_1}/{2}, \infty )} g_1(x_1,u_1) ^{\gamma_1-{\varepsilon}} \mathrm{d} x_1 \\ &\qquad = \bigg( \frac{2}{c_\mu} \bigg) ^{\gamma_1-{\varepsilon}} u_1 + \frac{2}{c_\mu^{\gamma_1-{\varepsilon}}} \int_{({u_1}/{2}, \infty )} ( {\rm e}^{-c_\mu x_1} ( {\rm e}^{c_\mu {u_1}/{2}} - {\rm e}^{-c_\mu {u_1}/{2}} ) )^{\gamma_1-{\varepsilon}} \mathrm{d} x_1 \\ &\qquad\leq C \bigg( u_1 + ( {\rm e}^{c_\mu {u_1}/{2}} - {\rm e}^{-c_\mu {u_1}/{2}} )^{\gamma_1-{\varepsilon}} \frac{1}{c_\mu({\gamma_1-{\varepsilon}})} {\rm e}^{-c_\mu ({\gamma_1-{\varepsilon}}) {u_1}/{2}}\bigg) \\ &\qquad\leq C ( u_1 + ( {\rm e}^{c_\mu {u_1}/{2}} )^{\gamma_1-{\varepsilon}} {\rm e}^{-c_\mu ({\gamma_1-{\varepsilon}}) {u_1}/{2}}) \\ &\qquad\leq C ( u_1 +1 ) \\ &\qquad\leq C u_1. \end{align*} $$

Finally, we can split the integral in (3.7) into two parts following this case distinction and see that these are bounded by

$$ \begin{equation*} \int_{( 0,1]} C u_1^{\gamma_1+{\varepsilon}} \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 < \infty \quad\text{and}\quad \int_{( 1, \infty )} C u_1 \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 < \infty, \end{equation*} $$

respectively. Therefore, the existence of the compensated Poisson integral $J_L(\mu)$ for $\mu \in \mathcal{M}_{L}$ is proven since the integral in (3.7) is finite. We note that in inequality (3.5) the particular exponent $\gamma_1-{\varepsilon}$ is not required for this proof and we could also replace $\gamma_1-{\varepsilon}$ by 1. However, we stick to the exponent $\gamma_1-{\varepsilon}$ because we will need the estimates here for later purposes, for instance, in the proof of Theorem 2.4.

• Since $f_\mu \in L^{\gamma_1}(\mathbb{R}^2)$ for $\mu \in \mathcal{M}_{P}$ , the random linear functional $S_{\gamma_1}(\mu)$ given in (2.9) is well defined. We refer the reader to Chapter 3 of [Reference Samorodnitsky and Taqqu27] for an extensive discussion.

• We can deduce from the proof of Theorem 2.6 that the integral in (2.12) is finite and serves to construct the covariance function of a centred Gaussian linear random field X.

3.3 Further useful lemmas

We continue with some useful lemmas that we use in the proofs of the main results in Section 4. These new lemmas are inspired by Lemma 2.4 of [Reference Biermé, Estrade and Kaj4], and Lemmas 4 and 6 of [Reference Kaj, Leskelä, Norros and Schmidt18].

Lemma 3.2

Let F be a measure on $\mathbb{R}_+^2$ according to (1.1) and to the asymptotic behaviour specified there. Furthermore, let g be a continuous function on $\smash{\mathbb{R}_+^2}$ such that there is a constant $C>0$ for some $\alpha_i>\gamma_i$ for $i=1,2$ such that

(3.8) $$ \begin{equation} \vert g(u) \vert \leq C \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ) \label{eqn21} \end{equation} $$

for all $u \in \mathbb{R}_+^2$ . Then, we have, as $\rho \to 0$ ,

$$ \begin{equation*} \int_{\mathbb{R}_+^2} g(u) F_\rho(\mathrm{d} u) \sim \rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2} g(u) \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u. \end{equation*} $$

Proof. The idea of the proof is to split the integral $\smash{\int_{\mathbb{R}_+^2} g(u) F_\rho(\mathrm{d} u)}$ into four parts and treat the four integrals separately.

Let ${\varepsilon}>0$ be given and define the constant $c_0$ by

(3.9) $$ \begin{equation} c_0\int_{\mathbb{R}_+^2} \vert g(u) \vert \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u = \bigg\vert \int_{\mathbb{R}_+^2}g(u) \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u \bigg\vert; \label{eqn22} \end{equation} $$

where we note that

$$ \begin{equation*} \int_{\mathbb{R}_+^2} \vert g(u) \vert \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u < \infty \end{equation*} $$

because of inequality (3.8) and that we have to treat the special case with

$$ \begin{equation*} \int_{\mathbb{R}_+^2}g(u) \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u = 0 \end{equation*} $$

slightly differently. Therefore, we can assume that $c_0>0$ .

Choose $N=N({\varepsilon})$ such that, for all $u_i>N$ for $i=1,2,$ we have

(3.10) $$ \begin{equation} f_i(u_i) \leq \frac{2}{u_i^{\gamma_i+1}} \label{eqn23} \end{equation} $$

and

(3.11) $$ \begin{equation} \bigg\vert \kern2ptf_1(u_1)f_2(u_2) - \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} \bigg\vert \leq c_0 \frac{{\varepsilon}}{2} \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}}, \label{eqn24} \end{equation} $$

which is feasible due to the power-law assumption on the measure F. We write $\mathbb{R}_+^2 = \smash{\bigcup_{k=1}^4 \Omega_k}$ with

(3.12) $$ \begin{equation} \begin{gathered} \Omega_1\,:\!= (\rho N, \infty)^2, \qquad \Omega_2\,:\!= (0, \rho N]^2, \\ \Omega_3\,:\!= (\rho N, \infty) \times (0, \rho N],\qquad \Omega_4\,:\!= (0, \rho N] \times (\rho N, \infty). \end{gathered} \label{eqn25} \end{equation} $$

From now on, we discuss the four corresponding integrals separately.

1. 1. Using (3.11), we obtain

   (3.13) $$ \begin{align} & \bigg\vert \int_{\Omega_1} g(u) F_\rho(\mathrm{d} u) - \rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2} g(u) \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u \bigg\vert \nonumber \\ &\qquad\leq \int_{\Omega_1} \vert g(u) \vert \bigg\vert \kern2ptf_1\bigg(\frac{u_1}{\rho}\bigg) \frac{1}{\rho} f_2\bigg(\frac{u_2}{\rho}\bigg)\frac{1}{\rho} - \rho^{\gamma_1+\gamma_2} \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} \bigg\vert \mathrm{d} u \nonumber \\ &\qquad + \rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2 \setminus \Omega_1} \vert g(u) \vert \frac{ 1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u \nonumber \\ &\qquad\leq c_0 \frac{{\varepsilon}}{2} \rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2} \vert g(u) \vert \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u \nonumber \\ &\qquad + \rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2 \setminus \Omega_1} \vert g(u) \vert \frac{ 1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u \label{eqn26}\end{align} $$
   $$ \begin{align} &\qquad\leq c_0 {\varepsilon} \rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2} \vert g(u) \vert \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u \nonumber \end{align} $$
   for small enough $\rho$ , where we also used the fact that the integral in (3.13) converges to zero by the dominated convergence theorem. Hence, we can deduce together with the definition of $c_0$ in (3.9) that there exists some $\rho_1 > 0$ such that, for all $\rho < \rho_1,$ we obtain
   $$ \begin{equation*} \bigg\vert\frac{ \int_{\Omega_1} g(u) F_\rho(\mathrm{d} u)}{\rho^{\gamma_1+\gamma_2} \int_{\mathbb{R}_+^2} g(u) (1/u_1^{\gamma_1+1})(1/u_2^{\gamma_2+1}) {\rm d} u} -1\bigg\vert \leq {\varepsilon}. \end{equation*} $$

2. 2. We show that $\smash{\vert \int_{\Omega_2} g(u) F_\rho(\mathrm{d} u) \vert} = o(\rho^{\gamma_1+\gamma_2}).$ Indeed, using (3.8), we obtain

   $$ \begin{align*} \bigg\vert \int_{\Omega_2} g(u) F_\rho(\mathrm{d} u) \bigg\vert & \leq C \int_0^{\rho N} \int_0^{\rho N} u_1^{\alpha_1} u_2^{\alpha_2} f_1\bigg(\frac{u_1}{\rho} \bigg) f_2\bigg(\frac{u_2}{\rho}\bigg)\frac{1}{\rho^2} {\rm d} u_1 {\rm d} u_2 \\ &= C \rho^{\alpha_1+\alpha_2} \int_0^N \int_0^N u_1^{\alpha_1} u_2^{\alpha_2} f_1(u_1)f_2(u_2) {\rm d} u_1 {\rm d} u_2 \\ &\leq C \rho^{\alpha_1+\alpha_2} N^{\alpha_1+\alpha_2}, \end{align*} $$
   where we recall that $ \smash{\int_0^\infty \int_0^\infty f_1(u_1)f_2(u_2) {\rm d} u_1 {\rm d} u_2}=1$ according to (1.1) and the assumption that $c_F=1$ there. Since $\alpha_1+\alpha_2 > \gamma_1+\gamma_2$ , the assertion of 2 follows upon $\rho \to 0$ .

3. 3. We show that $\smash{\vert \int_{\Omega_3} g(u) F_\rho(\mathrm{d} u) \vert }= o(\rho^{\gamma_1+\gamma_2}).$ We obtain, for N satisfying (3.10),

   $$ \begin{align*} \bigg\vert \int_{\Omega_3} g(u) F_\rho(\mathrm{d} u) \bigg\vert & \leq \int_{\rho N}^\infty \int_0^{\rho N} \vert g(u) \vert \kern2ptf_2\bigg(\frac{u_2}{\rho}\bigg)\frac{1}{\rho} {\rm d} u_2 f_1 \bigg(\frac{u_1}{\rho}\bigg) \frac{1}{\rho} {\rm d} u_1 \\ &\leq C \int_{\rho N}^\infty \int_0^{\rho N} \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ) f_2 \bigg(\frac{u_2}{\rho}\bigg)\frac{1}{\rho} {\rm d} u_2 \frac{\rho^{\gamma_1}}{u_1^{\gamma_1+1}} {\rm d} u_1 \\ &\leq C \rho^{\gamma_1} \int_{\rho N}^\infty \min(u_1,u_1^{\alpha_1}) \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 \int_0^{\rho N} u_2^{\alpha_2} f_2\bigg(\frac{u_2}{\rho}\bigg)\frac{1}{\rho} {\rm d} u_2 \\ & = C \rho^{\gamma_1} \rho^{\alpha_2} \int_0^{N} u_2^{\alpha_2} f_2(u_2) {\rm d} u_2 \\ &\leq C \rho^{\gamma_1+\alpha_2} N^{\alpha_2} \\ &\leq {\varepsilon} \rho^{\gamma_1+\gamma_2} \end{align*} $$
   for sufficiently small $\rho$ since $\gamma_1+\alpha_2 > \gamma_1+\gamma_2$ .

4. 4. Proceeding analogously to 3, we show that $\smash{\vert \int_{\Omega_4} g(u) F_\rho(\mathrm{d} u) \vert} = o(\rho^{\gamma_1+\gamma_2}).$

Finally, we are able to deduce the assertion of the lemma using the results from the four parts above.

The following lemma is also inspired by Lemma 2.4 of [Reference Biermé, Estrade and Kaj4].

Lemma 3.3

Let F be a measure on $\smash{\mathbb{R}_+^2}$ according to (1.1) and to the asymptotic behaviour specified there. Furthermore, let $( g_\rho )$ be a family of continuous functions on $\smash{\mathbb{R}_+^2}$ with

$$ \begin{equation*} \lim_{\rho \to 0} \rho^{\gamma_1 + \gamma_2} g_\rho(u) = 0 \end{equation*} $$

for all $u \in \mathbb{R}_+^2$ and

$$ \begin{equation*} \rho^{\gamma_1 + \gamma_2} \vert g_\rho(u) \vert \leq C \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ) \end{equation*} $$

for some constants $C>0$ and $\alpha_i>\gamma_i$ for $i=1,2$ for all $u \in \smash{\mathbb{R}_+^2}$ . Then, we have

(3.14) $$ \begin{equation} \lim_{\rho \to 0} \int_{\mathbb{R}_+^2} g_\rho(u) F_\rho(\mathrm{d} u) = 0. \label{eqn27} \end{equation} $$

Proof. The assumptions on $g_\rho$ ensure that, for all $\rho >0$ ,

$$ \begin{equation*} \int_{\mathbb{R}_+^2} \rho^{\gamma_1+\gamma_2} \vert g_\rho(u) \vert \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u < \infty, \end{equation*} $$

that there is an integrable majorant, and that we obtain

(3.15) $$ \begin{equation} \lim_{\rho \to 0} \int_{\mathbb{R}_+^2} \rho^{\gamma_1+\gamma_2} \vert g_\rho(u) \vert \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} u = 0 \label{eqn28} \end{equation} $$

by the dominated convergence theorem.

Due to the power-law assumption on F, we can choose $N>0$ such that, for all $u_i>N$ for $i=1,2,$ we have

(3.16) $$ \begin{equation} f_i(u_i) \leq \frac{2}{u_i^{\gamma_i+1}}. \label{eqn29} \end{equation} $$

We use the same definition of the domains $\Omega_k$ for $k=1,\ldots,4$ as in (3.12) and continue discussing the corresponding four integrals separately. First, using (3.16), we obtain

$$ \begin{align*} \bigg\vert \int_{\Omega_1} g_\rho(u) F_\rho(\mathrm{d} u) \bigg\vert &\leq \int_{\rho N}^\infty \int_{\rho N}^\infty \vert g_\rho(u) \vert \kern2ptf_1 \bigg(\frac{u_1}{\rho}\bigg) \frac{1}{\rho} f_2\bigg(\frac{u_2}{\rho}\bigg)\frac{1}{\rho} {\rm d} u_1 {\rm d} u_2 \\ &\leq \int_{0}^\infty \int_{0}^\infty \rho^{\gamma_1+\gamma_2} \vert g_\rho(u) \vert\frac{2}{u_1^{\gamma_1+1}} \frac{2}{u_2^{\gamma_2+1}} {\rm d} u_1 {\rm d} u_2. \end{align*} $$

Therefore, we obtain, together with (3.15),

$$ \begin{equation*} \lim_{\rho \to 0} \int_{\Omega_1} g_\rho(u) F_\rho(\mathrm{d} u) =0. \end{equation*} $$

Using the second assumption on $g_\rho$ and (3.16), we can check that

$$ \begin{equation*} \bigg\vert \int_{\Omega_k} g_\rho(u) F_\rho(\mathrm{d} u) \bigg\vert \to 0 \end{equation*} $$

as $\rho \to 0$ for $k=2,3,4$ by proceeding analogously to the corresponding parts in the proof of Lemma 3.2. Combining all four partial results, we can deduce (3.14).

We introduce for a signed measure $\mu \in \mathcal{M}_{k}$ for $k \in \{L,P\}$ the local averages $m_\mu(x,u)$ by

(3.17) $$ \begin{equation} m_\mu(x,u)\,:\!= \frac{1}{u_1 u_2} \int_{B(x,u)} f_\mu(y) {\rm d} y \label{eqn30} \end{equation} $$

and the maximal function $m_\mu^*$ by

(3.18) $$ \begin{equation} m_\mu^*(x)\,:\!= \sup_{u \in \mathbb{R}^2_+} \frac{1}{u_1 u_2} \int_{B(x,u)} \vert \kern2ptf_\mu(y) \vert {\rm d} y. \label{eqn31} \end{equation} $$

Similar to Lemma 4 of [Reference Kaj, Leskelä, Norros and Schmidt18], we obtain the following facts.

Lemma 3.4

Let $n_i(\rho) \to 0$ as $\rho \to 0$ for $i=1,2$ .

1. (i) For $\mu \in \mathcal{M}_{P}$ , we have

   $$ \begin{equation*} \lim_{\rho \to 0} m_\mu \bigg( x, \binom{n_1(\rho) u_1}{n_2(\rho) u_2} \bigg)= f_\mu(x) \quad \text{for all $(x,u) \in \mathbb{R}^2 \times \mathbb{R}_+^2$.} \end{equation*} $$

2. (ii) Let $\beta>1$ . For $\mu \in \mathcal{M}_{k}$ for $k \in \{L,P\}$ , there is a function $g \in L^\beta(\mathbb{R}^2)$ such that $m_\mu^*(x) \leq g(x)$ for all $x \in \mathbb{R}^2$ .

Proof. (i) The assertion is true because the function $f_\mu$ is continuous and because there exists, for all $\delta>0,$ some small enough $\rho_0>0$ such that the set $\smash{B ( x, \binom{n_1(\rho) u_1}{n_2(\rho) u_2} )}$ is contained in the $\ell^\infty$ -ball with centre x and radius $\delta$ for all $\rho<\rho_0$ .

(ii) We only require assumption (2.4) on $\mu \in \mathcal{M}_{k}$ for $k \in \{L,P\}$ . We obtain

(3.19) $$ \begin{align} m_\mu^*(x) &\leq C_\mu \sup_{u \in \mathbb{R}^2_+} \frac{1}{u_1 u_2} \int_{B(x,u)} {\rm e}^{-c_\mu \vert y_1 \vert} {\rm e}^{-c_\mu \vert y_2 \vert} {\rm d} y \nonumber \\ &= C_\mu \prod_{i=1,2} \sup_{u_i \in \mathbb{R}_+} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu\vert y_i \vert} {\rm d} y_i, \label{eqn32}\end{align} $$

and study the supremum in (3.19) by a case distinction. Let $x_i>0$ . We estimate

$$ \begin{align*} & \sup_{u_i >0} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu\vert y_i \vert} {\rm d} y_i \\ &\qquad\leq \sup_{0< {u_i}/{2} \leq x_i} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu\vert y_i \vert} {\rm d} y_i + \sup_{{u_i}/{2} \geq x_i} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu\vert y_i \vert} {\rm d} y_i, \end{align*} $$

and treat the two terms in the last line separately. For $0< {u_i}/{2}\leq x_i$ , we obtain

(3.20) $$ \begin{align} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu\vert y_i \vert} \mathrm{d} y_i &= \frac{1}{u_i} \frac{1}{c_\mu} ( {\rm e}^{-c_\mu(x_i - {u_i}/{2})} - {\rm e}^{-c_\mu (x_i + {u_i}/{2})} ) \nonumber \\ &= \frac{{\rm e}^{-c_\mu x_i}}{c_\mu} \frac{{\rm e}^{{c_\mu u_i}/{2}} - {\rm e}^{-{c_\mu u_i}/{2}}}{u_i} \nonumber \\ &\leq \frac{{\rm e}^{-c_\mu x_i}}{c_\mu} \frac{{\rm e}^{c_\mu x_i} - {\rm e}^{-c_\mu x_i}}{2x_i} \label{eqn33}\end{align} $$

$$ \begin{align} &\leq \frac{1}{2c_\mu x_i}, \nonumber \end{align} $$

where we used the fact that the function

$$ \begin{equation*} h(u_i)\,:\!= \frac{{\rm e}^{c u_i} - {\rm e}^{-cu_i}}{u_i} \end{equation*} $$

is increasing for $u_i\geq 0$ in (3.20). This can be seen by

$$ \begin{equation*} h(u_i) = \frac{1}{u_i} \bigg( \sum_{k=0}^\infty \frac{(cu_i)^k}{k!}-\sum_{k=0}^\infty \frac{(-cu_i)^k}{k!} \bigg) = \frac{1}{u_i} \sum_{l=0}^\infty \frac{2(cu_i)^{2l+1}}{(2l+1)!} = \sum_{l=0}^\infty \frac{2c(cu_i)^{2l}}{(2l+1)!} \end{equation*} $$

because the last term is increasing in $u_i$ . For ${u_i}/{2} \geq x_i$ , we observe that

$$ \begin{equation*} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu \vert y_i \vert} {\rm d} y_i \leq \frac{1}{2x_i} \int_{\mathbb{R}} {\rm e}^{-c_\mu \vert y_i \vert} {\rm d} y_i \leq \frac{1}{c_\mu x_i}. \end{equation*} $$

Combining the estimates, we obtain

$$ \begin{equation*} \sup_{u_i >0} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu \vert y_i \vert} {\rm d} y_i \leq \frac{2}{c_\mu x_i}. \end{equation*} $$

The corresponding estimate with $\vert x_i \vert$ for $x_i<0$ follows directly because of symmetry. Furthermore, we can bound the supremum in (3.19) by

$$ \begin{equation*} \sup_{u_i >0} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} {\rm e}^{-c_\mu \vert y_i \vert} {\rm d} y_i \leq \sup_{u_i >0} \frac{1}{u_i} \int_{[x_i-{u_i}/{2}, x_i+{u_i}/{2}]} 1 {\rm d} y_i = 1. \end{equation*} $$

Hence, we are able to conclude that $m_\mu^*(x) \leq g(x)$ for all $x \in \mathbb{R}^2$ , where g is defined by

$$ \begin{equation*} g(x)\,:\!= C_\mu \prod_{i=1,2} \min \bigg( 1, \frac{2}{c_\mu \vert x_i \vert} \bigg), \end{equation*} $$

and we see that $g^\beta$ is integrable with respect to x for any $\beta>1$ .

Lemma 3.5. For each $\mu \in \mathcal{M}_2$ , the functions

$$ \begin{equation*} u \mapsto \int_{\mathbb{R}^2} \Psi(\mu(B(x,u))) {\rm d} x \quad\text{and}\quad u \mapsto \int_{\mathbb{R}^2} \mu(B(x,u))^2 {\rm d} x \end{equation*} $$

are continuous on $\mathbb{R}_+^2$ .

Proof. The idea and the steps of the proof are identical to those in the proof of Lemma 6 of [Reference Kaj, Leskelä, Norros and Schmidt18]. Therefore, we only point out the difference. We start to proceed as there and then obtain

$$ \begin{equation*} d(u,v)\,:\!= \int_{\mathbb{R}^2} \vert \mu(B(x,u))-\mu(B(x,v)) \vert {\rm d} x \leq \Vert \mu \Vert \vert B(0,u) \triangle B(0,v) \vert \end{equation*} $$

for $u,v \in \smash{\mathbb{R}_+^2}$ , where $B(0,u) \triangle B(0,v)$ denotes the symmetric difference of the sets B(0,u) and B(0,v). Since the sets are just rectangles, we can deduce that $\lim_{u \to v} d(u,v) = 0 $ for all $v \in \smash{\mathbb{R}_+^2}$ and complete the proof together with the other parts.

4.1 Intermediate-intensity regime

Proof of Theorem 2.2. We recall the characteristic function of $\smash{\skew5\widetilde{J}_\rho(\mu)}$ :

$$ \begin{equation*} \mathbb{E} ({\rm e}^{{\rm i}\skew5\widetilde{J}_\rho(\mu)} ) = \exp \bigg( \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi(\mu(B(x,u))) \lambda_\rho {\rm d} x F_\rho(\mathrm{d} u) \bigg). \end{equation*} $$

The characteristic function of $J_I(\mu)$ is given by

$${\Bbb E}({{\text{e}}^{{\text{i}}{J_I}(\mu )}}) = \exp (\int_{{\mathbb {R}^2} \times \mathbb {R}_ + ^2} \Psi (\mu (B(x,u)))\frac{1}{{u_1^{{\gamma _1} + 1}}}\frac{1}{{u_2^{{\gamma _2} + 1}}}{\text{d}}x{\text{d}}u).$$

First, we define the function $\widetilde{\varphi}$ by

$$ \begin{equation*} \widetilde{\varphi}(u)\,:\!= \int_{\mathbb{R}^2} \Psi(\mu(B(x,u))) {\rm d} x \quad\text{for $u \in \mathbb{R}_+^2$.} \end{equation*} $$

We note that $\widetilde{\varphi}$ is continuous due to Lemma 3.5. Using $\vert \Psi(v) \vert \leq {v^2}/{2}$ and (2.1), there are constants $C>0$ and $\alpha_i$ with $\gamma_i < \alpha_i \leq 2$ for $i=1,2$ such that

$$ \begin{equation*} \vert \widetilde{\varphi}(u) \vert \leq C \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ). \end{equation*} $$

Now, we apply Lemma 3.2 with $g\,:\!= \widetilde{\varphi}$ to obtain

$$\int_{\mathbb{R}_ + ^2} {\tilde \varphi } (u){F_\rho }({\rm{d}}u){\rho ^{{\gamma _1} + {\gamma _2}}}\int_{\mathbb{R}_ + ^2} {\tilde \varphi } (u){1 \over {u_1^{{\gamma _1} + 1}}}{1 \over {u_2^{{\gamma _2} + 1}}}{\rm{d}}u.$$

Using this and the scaling $\lambda_\rho \rho^{\gamma_1+\gamma_2} \to 1$ shows the assertion.

4.2. High-intensity regime

Proof of Theorem 2.1. For the sake of simplicity, we introduce

$$ \begin{equation*} \varphi_\rho (u)\,:\!= \int_{\mathbb{R}^2} \Psi\bigg(\frac{\mu(B(x,u))}{n_\rho} \bigg) {\rm d} x \quad\text{for $u \in \mathbb{R}_+^2$,} \end{equation*} $$

with $n_\rho\,:\!= \smash{\sqrt{\lambda_\rho \rho^{\gamma_1+\gamma_2}}}$ and recall that the characteristic function of ${\skew5\widetilde{J}_\rho(\mu)} /{n_\rho}$ is given by

$$ \begin{equation*} \exp \bigg( \int_{\mathbb{R}_+^2} \varphi_\rho (u) \lambda_\rho F_\rho(\mathrm{d} u) \bigg). \end{equation*} $$

The goal is to show the convergence of this characteristic function to

$$ \begin{equation*} \exp \bigg( - \frac{1}{2} \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \mu(B(x,u))^2 \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}} {\rm d} x {\rm d} u \bigg), \end{equation*} $$

which corresponds to a centred Gaussian random variable. The covariance function given in (2.2) can then be obtained by the linearity of Z.

Since, by assumption, $n_\rho \to \infty$ as $\rho \to 0$ , we know that $\Psi({\mu(B(x,u))}/{n_\rho} ) $ can be approximated by $\smash{-\tfrac{1}{2}} ({\mu(B(x,u))}/{n_\rho} )^2.$ To be more precise, we write

(4.1) $$ \begin{equation} \int_{\mathbb{R}_+^2} \varphi_\rho (u) \lambda_\rho F_\rho(\mathrm{d} u) = - \frac{1}{2} \int_{\mathbb{R}_+^2} \varphi (u)\frac{\lambda_\rho}{n_\rho^2} F_\rho(\mathrm{d} u) + \int_{\mathbb{R}_+^2} \Delta_\rho(u) F_\rho(\mathrm{d} u), \label{eqn34} \end{equation} $$

where $\varphi$ is given in (3.2) and

$$ \begin{equation*} \Delta_\rho(u)\,:\!= \varphi_\rho(u) \lambda_\rho+ \frac{1}{2} \varphi(u) \frac{\lambda_\rho}{n_\rho^2} = \lambda_\rho \int_{\mathbb{R}^2}\bigg( \Psi\bigg(\frac{\mu(B(x,u))}{n_\rho} \bigg) + \frac{1}{2} \bigg( \frac{\mu(B(x,u))}{n_\rho} \bigg)^2 \bigg) {\rm d} x. \end{equation*} $$

Using Lemma 3.2 together with (2.1), the first integral on the right-hand side of (4.1) converges to

$$ \begin{equation*} {\int_{\mathbb{R}_+^2} {\varphi(u)} \frac{1}{u_1^{\gamma_1+1}} \frac{1}{u_2^{\gamma_2+1}}} {\rm d} u. \end{equation*} $$

Here, we again refer to Lemma 3.5 for the continuity of $\varphi$ .

It remains to show that the second integral on the right-hand side of (4.1) converges to zero. For this purpose, we show that $\Delta_\rho$ satisfies the assumptions on $g_\rho$ in Lemma 3.3.

First, we can show that the estimates $\vert \Psi(v) + {v^2}/{2}\vert \leq \vert v \vert ^3$ and

$$ \begin{equation*} \int_{\mathbb{R}^2} \vert \mu(B(x,u))\vert^3 {\rm d} x \leq \Vert \mu \Vert ^2 \int_{\mathbb{R}^2} \vert \mu(B(x,u))\vert {\rm d} x \leq \Vert \mu \Vert ^3 u_1 u_2 \end{equation*} $$

hold. Therefore, we obtain

$$ \begin{equation*} \vert \rho^{\gamma_1+\gamma_2} \Delta_\rho(u) \vert = \bigg\vert \frac{n_\rho^2}{\lambda_\rho} \Delta_\rho(u) \bigg\vert \leq \frac{\Vert \mu \Vert^3}{n_\rho} u_1 u_2 \to 0 \end{equation*} $$

as $\rho \to 0$ , which shows that the first assumption of Lemma 3.3 is satisfied. Using $\vert \Psi(v)\vert \leq {v^2}/{2}$ and (2.1), the second assumption is also satisfied because we obtain

$$ \begin{align*} \rho^{\gamma_1+\gamma_2}\vert \Delta_\rho(u) \vert &= \bigg\vert\frac{n_\rho^2}{\lambda_\rho} \Delta_\rho(u) \bigg\vert \\ &\leq n_\rho^2 \int_{\mathbb{R}^2}\bigg( \bigg\vert \Psi\bigg(\frac{\mu(B(x,u))}{n_\rho} \bigg) \bigg\vert+ \frac{1}{2} \bigg( \frac{\mu(B(x,u))}{n_\rho} \bigg)^2 \bigg) {\rm d} x \\ &\leq n_\rho^2 \int_{\mathbb{R}^2} \bigg( \frac{\mu(B(x,u))}{n_\rho} \bigg)^2 {\rm d} x \\ &= \int_{\mathbb{R}^2} \mu(B(x,u))^2 {\rm d} x \\ & \leq C \min(u_1,u_1^{\alpha_1} ) \min(u_2,u_2^{\alpha_2} ).\tag*{\qedhere} \end{align*} $$

4.3. Low-intensity regime

4.3.1 Points scaling regime

Proof of Theorem 2.5. In a first step, we prove that

$$ \begin{equation*} \lim_{\rho \to 0} \mathbb{E} \exp\bigg({{\rm i}\frac{\skew5\widetilde{J}_\rho(\mu)} {\lambda_\rho^{{1}/{\gamma_1}} \rho^2}}\bigg) = \exp \bigg( c_2^{\gamma_1} \int_{\mathbb{R}^2} \int_{\mathbb{R}_+} \Psi(u_1 f_\mu(x)) \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 {\rm d} x \bigg), \end{equation*} $$

where $c_2$ is defined in (4.4) below. In a second step, we show that the right-hand side is the characteristic function of an integral with respect to a stable random measure.

1. Step 1. We recall that the characteristic function of $\,{\skew5\widetilde{J}}_\rho(\mu) / (\lambda_\rho^{{1}/{\gamma_1}}\rho^2)$ can be written as

   (4.2) $$ \begin{equation} \exp \bigg( \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{1}{\lambda_\rho^{{1}/{\gamma_1}} \rho^2} \int_{B(x,u)} f_\mu(y) {\rm d} y\bigg) \lambda_\rho {\rm d} x\ F_\rho(\mathrm{d} u) \bigg). \label{eqn35} \end{equation} $$
   We use the definition of $m_\mu(x,u)$ in (3.17) and the density of the scaled measure F from (1.1) to obtain
   (4.3) $$ \begin{align} & \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{1}{\lambda_\rho^{{1}/{\gamma_1}} \rho^2} \int_{B(x,u)} f_\mu(y) \mathrm{d} y\bigg) \lambda_\rho {\rm d} x\ F_\rho(\mathrm{d} u) \nonumber \\ &\qquad= \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{u_1 u_2}{\lambda_\rho^{{1}/{\gamma_1}} \rho^2} m_\mu(x,u) \bigg) \lambda_\rho f_1 \bigg( \frac{u_1}{\rho}\bigg) \frac{1}{\rho} f_2 \bigg( \frac{u_2}{\rho}\bigg) \frac{1}{\rho} {\rm d} x {\rm d} u \nonumber \\ &\qquad= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \Psi \bigg( u_1 m_\mu \bigg (x,\binom{\lambda_\rho^{{1}/{\gamma_1}}\rho {u_1}/ {u_2}}{\rho u_2} \bigg) \bigg) \frac{ \lambda_\rho^{1+{1}/{\gamma_1}}}{ u_2} f_1 \bigg( \lambda_\rho^{{1}/{\gamma_1}} \frac{u_1}{ u_2}\bigg)\ f_2(u_2) {\rm d} x {\rm d} u, \label{eqn36}\end{align} $$
   where we substituted first $u_2 = \rho \widetilde{u}_2$ and then $u_1 = \smash{{\lambda_\rho^{{1}/{\gamma_1}} \rho} {\widetilde{u}_1}/{\widetilde{u}_2}}$ in the last line. We note that
   $$ \begin{equation*} \lim_{\rho \to 0} m_\mu \bigg (x,\binom{ \lambda_\rho^{{1}/{\gamma_1}}\rho {u_1}/ {u_2}}{\rho u_2} \bigg) = f_\mu(x) \end{equation*} $$
   because of Lemma 3.4(i) and that
   $$ \begin{align*} \frac{ \lambda_\rho^{1+{1}/{\gamma_1}}}{ u_2} f_1 \bigg( \lambda_\rho^{{1}/{\gamma_1}} \frac{u_1}{ u_2}\bigg) &= \frac{ \lambda_\rho^{1+{1}/{\gamma_1}}}{ u_2} f_1 \bigg( \lambda_\rho^{{1}/{\gamma_1}} \frac{u_1}{ u_2}\bigg) \bigg( \lambda_\rho^{{1}/{\gamma_1}} \frac{u_1}{ u_2}\bigg)^{\gamma_1+1-\gamma_1-1} \\ &= f_1 \bigg( \lambda_\rho^{{1}/{\gamma_1}} \frac{u_1}{ u_2}\bigg) \bigg( \lambda_\rho^{{1}/{\gamma_1}} \frac{u_1}{ u_2}\bigg)^{\gamma_1+1} \frac{ u_2^{\gamma_1}}{ u_1^{\gamma_1+1}} \\ &\to \frac{ u_2^{\gamma_1}}{ u_1^{\gamma_1+1}} \end{align*} $$
   as $\rho \to 0$ because of $\smash{\lambda_\rho^{1/{\gamma_1}}} \to \infty$ and the asymptotic behaviour of $f_1$ . Therefore, the integrand in (4.3) converges to
   $$ \begin{equation*} \Psi ( u_1 f_\mu(x) ) \frac{1}{u_1^{\gamma_1+1}}u_2^{\gamma_1}f_2(u_2). \end{equation*} $$
   If we can also find an integrable majorant of the integrand in (4.3), we obtain
   $$ \begin{align*} & \lim_{\rho \to 0} \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{1}{\lambda_\rho^{{1}/{\gamma_1}} \rho^2} \int_{B(x,u)} f_\mu(y) {\rm d} y\bigg) \lambda_\rho {\rm d} x F_\rho(\mathrm{d} u) \\ &\qquad= c_2^{\gamma_1} \int_{\mathbb{R}^2} \int_{\mathbb{R}_+} \Psi(u_1 f_\mu(x)) \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 {\rm d} x \end{align*} $$
   by the dominated convergence theorem, where $c_2$ is defined by
   (4.4) $$ \begin{equation} c_2\,:\!= \bigg( \int_{\mathbb{R}_+} u_2^{\gamma_1}f_2(u_2) {\rm d} u_2 \bigg)^{{1}/{\gamma_1}}. \label{eqn37} \end{equation} $$
   In order to find such a majorant, we can show that
   (4.5) $$ \begin{equation} \vert \Psi(v) \vert \leq 2\min ( \vert v \vert , v^2 ), \label{eqn38} \end{equation} $$
   and we note that there is an ${\varepsilon} >0$ with $1<\gamma_1-{\varepsilon}<\gamma_1+{\varepsilon} <2$ such that (3.3) and (3.4) hold. For all $\rho<\rho_0$ with small enough $\rho_0$ , the integrand (see (4.3)) is therefore dominated by
   (4.6) $$ \begin{equation} 2\min (\vert u_1 \vert^{\gamma_1- {\varepsilon}} , \vert u_1 \vert^{\gamma_1 + {\varepsilon}} ) ( \vert m_\mu^*(x) \vert^{\gamma_1- {\varepsilon}} + \vert m_\mu^*(x) \vert^{\gamma_1+ {\varepsilon}}) \frac{c_{f_1}}{u_1^{\gamma_1+1}} u_2^{\gamma_1} f_2(u_2), \label{eqn39} \end{equation} $$
   where we also used the technical assumption in (2.3). Finally, we can see that (4.6) is integrable because of Lemma 3.4(ii) and $1<\gamma_1 - {\varepsilon}$ .

2. Step 2. We deal with the integral

   (4.7) $$ \begin{equation} \int_{\mathbb{R}^2} \int_{\mathbb{R}_+} \Psi(u_1 f_\mu(x)) \frac{1}{u_1^{\gamma_1+1}} {\rm d} u_1 {\rm d} x. \label{eqn40} \end{equation} $$
   We split the integration over $\mathbb{R}^2$ into $\{x \colon f_\mu(x) \geq 0 \}$ and $\{x \colon f_\mu(x) < 0 \},$ and note that $\Psi(0)=0$ . We recall that ${f_\mu}_+\,:\!= \max({\kern2ptf_\mu},0)$ and ${f_\mu}_-\,:\!= - \min({\kern2ptf_\mu},0)$ . The substitution $\widetilde{u}_1=u_1 f_\mu(x)$ shows that (4.7) equals
   $$ \begin{equation*} d_{\gamma_1} \Vert {f_\mu}_+ \Vert_{\gamma_1}^{\gamma_1} + \bar{d}_{\gamma_1}\Vert {f_\mu}_- \Vert_{\gamma_1}^{\gamma_1}, \end{equation*} $$
   where $\smash{\bar{d}_{\gamma_1}}$ is the complex conjugate of $d_{\gamma_1}\,:\!= \int_{\mathbb{R}_+} ({\Psi(u_1)}/{u_1^{\gamma_1+1}}) {\rm d} u_1 $ . We obtain
   $$ \begin{equation*} d_{\gamma_1}= \frac{\Gamma(2-\gamma_1)}{\gamma_1(\gamma_1-1)} \cos \bigg( \frac{\pi \gamma_1}{2}\bigg) \bigg(1-{\rm i} \tan \bigg( \frac{\pi \gamma_1}{2} \bigg) \bigg) \end{equation*} $$
   due to [Reference Samorodnitsky and Taqqu27, p.170]. Therefore, we can finally conclude that
   $$ \begin{align*} & \lim_{\rho \to 0} \log \mathbb{E} \exp \bigg({{\rm i}\frac{\skew5\widetilde{J}_\rho(\mu)}{ c_{\gamma_1,\gamma_2} \lambda_\rho^{{1}/{\gamma_1}} \rho^2}}\bigg) \\ &\qquad= c_2^{\gamma_1} \bigg( d_{\gamma_1} \bigg\Vert \frac{{f_\mu}_+}{c_{\gamma_1} c_2} \bigg\Vert_{\gamma_1}^{\gamma_1} + \bar{d}_{\gamma_1} \bigg\Vert \frac{{f_\mu}_-}{c_{\gamma_1} c_2} \bigg\Vert_{\gamma_1}^{\gamma_1} \bigg) \\ &\qquad = - ( \Vert {f_\mu}_+ \Vert_{\gamma_1}^{\gamma_1} + \Vert {f_\mu}_- \Vert_{\gamma_1}^{\gamma_1} ) + {\rm i} \tan \bigg( \frac{\pi \gamma_1}{2} \bigg) ( \Vert {f_\mu}_+ \Vert_{\gamma_1}^{\gamma_1} - \Vert {f_\mu}_- \Vert_{\gamma_1}^{\gamma_1} ) \\ &\qquad= -\sigma_\mu^{\gamma_1} \bigg(1-{\rm i} \beta_\mu \tan \bigg( \frac{\pi \gamma_1}{2} \bigg) \bigg), \end{align*} $$
   where
   (4.8) $$ \begin{equation} c_{\gamma_1,\gamma_2}\,:\!= c_{\gamma_1} c_2, \qquad c_{\gamma_1}\,:\!= \bigg( - \frac{\Gamma(2-\gamma_1)}{\gamma_1(\gamma_1-1)} \cos \bigg( \frac{\pi \gamma_1}{2}\bigg) \bigg)^{{1}/{\gamma_1}}, \label{eqn41} \end{equation} $$
   $c_2$ is given in (4.4), and $\sigma_\mu$ , $ \beta_\mu$ are given in (2.10).

4.3.2 Poisson-lines scaling regime

Proof of Theorem 2.4. We recall the characteristic function of $\,\smash{{\skew5\widetilde{J}_\rho(\mu)}/{\rho}}$ given in (4.2). We proceed as in the proof of Theorem 2.5. Using the definition of $m_\mu(x,u)$ in (3.17) and the density of the scaled measure F from (1.1), we obtain

(4.9) $$ \begin{align} & \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{1}{\rho} \int_{B(x,u)} f_\mu(y) {\rm d} y\bigg) \lambda_\rho {\rm d} x F_\rho(\mathrm{d} u) \nonumber \\ &\qquad= \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{u_1 u_2}{\rho} m_\mu(x,u) \bigg) \lambda_\rho f_1 \bigg( \frac{u_1}{\rho}\bigg) \frac{1}{\rho} f_2 \bigg( \frac{u_2}{\rho}\bigg) \frac{1}{\rho} {\rm d} x {\rm d} u \nonumber \\ &\qquad= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \Psi \bigg( u_1 u_2 m_\mu \bigg(x,\binom{u_1}{\rho u_2} \bigg) \bigg) \frac{\lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho}\bigg) f_2 ( u_2 ) {\rm d} x {\rm d} u, \label{eqn42}\end{align} $$

where we substituted $u_2 = \rho \widetilde{u}_2$ in the last line. We note that, due to (2.5) in Definition 2.2 of the space $\mathcal{M}_{L}$ ,

$$ \begin{equation*} \lim_{\rho \to 0} u_1 u_2 m_\mu \bigg (x,\binom{u_1}{\rho u_2} \bigg) = u_2 \int_{[x_1-{u_1}/{2},x_1+{u_1}/{2}]} f_\mu(y_1,x_2) {\rm d} y_1 \end{equation*} $$

(pointwise for all $(x,u) \in \mathbb{R}^2 \times \mathbb{R}_+^2$ ) and that

$$ \begin{align*} \frac{\lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho}\bigg)& = \frac{\lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho}\bigg) \bigg( \frac{u_1}{\rho}\bigg) ^{\gamma_1+1} \bigg( \frac{\rho}{u_1}\bigg)^{\gamma_1+1} \\ &= f_1 \bigg( \frac{u_1}{\rho}\bigg) \bigg( \frac{u_1}{\rho}\bigg) ^{\gamma_1+1} \lambda_\rho \rho^{\gamma_1} \frac{1}{u_1^{\gamma_1+1}} \\ & \to \frac{1}{u_1^{\gamma_1+1}} \end{align*} $$

as $\rho \to 0$ because of $1 / \rho \to \infty$ , the asymptotic behaviour of $f_1,$ and the fact that $\lambda_\rho \rho^{\gamma_1} \to 1$ . Therefore, the integrand in (4.9) converges to

$$\Psi ({u_2}\int_{[{x_1} - {u_1}/2,{x_1} + {u_1}/2]} {{f_\mu }} ({y_1},{x_2}){\text{d}}{y_1})\frac{1}{{u_1^{{\gamma _1} + 1}}}{f_2}({u_2}).$$

If we can also find an integrable majorant of the integrand in (4.9), we obtain

(4.10) $$ \begin{align} & \lim_{\rho \to 0} \int_{\mathbb{R}_+^2}\int_{\mathbb{R}^2} \Psi \bigg( \frac{1}{\rho} \int_{B(x,u)} f_\mu(y) {\rm d} y\bigg) \lambda_\rho {\rm d} x F_\rho(\mathrm{d} u) \nonumber \\ &\qquad= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \Psi \bigg( u_2 \int_{[x_1-{u_1}/{2},x_1+{u_1}/{2}]} f_\mu(y_1,x_2){\rm d} y_1 \bigg)\frac{1}{u_1^{\gamma_1+1}} f_2 ( u_2 ) {\rm d} x {\rm d} u \label{eqn43}\end{align} $$

by the dominated convergence theorem. Using the estimates in (4.5) and (3.3), an extended version of (3.4) and the technical assumption in (2.3), we see that the integrand in (4.9) is dominated by

(4.11) $$ \begin{equation} 2 \min (\vert u_1 \vert^{\gamma_1- {\varepsilon}} , \vert u_1 \vert^{\gamma_1 + {\varepsilon}} ) ( \vert u_2 \vert^{\gamma_1- {\varepsilon}} + \vert u_2 \vert^{\gamma_1+ {\varepsilon}}) ( \vert m_\mu^*(x) \vert^{\gamma_1- {\varepsilon}} + \vert m_\mu^*(x) \vert^{\gamma_1+ {\varepsilon}}) \frac{c_{f_1}}{u_1^{\gamma_1+1}} f_2(u_2) \label{eqn44} \end{equation} $$

for all $\rho<\rho_0$ with small enough $\rho_0$ . Here, we have to choose ${\varepsilon} >0$ such that ${1<\gamma_1 - {\varepsilon}}$ , ${\gamma_1 + {\varepsilon} < 2},$ as well as $\gamma_1 + {\varepsilon} < \gamma_2$ . These conditions together with Lemma 3.4(ii) ensure that (4.11) is integrable.

Since the characteristic function $\mathbb{E} ({\rm e}^{{\rm i} J_L(\mu)} )$ of the limit $J_L(\mu)$ is given by the exponential of (4.10), the convergence of the characteristic function is proven.

Remark 4.2. In the general case, let us say $\lambda_\rho \rho^{\gamma_1} \to a^{2-\gamma_1} \in (0,\infty)$ with $a>0$ as $\rho \to 0$ , we obtain $\,\smash{\skew5\widetilde{J}_\rho(\mu) / (a \rho)} \to J_L(\mu_{a}), $ where we recall that $\mu_a(\cdot)\,:\!= \mu(a^{-1} \, \cdot)$ . In order to prove this, we note that we obtain (4.10) with the additional factor $a^{2-\gamma_1}$ for the logarithm of the characteristic function of the limit in the general case. Then, we can deduce the result after an appropriate substitution.

4.3.3 Gaussian-lines scaling regime

Proof of Theorem 2.3. We recall the characteristic function of $\,\smash{{\skew5\widetilde{J}_\rho(\mu) }/ {{\rho}^{1-\eta / 2}}}$ , which, after the substitution $u_2 = \rho \widetilde{u}_2$ , equals

$$ \begin{equation*} \exp \bigg( \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \Psi\bigg( \frac{\mu (B (x,\binom{u_1}{\rho u_2} ))}{{\rho}^{1-\eta / 2}} \bigg) \frac{ \lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho} \bigg)\ f_2(u_2) {\rm d} x {\rm d} u \bigg). \end{equation*} $$

The goal is to show for some $\sigma^2>0$ the convergence of this characteristic function to $\exp ( - \sigma^2 /2 )$ , which corresponds to a centred Gaussian random variable.

To be more precise, we write

(4.12) $$ \begin{align} & \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \Psi\bigg( \frac{ \mu (B (x,\binom{u_1}{\rho u_2} ))}{{\rho}^{1-\eta / 2}} \bigg) \frac{ \lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho} \bigg) f_2(u_2) {\rm d} x {\rm d} u \nonumber \\ &\qquad= - \frac{1}{2} \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} u_2^2 \bigg( \frac{\mu (B (x,\binom{u_1}{\rho u_2} ))}{\rho u_2} \bigg)^2 \rho^\eta \frac{\lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho} \bigg) f_2(u_2) {\rm d} x {\rm d} u \label{eqn45}\end{align} $$

(4.13) $$ \begin{align} &\qquad +\int_{\mathbb{R}^2 \times \mathbb{R}_+^2} \Delta_\rho(u,x)\frac{ \lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho} \bigg) f_2(u_2) {\rm d} x {\rm d} u, \label{eqn46}\end{align} $$

where

(4.14) $$ \begin{equation} \Delta_\rho(u,x)\,:\!= \Psi\bigg( \frac{ \mu (B (x,\binom{u_1}{\rho u_2})) }{\rho^{1-\eta/2}}\bigg)+ \frac{1}{2}\bigg(\frac{ \mu (B (x,\binom{u_1}{\rho u_2} )) }{\rho^{1-\eta/2}} \bigg)^2. \label{eqn47} \end{equation} $$

First, we discuss the integral in (4.13) in the case of $\gamma_2>3$ . Due to $\vert \Psi(v) + {v^2}/{2}\vert \leq \vert v \vert ^3$ , we can bound (4.14) and can thus bound the integrand by

(4.15) $$ \begin{align} & \rho^{3 \eta/2} u_2^3 \bigg( \frac{ \vert \mu (B (x,\binom{u_1}{\rho u_2} )) \vert}{\rho u_2} \bigg)^3 \frac{ \lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho} \bigg) f_2(u_2) \nonumber \\ &\qquad\leq \rho^{\eta/2} u_2^3 \bigg( \frac{C_\mu}{\rho u_2} \int_{B (x,\binom{u_1}{\rho u_2})} {\rm e}^{-c_\mu \vert y_1 \vert} {\rm e}^{-c_\mu \vert y_2 \vert} {\rm d} y \bigg) ^3 \lambda_\rho \rho^{\eta-1} c_{f_1} \bigg( \frac{\rho}{u_1} \bigg)^{\gamma_1+1} f_2(u_2) \nonumber \\ &\qquad\leq C \rho^{\eta/2} u_2^3 g_1(x_1,u_1) ^3 g_2(x_2) ^3 \frac{1}{u_1^{\gamma_1+1}} f_2(u_2) \label{eqn48}\end{align} $$

for $\rho<\rho_0$ with small enough $\rho_0$ , where $g_1$ is given in (3.6) and

$$ \begin{equation*} g_2(x_2)\,:\!= \min \bigg( 1, \frac{2}{c_\mu \vert x_2 \vert} \bigg). \end{equation*} $$

Here, we used assumption (2.4) from Definition 2.2, the technical assumption in (2.3), and the fact that $\lambda_\rho \rho^{\gamma_1+\eta} \to 1$ . Furthermore, we used

$$ \begin{equation*} \sup_{\rho > 0} \frac{1}{\rho u_2} \int_{[x_2-{\rho u_2}/{2}, x_2+{\rho u_2}/{2}]} {\rm e}^{-c_\mu\vert y_2 \vert} {\rm d} y_2 \leq g_2(x_2) \end{equation*} $$

from the proof of Lemma 3.4(ii). By (4.15), we see that the integrand in (4.13) has an integrable majorant since we assumed that $\gamma_2 > 3,$ and because $g_2^3$ is integrable with respect to $x_2$ and $\smash{g_1(x_1,u_1)^3 /{u_1^{\gamma_1+1}}} $ is also integrable (in order to check this, we just have to follow the lines below (3.7)). Moreover, the majorant converges to zero because of $\rho^{\eta/2} \to 0$ .

In the case of $2 < \gamma_2 \leq 3$ , we note that there is an ${\varepsilon}>0$ such that $2 < \gamma_2 - {\varepsilon} < 3$ as well as $\vert \Psi(v) + {v^2}/{2}\vert \leq \vert v \vert ^{\gamma_2 - {\varepsilon}}$ . The last-mentioned estimate can be deduced from Lemma 1 of [Reference Kaj, Leskelä, Norros and Schmidt18] by a case distinction (cf. (3.3)). Similar to above, we can bound the integrand in (4.13) by

(4.16) $$ \begin{align} & \rho^{{(\gamma_2 - {\varepsilon})} \eta/2} \bigg( \frac{ \vert \mu (B (x,\binom{u_1}{\rho u_2} )) \vert}{\rho} \bigg)^{\gamma_2 - {\varepsilon}} \frac{ \lambda_\rho}{\rho} f_1 \bigg( \frac{u_1}{\rho} \bigg) f_2(u_2) \nonumber \\ &\qquad\leq \rho^{{(\gamma_2 - {\varepsilon}-2+2)} \eta/2} u_2^{\gamma_2-{\varepsilon}} \bigg( \frac{\vert \mu (B (x,\binom{u_1}{\rho u_2} )) \vert}{\rho u_2} \bigg)^{\gamma_2-{\varepsilon}} \frac{ \lambda_\rho}{\rho} c_{f_1} \bigg( \frac{\rho}{u_1} \bigg)^{\gamma_1+1} f_2(u_2) \nonumber \\ &\qquad\leq C \rho^{{(\gamma_2 - {\varepsilon}-2)} \eta/2} u_2^{\gamma_2-{\varepsilon}} ( g_1(x_1,u_1) g_2(x_2) )^{\gamma_2-{\varepsilon}} \lambda_\rho \rho^{\gamma_1+\eta} \frac{1}{u_1^{\gamma_1+1}} f_2(u_2) \nonumber \\ &\qquad\leq C \rho^{{(\gamma_2 - {\varepsilon}-2)} \eta/2} u_2^{\gamma_2-{\varepsilon}} g_1(x_1,u_1) ^{\gamma_2-{\varepsilon}} g_2(x_2)^{\gamma_2-{\varepsilon}} \frac{1}{u_1^{\gamma_1+1}} f_2(u_2) \label{eqn49}\end{align} $$

for $\rho<\rho_0$ with small enough $\rho_0$ . Using $\gamma_1<\gamma_2-{\varepsilon}$ , we can see by (4.16) that the integrand in (4.13) has an integrable majorant because $g_2^{\gamma_2-{\varepsilon}}$ and $\smash{g_1(x_1,u_1)^{\gamma_2-{\varepsilon}}/{u_1^{\gamma_1+1}}}$ are integrable (with the same reasons as above) and that it converges to zero because of ${\gamma_2-{\varepsilon}-2}>0$ . Therefore, we obtain in both cases that the integral in (4.13) converges to zero by the dominated convergence theorem.

Next, we deal with the integral in (4.12) and show that it converges to

(4.17) $$ \begin{equation} \sigma^2\,:\!= \int_{\mathbb{R}^2 \times \mathbb{R}_+^2} u_2^2 \bigg( \int_{[x_1-{u_1}/{2},x_1+{u_1}/{2}]} f_\mu(y_1,x_2) {\rm d} y_1 \bigg)^2 \frac{f_2(u_2)}{u_1^{\gamma_1+1}} {\rm d} x {\rm d} u. \label{eqn50} \end{equation} $$

The convergence of the integrand can be seen similar to above using Definition 2.2 of the space $\mathcal{M}_{L}$ , the asymptotic behaviour of $f_1,$ and the fact that $\lambda_\rho \rho^{\gamma_1+\eta} \to 1$ . A majorant of the integrand is given by

$$ \begin{equation*} C u_2^2 g_1(x_1,u_1) ^2 g_2(x_2)^2 \frac{1}{u_1^{\gamma_1+1}} f_2(u_2), \end{equation*} $$

which is integrable for $\gamma_2>2$ . Applying the dominated convergence theorem, the convergence of the characteristic function is proven. By linearity, the covariance function given in (2.6) follows from (4.17).

Remark 4.3. In the general case, let us say $\lambda_\rho \rho^{\gamma_1+\eta} \to a^{2} \in (0,\infty)$ with $a>0$ as $\rho \to 0$ , the limit is a centred Gaussian linear random field which is given by $( Y(a \mu))_\mu$ , where $a \mu$ has the density $a f_\mu$ and the variance of $ Y(a \mu)$ is just $ a^2 \sigma^2$ . This can be seen since we obtain the additional factor $a^{2}$ in (4.17).

4.4 The finite-variance case

Proof of Theorem 2.6. We use the definition of $m_\mu(x,u)$ in (3.17) to obtain, for the logarithm of the characteristic function of $\,\smash{\skew5\widetilde{J}_\rho(\mu) / (\rho^2 \sqrt{\lambda_\rho v_1v_2})}$ ,

$$\int_{\mathbb {R}_ + ^2} {\int_{{\mathbb {R}^2}} \Psi } (\frac{1}{{\lambda _\rho ^{1/{\gamma _1}}{\rho ^2}}}\int_{B(x,u)} {{f_\mu }} (y){\text{d}}y){\lambda _\rho }{\text{d}}x\;{F_\rho }({\text{d}}u){\text{ }} = \int_{\mathbb {R}_ + ^2} {\int_{{\mathbb {R}^2}} \Psi } (\frac{{{u_1}{u_2}}}{{\lambda _\rho ^{1/{\gamma _1}}{\rho ^2}}}{m_\mu }(x,u)){\lambda _\rho }{f_1}(\frac{{{u_1}}}{\rho })\frac{1}{\rho }{f_2}(\frac{{{u_2}}}{\rho })\frac{1}{\rho }{\text{d}}x{\text{d}}u{\text{ }} = \int_{{\mathbb {R}^2} \times \mathbb {R}_ + ^2} \Psi ({u_1}{m_\mu }(x,\left( {{\text{ }}\lambda _\rho ^{1/{\gamma _1}}\rho {u_1}/{u_2}{\text{ }}\rho {u_2}{\text{ }}} \right)))\frac{{\lambda _\rho ^{1 + 1/{\gamma _1}}}}{{{u_2}}}{f_1}(\lambda _\rho ^{1/{\gamma _1}}\frac{{{u_1}}}{{{u_2}}})\;{f_2}({u_2}){\text{d}}x{\text{d}}u,$$

where we substituted $u_2 = \rho \widetilde{u}_2$ and $u_1 = \rho \widetilde{u}_1$ in the last line. We note that

$$ \begin{equation*} \lim_{\rho \to 0} m_\mu \bigg (x,\binom{\rho u_1}{\rho u_2} \bigg) = f_\mu(x) \end{equation*} $$

because of Lemma 3.4(i). Due to the estimate $\vert \Psi(v) + {v^2}/{2}\vert \leq \vert v \vert ^3$ and $\lambda_\rho \to \infty$ , we obtain

$$ \begin{equation*} \lim_{\rho \to 0} \Psi \bigg(\frac{u_1 u_2}{\sqrt{ \lambda_\rho v_1 v_2}\,} m_\mu \bigg (x,\binom{\rho u_1}{\rho u_2} \bigg) \bigg) \lambda_\rho =-\frac{u_1^2 u_2^2\ f_\mu(x)^2}{2 v_1 v_2}. \end{equation*} $$

Furthermore, we use $\vert \Psi(v)\vert \leq {v^2}/{2}$ and the definition of $m_\mu^*(x)$ in (3.18) to obtain

$$ \begin{equation*} \Psi \bigg(\frac{u_1 u_2}{\sqrt{ \lambda_\rho v_1 v_2}\,} m_\mu \bigg (x,\binom{\rho u_1}{\rho u_2} \bigg) \bigg) \lambda_\rho \leq \frac{u_1^2 u_2^2 m_\mu^*(x)^2}{2 v_1 v_2}. \end{equation*} $$

Since the right-hand side can serve as an integrable majorant, we can apply the dominated convergence theorem and obtain

$$ \begin{equation*} \lim_{\rho \to 0} \mathbb{E} \exp \bigg({{\rm i}\frac{\skew5\widetilde{J}_\rho(\mu)}{{\rho^2 \sqrt{ \lambda_\rho v_1 v_2}}\,}} \bigg) = \exp \bigg(-\frac{1}{2} \int_{\mathbb{R}^2} f_\mu(x)^2 {\rm d} x\bigg), \end{equation*} $$

which is the characteristic function of a Gaussian random variable. Finally, we can conclude that the limiting random field is a centred Gaussian linear random field with covariance function given in (2.12).