2.1 Regular variation and St $\alpha$ S point processes

We begin with defining Hult–Lindskog (HL) convergence of measures on a Polish space with a zero element. We also recall the definition of regularly varying measures on a Polish space.

Let $(\mathbb{S},d)$ be a Polish space with a continuous scalar multiplication $\bullet\colon (0, \infty) \times \mathbb{S} \to \mathbb{S}$ satisfying $1 \bullet s = s$ for all $s \in \mathbb{S}$ and $b_1 \bullet (b_2 \bullet s) = (b_1 b_2) \bullet s$ for all $b_1, b_2 > 0$ and all $s \in \mathbb{S}$ . We also assume the existence of a zero element $s_0 \in \mathbb{S}$ such that $ b \bullet s_0 = s_0 $ for all $b > 0$ . Equip the punctured space $\mathbb{S}_0 = \mathbb{S} \setminus \{s_0\}$ with the relative topology coming from $\mathbb{S}$ . We denote by $\mathbb{M}(\mathbb{S}_0)$ the space of all Borel measures on $\mathbb{S}_0$ which are finite outside every neighbourhood of $s_0$ , and by ${\char103}_0$ the collection of all bounded continuous functions $f \colon \mathbb{S}_0 \to [0, \infty)$ that vanishes on some neighbourhood of $s_0$ .

Definition 2.1. [Hult–Lindskog convergence] A sequence of measures $\{\nu_n (\!\cdot\!) \colon n \ge 1\} \subset \mathbb{M}(\mathbb{S}_0)$ is said to converge in the Hult–Lindskog sense (see [Reference Hult and Lindskog25]) to a measure $\nu(\!\cdot\!) \in \mathbb{M}(\mathbb{S}_0)$ if

\begin{align*}% \lim_{n \to \infty} \int_{\mathbb{S}_0} f(x) \nu_n(\textrm{d} x) = \int_{\mathbb{S}_0} f(x) \nu(\textrm{d} x) \quad\text{for all }f \in {\char103}_0. \end{align*}

This convergence will be denoted by $\nu_n \stackrel{{{HL}}}{\longrightarrow} \nu$ .

Using this notion of Hult–Lindskog convergence, next we define the regularly varying measures.

Definition 2.2 [Regularly varying measure] A measure $\nu \in \mathbb{M}(\mathbb{S}_0)$ is called regularly varying if there exist $\alpha > 0$ , an increasing sequence of positive scalars $\{c_n\}$ satisfying $c_{k n}/ c_n \to k^{1/\alpha}$ for every $k \in \mathbb{N}$ , and a nonzero measure $\lambda \in \mathbb{M}(\mathbb{S}_0)$ such that

\begin{align*} n \nu (c_n \bullet \cdot) \stackrel{{{HL}}}{\longrightarrow} \lambda(\!\cdot\!) \quad\text{as }n \to \infty. \end{align*}

We shall denote this by $\nu \in \text{RV}(\mathbb{S}_0, \alpha, \lambda)$ .

It is not difficult to check that the limit measure $\lambda$ satisfies the scaling property $\lambda( b \bullet \cdot) = b^{-\alpha} \lambda(\!\cdot\!)$ for all $b>0$ .

Example 2.1. We take $\mathbb{S} = \mathbb{R}^\mathbb{N} = \{ {{\boldsymbol{x}}}=(x_1, x_2, \ldots) \colon x_i \in \mathbb{R}, i \ge 1\}$ and $s_0 ={\textbf{0}}_\infty :\!= (0, 0, \ldots)$ . Let ${{\boldsymbol{X}}} = (X_1, X_2, \ldots )$ be the independent and identically distributed (i.i.d.) process with ${\mathbb{P}}(X_1 \in \cdot ) \in \text{RV}(\bar{\mathbb{R}}_0 = [{-}\infty, \infty] \setminus \{0\}, \alpha, \nu_\alpha(\!\cdot\!) ),$ where

(2.1) \begin{align}% \nu_\alpha (\!\text{d} x) = \beta \alpha x^{-\alpha -1} {\textrm{I}}(x>0) \text{d} x + (1- \beta) \alpha ({-}x)^{-\alpha -1} {\textrm{I}}(x<0) \text{d} x \label{eqn2} \end{align}

with $0 \le \beta \le 1$ being the tail balancing constant. If $\beta=1$ then $\nu_\alpha$ is denoted by $m_\alpha(\!\cdot\!)$ . It was shown in [Reference Lindskog, Resnick and Roy32] that $\mathbb{P} ({{\boldsymbol{X}}} \in \cdot) \in \text{RV}(\mathbb{R}_0^{\mathbb{N}}= \mathbb{R}^\mathbb{N} \setminus \{\textbf{0}_\infty\}, \alpha, \lambda_{iid} (\!\cdot\!)),$ where $\lambda_{iid}(\!\cdot\!)$ is a measure on $\mathbb{R}_0^{\mathbb{N}}$ (concentrated on axes) such that

\begin{equation*}%\label{chap4_eq:defn_lamiid} \lambda_{iid} (\!\text{d} {{\boldsymbol{x}}}) = \sum_{i=1}^\infty \bigotimes_{j=1}^{i-1} \delta_0(\!\text{d} x_j) \otimes\nu_\alpha(\!\text{d} x_i) \bigotimes_{j=i+1}^\infty \delta_0(\!\text{d} x_j). \end{equation*}

Here $\delta_0$ denotes the Dirac measure putting unit mass at 0. More generally, we can obtain other limit measures $\lambda$ (that are not necessarily concentrated on the axes) by introducing dependence among $X_1, X_2, \ldots $ ; see, for example, [Reference Resnick and Roy37].

Example 2.2. Here we consider the space of all Radon point measures $\mathbb{S} = {{\char77}}(\bar{\mathbb{R}}_0)$ on $\bar{\mathbb{R}}_0 = [{-}\infty, \infty] \setminus \{0\}$ endowed with vague topology (the corresponding vague metric is denoted by $d_{vg}$ ). Take $s_0 = \emptyset$ to be the null measure. The scalar multiplication by $b>0$ is denoted by ${{\boldsymbol{S}}}_b$ and defined as follows: if ${\char120} = \sum_i \delta_{u_i} \in {{\char77}}(\bar{\mathbb{R}}_0)$ then

\begin{align*} {{\boldsymbol{S}}}_b {\char120} = \sum_i \delta_{b u_i}. \end{align*}

The HL convergence in ${{\char77}}_0 = {{\char77}}(\bar{\mathbb{R}}_0) \setminus \{ \emptyset \}$ is discussed in and used by [Reference Fasen and Roy21] and [Reference Hult and Samorodnitsky26] in the context of large deviation for point processes, and in [Reference Bhattacharya, Hazra and Roy10] in the context of a branching random walk.

A point process on $\bar{\mathbb{R}}_0$ is an ${{\char77}}(\bar{\mathbb{R}}_0)$ -valued random variable defined on $(\Omega, {\char106}, \mathbb{P}\!)$ that does not charge any mass to $\pm \infty$ . The definition of strict stability of such point processes is introduced in [Reference Davydov, Molchanov and Zuyev18] and has been shown to be connected to regular variation on ${{\char77}}_0$ in [Reference Bhattacharya, Hazra and Roy10].

Proposition 2.1. ([Reference Bhattacharya, Hazra and Roy10, Theorem 2.3].) Let ${\char114}$ be a point process on $\bar{\mathbb{R}}_0$ and ${\char114}_i$ s are independent copies of ${\char114}$ . Suppose that there exists $\alpha >0$ and a nonnull measure $m^*(\!\cdot\!) \in \mathbb{M}({{\char77}}_0)$ such that $\mathbb{P}({\char114} \in \cdot) \in \text{RV}({{\char77}}_0, \alpha, m^*)$ with $\{c_n \colon n \ge 1\}$ as the scaling. Then ${\char114}$ is in the superposition domain of attraction of a St $\alpha$ S point process ${\char121}$ , i.e. ${{\boldsymbol{S}}}_{c_n^{-1}} {\sum_{i=1}^n {\char114}_i} \Rightarrow {\char121}$ . Furthermore, in the above situation, the Laplace functional of the limiting point process is given by

\begin{align*} \Psi_{{{\char121}}}({\kern2.1pt}f) = \mathbb{E} ({\exp} \{ - {{\char121}}({\kern2.1pt}f)\})= \exp \bigg\{ - \int_{{{\char77}}_0} \bigg( 1- \exp \bigg\{ - \int f \,\textrm{d} \nu \bigg\} \bigg) m^*(\textrm{d} \nu) \bigg\} \end{align*}

for all nonnegative real-valued measurable functions f.

For the proof of this proposition, we refer the reader to [Reference Bhattacharya, Hazra and Roy10]. We shall exploit this proposition for computing the weak limit for a sequence of point processes.

It is established in Example 8.6 of [Reference Davydov, Molchanov and Zuyev18] that the St $\alpha$ S point process admits a special kind of representation. A slightly general notation is introduced in [Reference Bhattacharya, Hazra and Roy9] in parallel to the notation introduced in [Reference Subag and Zeitouni38].

Definition 2.4. (Randomly scaled scale-decorated Poisson point process.) A point process $\daleth$ is called a randomly scaled scale-decorated Poisson point process (SScDPPP) with intensity measure $m(\!\cdot\!)$ , scale-decoration ${\char120}$ and a positive random scale U if

\begin{align*} \daleth \stackrel{\text{D}}{=} {{\boldsymbol{S}}}_U \sum_{i=1}^\infty {{\boldsymbol{S}}}_{\lambda_i} {\char120}_i, \end{align*}

where U is a positive random variable, $\Lambda = {\sum_{i=1}^\infty \delta_{\lambda_i}}$ is a Poisson random measure on $(0, \infty)$ with intensity measure m and independent of U, and $\{{\char120}_i \colon i \ge 1\}$ is a collection of independent copies of the point process ${\char120}$ , which is also independent of U and $\Lambda$ .

Bhattacharya et al. [Reference Bhattacharya, Hazra and Roy9], [Reference Bhattacharya, Hazra and Roy10] established that the limiting point process associated to the branching random walk with marginally regularly varying displacements admits a SScDPPP representation.

3.1 Model

We consider a multitype branching process with Q types of particles. The root is of type p with probability $\pi(p)$ for $1 \le p \le Q,$ where $\pmb{\pi}^t =(\pi(1), \ldots, \pi(Q))$ is a probability vector. Each particle of type q produces $k_p$ children of type p (for all $1 \le p \le Q$ ) with probability $p^{(q)}(k_1, \ldots, k_Q)$ .

Assumptions 3.1. (Assumptions on the branching mechanism) The assumptions on the branching process are stated as follows.

(B1) We assume that the multitype Galton–Watson tree does not have any leaves, i.e. ${p^{(q)}} (k_1, \ldots, k_Q) = 0$ whenever $k_j=0$ for some $1\le j\le Q$ .

(B2) We assume that $\mu_{p,q} = {{\mathbb{E}}(Z_1^{(p)}(q))} < \infty$ for all $p,q \in \{1, \ldots, Q\}$ .

(B3) (Kesten–Stigum conditions.) For all $p,q \in \{1, \ldots, Q\}$ , we assume that

$$ {\mathbb{E}}(Z_1^{(p)}(q) \log Z_1^{(p)}(q) ) < \infty. $$

Remark 3.2. Let $M= ((\mu_{p,q}))$ be the mean matrix. Since the tree does not have any leaves, $\mu_{p,q}\ge 1$ and, hence, M has all entries positive. Let $\rho > 0$ denote the maximal (in terms of modulus) eigenvalue of M (such that $\rho$ exists and is unique by the Perron–Frobenius theory [27, Theorem 2.1, p.543]) is called the Perron–Frobenius eigenvalue of M. Let $\varsigma$ and $\vartheta$ be the left and right eigenvectors corresponding to $\rho$ satisfying $\varsigma^t\textbf{1} =1$ and $\varsigma^t \vartheta =1,$ where $\textbf{1} = (1, \ldots, 1)^t$ (existence and uniqueness of such vectors are again guaranteed by Perron–Frobenius theory). Then we have

\begin{align*} M^n= \rho^n P + R^n \text{ such that } \limsup_{n \to \infty} \max_{i,j} |r_{ij}^{(n)}|^{1/n}< \rho_0 \quad\text{for $1 < \rho_0 < \rho$, } \end{align*}

where $P = \vartheta. \varsigma^t$ i.e. $P_{ij} = \vartheta_i \varsigma_j$ and ${r_{ij}^{(n)}}$ denotes the (i,j)th elements of $R^n$ ; see [27, Theorem 2.3, p.

546]. Since each $\mu_{pq} \ge 1$ , it follows that $\rho \in (1, \infty)$ .

Remark 3.3. Using (B3), from [Reference Kesten and Stigum28] we obtain

(3.1) \begin{align}% \lim_{n \to \infty} \frac{1}{\rho^n} {{\boldsymbol{Z}}}_n = W \varsigma \label{eqn3} \end{align}

almost surely for some positive random variable W.

We are now ready to describe the model formally in terms of the tree given in the assumptions above and the displacements of each of its vertices. We recall the notations for the position of the particles born in the branching process from Section 1. Let $\mathbb{T} = (\mathbb{V},\mathbb{E})$ denote the genealogical tree obtained from the supercritical multitype Galton–Watson process, where $\mathbb{V}$ and $\mathbb{E}$ denote the collection of all vertices and edges (of $\mathbb{T}$ ) respectively. We use the Harris–Ulam labels for the vertices of $\mathbb{T}$ as described in [Reference Mode35]. The root is denoted by o. We also identify each edge with its vertex away from the root and assign its type to the edge. The displacements of the particles are described next. We assume that each particle of type p produces an independent copy of ${({{\char76}}^{(p)}(1), {{\char76}}^{(p)}(2), \ldots, {{\char76}}^{(p)}(Q))}$ , where

(3.2) \begin{align} {{\char76}}^{(p)}(q) = \sum_{i=1}^{Z_1^{(p)}(q)} \delta_{X_i^{(q)}}, \label{eqn4} \end{align}

with ${{\boldsymbol{X}}^{(q)} = (X^{(q)}_1, X^{(q)}_2, \ldots\!)}$ being a random element of $\mathbb{R}^\mathbb{N}$ which is independent of the random vector ${{{\boldsymbol{Z}}}_1^{(p)}}$ . This means the following: if a particle has $k_q (\!\in \mathbb{N})$ children of type q in the next generation (for all $1 \le q \le Q$ ), then the displacements associated to the children are independent copies of ${( X_1^{(1)}, \ldots, X_{k_1}^{(1)}, \ldots, X^{(Q)}_{1}, \ldots, X^{(Q)}_{k_Q} )}$ .

Now we define the position of a particle as the sum of its own displacement and displacements of its parents up to root o. For mathematical description, let $I_{{{\boldsymbol{v}}}}$ denote the path from root o to the vertex ${\boldsymbol{v}} \in \mathbb{V}$ and let ${{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}$ denote the displacement attached to the ${\boldsymbol{\varpi}}$ th particle (see the convention mentioned above). Then the position of the ${\boldsymbol{v}}$ th particle is denoted by $S_{{{\boldsymbol{v}}}}$ and defined in (1.1).

Assumptions 3.2. (Assumptions on displacements.) For each $p \in \{1, \ldots, Q\}$ , $ {{{\boldsymbol{X}}}^{(p)}} = {(X_1^{(p)}, X_2^{(p)}, \ldots) }$ is an $\mathbb{R}^\mathbb{N}$ -valued random variable such that the following assumptions are satisfied.

(D1) For every fixed p, ${\{X_i^{(p)} \colon i \ge 1\}}$ is a collection of marginally identically distributed random variables.

(D2) There exists an increasing sequence of positive constants $\{b_n \colon n \ge 1\}$ such that

(a) (condition on marginal distribution)

(3.3) \begin{align}% \rho^n \mathbb{P}(b_n^{-1} X_1^{(Q)} \in \cdot) \stackrel{v}{\longrightarrow} \nu_\alpha(\!\cdot\!), \label{eqn5} \end{align}

where $\nu_\alpha(\!\cdot\!)$ is a measure on $\bar{\mathbb{R}}_0$ as defined in (2.1).

(b) (condition on joint distribution of the displacements associated to the particles of type Q)

(3.4) \begin{align}% \rho^n \mathbb{P}( b_n^{-1} {\boldsymbol{X}}^{(Q)} \in \cdot ) \stackrel{{{HL}}}{\longrightarrow} \lambda(\!\cdot\!), \label{eqn6} \end{align}

where $\lambda$ is a measure on $\mathbb{R}_0^{\mathbb{N}} = \mathbb{R}^\mathbb{N} \setminus \{0_\infty\}$ , such that $\lambda(a \cdot) = a^{-\alpha} \lambda(\!\cdot\!)$ for every $a > 0$ , and $\lambda(A) < \infty$ if $0_\infty\notin \bar{A}$ .

(D3) For all $1 \le p \le Q-1$ , there exists $\gamma >0$ such that

\begin{equation*}%\label{chap4_ass:disp_otherthanQ} \mathbb{P}(|X^{(p)}_1|>x) \le \varepsilon(x) \mathbb{P}(|X_1^{(Q)}| > x) \end{equation*}

such that $\varepsilon \colon \mathbb{R}_+ \to \mathbb{R}_+ $ is a map satisfying

(3.5) \begin{align}% \lim_{t \to \infty}\dfrac{\varepsilon(tx)}{\varepsilon(t)} = x^{-\gamma} \label{eqn7} \end{align}

for all $x>0$ . This assumption precisely states that displacements corresponding to the Qth type of particles dominate the tail of displacements corresponding to other particles.

Let $|{\boldsymbol{v}}|$ denote the generation of the vertex ${\boldsymbol{v}} \in \mathbb{V}$ in the genealogical tree $\mathbb{T}$ . In this paper we obtain the weak limit of the point process

(3.6) \begin{align}% N_n = \sum_{|{\boldsymbol{v}}| =n} \delta_{b_n^{{-1}} S_{{{\boldsymbol{v}}}}}. \label{eqn8} \end{align}

5.1 Heaviest tail wins

We start with the following consequence of Karamata’s theorem.

Lemma 5.1. Fix $\gamma' \in (0, \gamma)$ and $\beta \in (0, \alpha^{-1})$ . For $\varepsilon$ defined in (3.5), there exists $N \in \mathbb{N}$ such that $n \ge N$ and $b_n$ as chosen in (3.4) and (3.3),

(5.1) \begin{align}% \varepsilon (b_n) \le {\text{const.} }\rho^{-n(\gamma - \gamma')(1/\alpha - \beta)}, \label{eqn12} \end{align}

and some $0 < \gamma' < \gamma$ and $0 < \beta < 1/ \alpha$ .

Proof. For any $t\geq 0$ , let b(t) be a step function such that $b(n)=b_n$ . Let $c(t) = b(\log_\rho t)$ and ${\bar{F}_Q(x) = \mathbb{P}(|X_1^{(Q)}| > x)}$ . From (3.3) it follows that

\begin{align*} \rho^n \bar{F}_Q(b_n x) \to \beta x^{-\alpha} \quad\text{for all $x > 0$. } \end{align*}

Now using Proposition 0.1 of [Reference Resnick36], we obtain, for all $x>0$ ,

\begin{align*} \dfrac{1}{b_n} U (\rho^n x) \to (\beta x)^{1/\alpha}, \end{align*}

where $U = ( {1}/{\bar{F}_Q} )^{\leftarrow}$ . Observe that

\begin{align*} \lim_{t \to \infty} \dfrac{b(t+x)}{b(t)} = \lim_{t \to \infty} \dfrac{b(t+x)}{U(\rho^{t+x})} \dfrac{U(\rho^t \rho^x)}{b(t)} = \rho^{x/ \alpha} \end{align*}

and

\begin{align*} \lim_{t \to \infty} \dfrac{c(tx)}{c(t)} = \lim_{t \to \infty} \dfrac{b({\log}_\rho t + \log_\rho x)}{b(\log_\rho t)} = \rho^{\alpha^{-1} \log_\rho x} = x ^{1/\alpha}. \end{align*}

Hence, $c(\!\cdot\!) \in \text{RV}_{1/\alpha}$ .

Using Karamata’s representation (see Theorem 0.6 of [Reference Resnick36]), we can write

\begin{align*} c(t) = t^{1/\alpha} \phi_1(t) \exp \bigg\{ \int_{1}^t \dfrac{1}{x} \eta_1(x ) \,\textrm{d} x \bigg\}, \end{align*}

where $ \phi_1 \colon \mathbb{R}_+ \to \mathbb{R}_+$ and $\eta_1 \colon \mathbb{R} _+ \to \mathbb{R}_+$ such that

\begin{align*} \lim_{t \to \infty} \phi_1(t ) = \phi_1 \in (0, \infty) \quad\text{and}\quad \lim_{t \to \infty} \eta_1(t) =0. \end{align*}

We have also assumed that $\varepsilon( \cdot) \in \text{RV}_\gamma$ for some $\gamma > 0$ . Thus,

\begin{align*} \varepsilon(t) = t^{-\gamma} \phi_2(t) \exp \bigg\{ \int_1^t \dfrac{1}{x} \eta_2(x) \,\textrm{d} x \bigg\}, \end{align*}

where $\phi_2 \colon \mathbb{R}_+ \to \mathbb{R}_+$ and $\eta_2 \colon \mathbb{R}_+ \to \mathbb{R}_+$ such that

\begin{align*} \lim_{t \to \infty} \phi_2(t) = \phi_2 \in (0, \infty) \quad\text{and}\quad \lim_{t \to \infty} \eta_2(t) =0. \end{align*}

We want to obtain an upper bound of

\begin{align*} \varepsilon(b(n)) &= \varepsilon( b(\log_\rho \rho^n)) \nonumber \\ & = \varepsilon(c(\rho^n)) \nonumber \\ &= (c(\rho^n))^{-\gamma} \phi_2(c(\rho^n)) \exp \bigg\{ \int_1^{(c(\rho^n))} \frac{1}{x} \eta_2(x) \,\textrm{d} x \bigg\}. \end{align*}

Now fix $\delta_1 > 0$ and $ \gamma' \in (0, \gamma)$ . Then there exits $N_1 \in \mathbb{N}$ such that, for all $n \ge N_1$ ,

\begin{align*} \phi_2(c \rho^n) \exp \bigg\{ \int_1^{(c(\rho^n))} \frac{1}{x} \eta_2(x) \,\textrm{d} x \bigg\} < \text{ const. }(\phi_2 + \delta_1) (c(\rho^n) )^{\gamma'}. \end{align*}

Now we derive the following inequality, for all $n \ge N_1$ :

(5.2) \begin{align}% \varepsilon(b(n)) < \text{ const. } (c(\rho^n) )^{-(\gamma - \gamma')} = \text{ const. } \bigg( \rho^{n/\alpha} \phi_1(\rho^n) \exp \bigg\{ \int_1^{\rho^n} \frac{1}{x} \eta_1(x) \,\textrm{d} x \bigg\} \bigg)^{-(\gamma -\gamma')}. \label{eqn13} \end{align}

Fixing $\delta_2 \in (0, \phi_1), \beta \in (0, \alpha^{-1}),$ and using similar arguments we can prove that there exists $N_2 \in \mathbb{N}$ such that, for all $n \ge N_2$ ,

\begin{align*} \phi_1(\rho^n) \exp \bigg\{ \int_1^{\rho^n} \frac{1}{x} \eta_1(x) \,\textrm{d} x \bigg\} > (\phi_1- \delta_2) \rho^{-n\beta}. \end{align*}

Combining this inequality with (5.2) proves (5.1) with $N= \max(N_1, N_2)$ .

Proof of Lemma \refchap4_lemma:one_large_jump. Consider a Lipschitz function $f \in C_c^+ (\bar{\mathbb{R}}_0)$ with support $(f) = \{x \colon |x| > \zeta \}$ , where $C_c^+ (\bar{\mathbb{R}}_0)$ denotes the class of all nonnegative, bounded, and continuous functions. To prove (4.1), it is enough to show that

(5.3) \begin{align}% \limsup_{n \to \infty} \mathbb{P} ( |N_n(f) - \tilde{N}_n(f)| > \epsilon ) = 0 \label{eqn14} \end{align}

for every $\epsilon > 0$ .

In order to prove this, at the beginning we will formalize the fact that on a fixed path the contribution of the pth type of particles is asymptotically negligible for $p=1, \ldots, Q-1$ . Fix $0 < \theta < \zeta/2$ . For $p= 1, \ldots, Q-1,$ we define

\begin{align*} A_n^{(p)}(\theta) = \bigg[ \bigcup_{|{\boldsymbol{v}}|=n} \bigg( \sum_{{\boldsymbol{\varpi}} \in I_{{{\boldsymbol{v}}}}^{(p)}} \delta_{b_n^{-1} |{{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}|} \bigg(\frac{\theta}{n}, \infty \bigg) \ge 1 \bigg) \bigg]^c. \end{align*}

We will show that

(5.4) \begin{align}% \lim_{n \to \infty} \mathbb{P} ((A_n^{(p)}(\theta) )^c ) =0 \label{eqn15} \end{align}

for every $p=1, \ldots , Q-1$ . Fix p. To prove (5.4) it suffices to show that

\begin{align*} \lim_{n \to \infty} \mathbb{P}_{\mathbb{T}}((A_n^{(p)}(\theta) )^c ) =0 \quad\text{almost surely}, \end{align*}

where $\mathbb{P}_{\mathbb{T}}(\!\cdot\!)$ denotes the probability conditioned on the underlying Galton–Watson tree. Observe that

(5.5) \begin{align}% \mathbb{P}_{\mathbb{T}} [ ( A_n^{(p)}(\theta) )^c ] \le \sum_{|{\boldsymbol{v}}|=n} \mathbb{P}_{\mathbb{T}} \bigg( \sum_{{\boldsymbol{\varpi}} \in I_{{{\boldsymbol{v}}}}^{(p)}} \delta_{n b_n^{-1} |{{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}|} (\theta, \infty) \ge 1 \bigg) . \label{eqn16} \end{align}

Note that on a path $I_{{{\boldsymbol{v}}}}$ , the displacements corresponding to the pth type of particles are independent and they are also identically distributed. Thus, conditioned on $\mathbb{T}$ ,

\begin{align*} \sum_{{\boldsymbol{\varpi}} \in I_{{{\boldsymbol{v}}}}^{(p)}} \delta_{n b_n^{-1} |{{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}|} (\theta, \infty) \sim \text{ Binomial} (|I_{{{\boldsymbol{v}}}}^{(p)}|, \mathbb{P} (n |X^{(p)}_1| > b_n \theta ) ). \end{align*}

Using ${|I_{{{\boldsymbol{v}}}}^{(p)}| \le n}$ , Assumption 2, and taking a sequence $\{{X^{(p,i)}} \colon i \ge 1\}$ of i.i.d. copies of ${X_1^{(p)}}$ , we obtain the following upper bound to the right-hand side of (5.5):

(5.6) \begin{align} \sum_{|{\boldsymbol{v}}| = n} \mathbb{P} \bigg( \sum_{i=1}^n \delta_{n b_n ^{-1} |X^{(p,i)}|} (\theta, \infty) \ge 1 \bigg) & = |D_n | \mathbb{P} \bigg( \sum_{i=1}^n \delta_{n b_n ^{-1} |X^{(p,i)}|} (\theta, \infty) \ge 1 \bigg) \nonumber \\[4pt] & \le (\text{const.}) |D_n| n \mathbb{P}(n |X_1^{(p)}| > b_n \theta) \nonumber \\[4pt] & \le (\text{const.}) |D_n| n \varepsilon(n^{-1} b_n \theta) \mathbb{P}( n |X_1^{(Q)}| > b_n \theta). \label{eqn17}\end{align}

Here ${\{X^{(p,i)} \colon i \ge 1\}}$ is the collection of independent copies of ${X_1^{(p)}}$ .

Fix $\eta_1 > 0$ and $\eta_2 > 0$ . It follows from (3.3) that $b_n = \rho^{n/\alpha} L'(\mu^{n/\alpha}),$ where $L'(\!\cdot\!)$ is a slowly varying function. As $\{n^{-1} b_n\}$ is an increasing sequence, we can use Potter’s bound combined with the upper bound in Lemma 3 to obtain

(5.7) \begin{align}% \varepsilon(n^{-1} b_n \theta) \le (\text{const.}) n^{\gamma + \eta_1} \rho^{-n(\gamma- \gamma')(\alpha^{-1} - \beta)} \frac{\varepsilon(b_n \theta)}{\varepsilon(b_n)} \label{eqn18} \end{align}

for large enough n. Again using Potter’s bound (see Proposition 0.8(ii) of [Reference Resnick36]), we obtain

(5.8) \begin{align}% \mathbb{P} (n |X_1^{(Q)} | > b_n \theta ) \le (1- \eta_2)^{-1} n^{\alpha + \eta_2} \mathbb{P}(|X_1^{(Q)}| > b_n \theta) \label{eqn19} \end{align}

for sufficiently large n. Finally, combining (5.7) and (5.8), we obtain, for (5.6), the upper bound

(5.9) \begin{align}% \text{ const.} \times \frac{|D_n|}{\rho^n} \times \frac{\varepsilon(b_n \theta)}{\varepsilon(b_n)} \times \rho^n \mathbb{P} ( |X_1^{(Q)}| > b_n \theta ) \times n^{1 + \gamma + \alpha + \eta_1 + \eta_2} \rho^{-n(\gamma- \gamma')(\alpha^{-1} - \beta)} \label{eqn20} \end{align}

for $ \gamma' \in (0, \gamma)$ and $\beta \in (0, \alpha^{-1})$ . Note that by (3.1),

\begin{align*} \lim_{n \to \infty} \frac{|D_n|}{\rho^n} = \lim_{n \to \infty} \dfrac{\sum_{p=1}^Q |D_n^{(p)}|}{\rho^n} = \lim_{n \to \infty} \frac{|\sum_{p=1}^Q Z_n^{(p)}|}{\rho^n} = W \end{align*}

almost surely. The other factors in (5.9) converge to finite positive limits except the last factor which converges to 0. This completes the proof of (5.4).

To formalize the fact that on a path at most one displacement associated with the Qth type of particles can be large, we define

\begin{align*} A_n^{(Q)}(\theta)= \bigg[ \bigcup_{|{\boldsymbol{v}}|=n} \bigg( \sum_{{\boldsymbol{\varpi}} \in I_{{{\boldsymbol{v}}}}^{(Q)}} \delta_{b_n^{-1} |{{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}|} \bigg(\frac{\theta}{n}, \infty \bigg) \ge 2 \bigg) \bigg]^c. \end{align*}

We want to show that

(5.10) \begin{align}% \lim_{n \to \infty} \mathbb{P} [ ( A_n^{(Q)}(\theta) )^c ]=0. \label{eqn21} \end{align}

Following the arguments used for ${\mathbb{P} ( A_n^{(p)}(\theta) )}$ , we obtain

\begin{align*} \mathbb{P}_{\mathbb{T}}(A_n^{{(Q)}}(\theta)) & \le \sum_{|{\boldsymbol{v}}| = n} \mathbb{P} \bigg( \sum_{i=1}^n \delta_{n b_n^{-1} |X_1^{(Q)}|} (\theta, \infty) \ge 2 \bigg) \nonumber \\ & \sim {\textrm{const.}} ( \rho^{-2n} n^{2} |D_n|) [ \rho^n \mathbb{P} ( |X_1^{(Q)}| > n^{-1} b_n \theta ) ]^2 \nonumber \\ & \le {\textrm{const.}} ( \rho^{-2n} n^{2 + 2 \alpha + 2 \eta_2} |D_n| ) [ \rho^n \mathbb{P} ( |X_1^{(Q)}| > n^{-1} b_n \theta ) ]^2 \end{align*}

almost surely for large enough n. We have used Potter’s bound for ${\mathbb{P} ( |X_1^{(Q)}| > n^{-1} b_n \theta )}$ derived in (5.8) to obtain the last upper bound. It is clear that the second term inside braces converges to a finite and positive constant as $n \to \infty$ by choice of the sequence $(b_n \colon n \ge 1)$ (see (D2)). The first term converges to 0 almost surely as $\rho^{-n} |D_n|$ converges to a finite positive random variable almost surely. Hence, the assertion stated in (5.10) holds.

Now we are ready to prove (5.3) using (5.4) and (5.10). We can bound the probability in (5.3) by

(5.11) \begin{align}% \mathbb{P} \bigg(\sum_{|{\boldsymbol{v}}=n|} \bigg|\, f(b_n ^{-1} S_{{{\boldsymbol{v}}}}) - \sum_{{\boldsymbol{\varpi}} \in I_{{{\boldsymbol{v}}}}^{(Q)}} f(b_n^{-1} {{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}\!) \bigg| > \epsilon, \bigcap_{p=1}^Q A_n^{(p)}(\theta) \bigg) + o(1). \label{eqn22} \end{align}

Let ${T_{\boldsymbol{v}}^{(p)}}$ denote the maxima in modulus among the displacements on the set ${I_{{{\boldsymbol{v}}}}^{(p)}}$ ( $p=1, \ldots, Q$ ), where maxima in modulus on an empty set is assumed to be 0. We now discuss the possible scenarios appearing on the event considered in (5.11). One possibility is that ${\max_{1 \le p \le Q} \max_{|{\boldsymbol{v}}| = n} n T_{\boldsymbol{v}}^{(p)}} < b_n \theta$ and it is clear that in this case its probability is 0. Another possibility is that ${\max_{1 \le p \le Q-1} \max_{|{\boldsymbol{v}}|=n} n T_{\boldsymbol{v}}^{(p)}} &lt b_n \theta $ and ${\max_{|{\boldsymbol{v}}| =n} n T_{\boldsymbol{v}}^{(Q)} }> b_n \theta$ . Then the probability in (5.11) equals

\begin{align*} \mathbb{P} \bigg( \sum_{|{\boldsymbol{v}}|=n} \bigg| f(b_n^{-1} S_{{{\boldsymbol{v}}}}) - f(b_n^{-1} T_{\boldsymbol{v}}^{(Q)}) \bigg| > \epsilon, \bigcap_{p=1}^Q A_n^{(p)}(\theta) \bigg). \end{align*}

Starting from this step, by molding the proof discussed in Subsection 4.4 of [Reference Bhattacharya, Hazra and Roy9], it is easy to conclude the assertion of the lemma.

5.2 Computation of the weak limit

To compute the weak limit of the sequence of point processes ${\tilde{N}_n^{(K,B)}}$ , we follow the method of ‘regularization’ as used in [Reference Bhattacharya, Hazra and Roy10]; see Figure 3. Recall that we have decided to identify each edge with its vertex away from the root and assign its type to the edge. Also recall that ${A_{\boldsymbol{\varpi}}^{(B)}}$ denotes the number of descendants of the vertex $\boldsymbol{\varpi}$ in the Kth generation of the pruned subtree containing $\boldsymbol{\varpi}$ . Keeping this abuse of notation in mind, we modify the pruned subtrees $\mathbb{T}_{pj}(B)$ for $j= 1, \ldots, {Z_{n-K}^{(p)}}$ and $p=1, \ldots, Q$ according to the following algorithm.

Figure 3.

Regularizing pruned the Galton–Watson tree ( $B=2, K=1$ ) at generation 2. The newly addded vertices in prunning are denoted by - -.

(R1) Fix $p=1$ and $j=1$ , i.e. look at $\mathbb{T}_{11}(B)$ .

(R2) Look at the root of $\mathbb{T}_{11}(B)$ . If it has exactly B descendants of[-50pt]please specifically cite Figure 3 somewhere in the text. type 1 in the next generation then keep it as it is. If it has l ( $&lt B$ ) descendants of type 1, then add $(B-l) $ descendants (of type 1) and define ${A_{\boldsymbol{\varpi}}^{(B)}}=0$ , where ${\boldsymbol{\varpi}}$ is a newly added edge. Next do the same for descendants of other types. Now the root has QB descendants in total in the next generation. Now attach an independent copy of ${(X^{(1)}_1, \ldots, X^{(1)}_B, \ldots, X^{(Q)}_1, \ldots, X^{(Q)}_B)}$ to the descendants.

(R3) Repeat step (R2) for the particles at the first generation of $\mathbb{T}_{11}(B)$ . Continue till the Kth generation.

(R4) Repeat steps (R2) and (R3) for the other trees in $\mathbb{F}_1$ .

(R5) Repeat steps (R2) (R3), and (R4) for the forests $\mathbb{F}_p$ for $2 \le p \le Q$ .

The regularized subtrees will be denoted by $\widetilde{\mathbb{T}}_{pj}(B)$ . We shall first work on $\widetilde{\mathbb{T}}_{11}(B)$ to show that the associated point process is regularly varying. To each edge ${\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}_{11}(B)$ , we associate the triplet (m,q,k), where m denotes the generation of the edge ${\boldsymbol{\varpi}}$ , q denotes the type of the edge ${\boldsymbol{\varpi}}$ and k denotes the enumeration of the vertex ${\boldsymbol{\varpi}}$ among all the vertices of type q at the mth generation of $\widetilde{\mathbb{T}}_{11}(B)$ . Using this notation, we can write down the vector of displacements associated to the particles of type Q in the tree $\widetilde{\mathbb{T}}_{11}(B)$ as follows:

\begin{align*} \tilde{X}_1 \!= ( X'_{(1,Q,1)}, \ldots, X'_{(1,Q,B)}, X'_{(2,Q,1)}, \ldots, X'_{(2, Q, QB^2)}, \ldots, X'_{(K,Q,1)}, \ldots, X'_{(K, Q, Q^{K-1}B^K)} ). \end{align*}

It takes values in

\begin{align*} \tilde{\mathbb{R}}^B = \prod_{t=1}^{B+ Q\ B^2 + \cdots + Q^{K-1}B^K} \mathbb{R}. \end{align*}

Similarly, we can write

\begin{align*} \tilde{A}_1 = (A^{(B)}_{(1,Q,1)}, \ldots, A^{(B)}_{(1,Q,B)}, A_{(2,B,1)}^{(B)}, \ldots, A_{(2,B,QB^2)}^{(B)}, \ldots, A^{(B)}_{(K,Q,1)}, \ldots, A^{(B)}_{(K,Q,Q^{K-1}B^K)}) \end{align*}

taking values in

\begin{align*} \tilde{S}^B = \prod_{t=1}^{B + QB^2 + \ldots + Q^{K-1}B^K} \{0, \ldots, B^K \} . \end{align*}

Let ${{\char71}}(\!\cdot\!)$ denote the law of $\tilde{\!A}_1$ . Recall from (3.4) that

\begin{align*} \rho^n \mathbb{P} ( b_n^{-1} (X^{(Q)}_1, \ldots, X^{(Q)}_B) \in \cdot ) \stackrel{{{HL}}}{\longrightarrow} \lambda^{(B)} (\!\cdot\!) \end{align*}

on the space $\mathbb{R}^B \setminus \{0_B\},$ where $0_B = (0,\ldots, 0) \in \mathbb{R}^B$ and $\lambda^{(B)} = \lambda \circ \text{PROJ}_B^{-1}$ with $\text{PROJ}_B\colon \mathbb{R}^\mathbb{N} \to \mathbb{R}^B$ defined by $\text{PROJ}_B ((u_i)_{i=1}^\infty) = (u_1, \ldots, u_B)$ . Using this combined with the fact that the branching mechanism and the displacements are independent, we obtain

(5.12) \begin{align} \rho^n \mathbb{P}(b_n^{-1} \tilde{X}_1 \in \cdot, \tilde{A}_1 \in \cdot ) \stackrel{{{HL}}}{\longrightarrow} \tau \otimes{{\char71}} \label{eqn23} \end{align}

on $\tilde{\mathbb{R}}^B \setminus \{\textbf{0}_B\},$ where $\text{0}_B \in \tilde{\mathbb{R}}^B$ with all its elements as 0 and

\begin{align*} \tau &= \sum_{m=1}^K \sum_{l \in J_m} \tau_{ml} \end{align*}

with

\begin{align} \tau_{ml} = \bigotimes_{t=1}^{B+QB^2 + \cdots + Q^{m-2}B^{m-1} +l -1} \delta_0 \otimes \lambda^{(B)} \bigotimes_{t=B+QB^2 + \ldots + Q^{m-2}B^{m-1} +l +B}^{B+QB^2 + \cdots + Q^{K-1}B^K} \delta_0, \label{eqn24} \end{align}

where $J_m = \{r \in \{1,2, \ldots, Q^{m-1}B^m\} \colon r \equiv 1 B(\text{mod}\})$ .

It is clear that after regularization, the point process associated to the tree ${\widetilde{\mathbb{T}}_{11}(B)}$ is modified. But an important point to note is that this modification does not affect the dependence structure prevailing among the displacements associated to a tree and also among the trees. In particular, it does not affect the joint distribution of the point processes associated to the trees and also to a specific tree. This can be done for each of the subtrees ${\widetilde{\mathbb{T}}_{pj}(B)}$ . The point process of displacements (without the scaling) associated to the tree ${\widetilde{\mathbb{T}}_{p1}(B)}$ will be denoted by ${N^{'(K,B,p,1)}}$ ,

\begin{align*} N^{'(K,B,p,1)} = \sum_{{\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}_{p1}^{(Q)}(B)} A_{\boldsymbol{\varpi}}^{(B)} \delta_{{{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}'} \quad\text{for every $p=1, \ldots, Q$.} \end{align*}

Proof of Lemma 4.3. Note that it suffices to prove that ${N^{'(K,B,p,1)}}$ is regularly varying in the space of all Radon point measures ${{\char77}}_0$ . In order to establish this fact, we have to show that

\begin{align*} m_n^{(p)} = \rho^n \mathbb{P} ( {{\boldsymbol{S}}}_{b_n^{-1}} N^{'(K,B,p,1)} \in \cdot ) \stackrel{{{HL}}}{\longrightarrow} m^*_p(\!\cdot\!) \end{align*}

for $m^*_p \in \mathbb{M}({{\char77}}_0),$ described in (5.16). Following [26, Theorem A.2], it is clear that then we have to establish that, for every ${f_1, f_2 \in C_c^+ (\bar{\mathbb{R}}_0)}$ and $\epsilon_1, \epsilon_2 > 0$ ,

(5.13) \begin{align}% &\int \prod_{i=1}^2 ( 1- \textrm{e}^{ - [\nu(f_i) - \epsilon_i ]_+ } ) m_n^{(p)}(\!\text{d} \nu) \to \int \prod_{i=1}^2 ( 1- \textrm{e}^{ - [\nu(f_i) - \epsilon_i ]_+ } ) m^*_p(\textrm{d} \nu) \label{eqn25} \end{align}

as $n\to\infty$ , where $\nu(f) = \int f(x) \nu(\!\text{d} x)$ . Now we shall consider the $p=1$ case and the other cases can be dealt with similarly. Following the steps discussed in [10, p.~197(last paragraph)] and using (5.12), we obtain

(5.14) \begin{align} &\int \prod_{i=1}^2 ( 1- \textrm{e}^{ - [\nu(f_i) - \epsilon_i ]_+ } ) m_n^{(1)}(\!\text{d} \nu) \nonumber \\ &\quad\to \!\sum_{m=1}^K \int\! \sum_{\tilde{a} \in \tilde{S}^B} \sum_{l \in J'_m } {{\char71}}(\tilde{a}) \prod_{i=1}^2 \bigg[ 1- \exp\bigg\{- \bigg( \sum_{j=l}^{l+B-1} a_{(m,Q,j)} f_i(x_{j-l+1}) -\epsilon_i \bigg)_+ \bigg\} \bigg] \lambda^{(B)}(\textrm{d} {{\boldsymbol{x}}}) \label{eqn26}\end{align}

with $J'_m = \{l \in J_m \colon (a_{(m,Q,l)}, \ldots, a_{(m,Q, l+B-1)}) \neq (0,\ldots, 0)\}$ for every $m =1, \ldots, K$ . In the above derivation, we have used the HL convergence on $\mathbb{R}$ to derive HL convergence in the space of all Radon measures. Hult–Lindskog convergence on these two spaces are connected by construction of an appropriate function $h \colon \tilde{\mathbb{R}} \to \mathbb{R}_+$ defined as

\begin{align*} h(\widetilde{x}, \widetilde{a}) = \prod_{i=1}^2 \bigg( 1 - \exp \bigg\{ - \bigg( \sum_{m=1}^K \sum_{l=1}^{Q^{m-1} B^m} a_{(m,Q,l)}\, f_i(x_{(m,Q,l)}) - \epsilon_i \bigg)_+ \bigg\} \bigg), \end{align*}

which is similar to the function specified in (4.20) of [Reference Bhattacharya, Hazra and Roy10]. To describe the above limit, we will introduce new variables as follows.

• • Let ${\{U_{1i}^{(p)}(Q,B)} \colon p=1,2, \ldots, Q \text{ and } i \ge 1\}$ be a collection of independent random variables such that, for every $p \in \{1,2, \ldots,Q\}$ , ${U_{1i}^{(p)}(Q,B) \stackrel{\text{D}}{=} Z_1^{(p)}(Q,B)}$ for all $i \ge 1$ .

• • We denote by ${(Z^{(p)}_{i} (1,B), \ldots, Z^{(p)}_{i}(Q,B))}$ the random vector of the number of particles at the ith generation for different types when the root is of type p in the pruned tree. Let $\{{\Delta_{K-m,i}^{(Q)} (B)} \colon m=1,2, \ldots, K, i \ge 1\}$ be a collection of independent random variables such that, for every fixed $m \in \{1,2, \ldots, K\}$ ,

  \begin{align*} \Delta^{(Q)}_{K-m, i} (B) \stackrel{\text{D}}{=} \sum_{q=1}^Q Z_{K-m}^{(Q)} (q,B)\quad\text{for all $i \ge 1$. } \end{align*}

Furthermore, assume that the collection $\{{\Delta_{K-m,i}^{(Q)} (B)} \colon m=1, \ldots, K, i \ge 1\}$ is independent of the collection $\{{U_{1i}^{(p)}(Q,B) }\colon p=1, \ldots, Q \text{ and } i \ge 1\}$ . We also define a collection of independent random variables $({\Delta_{K-m, i,l}^{(Q)}(B)} \colon i \ge 1, l \ge 1)$ such that ${\Delta_{K-m, i, l}^{(Q)}(B) \stackrel{\text{D}}{=} \Delta_{K-m,i}^{(Q)}(B)}$ for all $l \ge 1$ .

Fix generation m and $\mathbf{x} \in \mathbb{R}^B \setminus \{\mathbf{0}\}$ . We need to count the number of vertices in the mth generation of ${\widetilde{\mathbb{T}}_{11}^{(Q)}(B)}$ such that ${A_{{\boldsymbol{\varpi}}}^{(B)}} > 0$ . Due to the no leaf assumption, it is clear that the vertices in the mth generation of the tree $\mathbb{T}_{11}{(B)}$ before regularization. We also need to count the number of descendants of each of the particles of type Q at the mth generation. Also note that the displacements of corresponding to the vertices in the mth generation coming from the same ancestor in the $(m-1)$ th generation are dependent and so have been taken into account together in ${\lambda^{(B)}}$ . So we need to count the groups of vertices in the mth generation containing vertices of type Q which are independent. To perform this, we should count the number of particles in the $(m-1)$ th generation of $\mathbb{T}_{11}(B)$ as each of them has at least one descendant of type Q in the next generation (no leaf assumption in (B1)). It is clear that the configuration at the $(m-1)$ th generation is given by ${\mathbf{Z}_{m-1}^{(1)}(B) = (Z_{m-1}^{(1)}(1,B), Z_{m-1}^{(1)}(2,B),} \ldots, {Z_{m-1}^{(1)}(Q, B))}$ . Note that the ith particle of type q at the $(m-1)$ th generation has $U_{1i}^{(q)}(Q,B)$ descendants at the mth generation of type Q. Note that the number of descendants of the lth particle of type Q in the mth generation whose ancestor is the ith particle of type q in the $(m-1)$ th generation is independent of others and also independent of ${\mathbf{Z}_{m-1}^{(1)}(B)}$ . Also note that the ith particle of type Q has ${\Delta_{K-m,i}^{(Q)}(B)}$ descendants ${\boldsymbol{\varpi}}$ in the Kth generation of $\mathbb{T}_{11}(B)$ such that $A_{{\boldsymbol{\varpi}}}(B)$ is nonzero. Due to branching structure, we know that the number of descendants in the Kth generation of ${\mathbb{T}_{11}^{(B)}}$ of each of the particles in the mth generation of type Q are independent of the others and independent of the collection $({U_{1i}^{(p)}(Q, B)} \colon i \ge 1, 1 \le p \le Q)$ and ${\mathbf{Z}_{m-1}^{(1)}(B)}$ . Then we obtain the following expression for (5.14):

\begin{align*}%\label{eq:counting:regularization:hl:limit} & \sum_{m=1}^K \int_{\mathbb{R}^B \setminus \mathbf{0}} {\mathbb{E}}\bigg[ \sum_{q=1}^{Q} \sum_{l=1}^{Z_{m-1}^{(1)}(q,B)} \prod_{i=1}^2 \bigg( 1 - \exp \bigg\{ - \bigg( \sum_{j=1}^{U_{1l}^{(q)}} \Delta^{(Q)}_{K-m, j, l} (B) f_i(x_j) - \epsilon_i \bigg)_+ \bigg\} \bigg) \bigg] \lambda(\textrm{d} {{\boldsymbol{x}}}). \end{align*}

It is clear that expectation inside the expression gives

\begin{align*} & {\mathbb{E}}\bigg[ \sum_{l=1}^{Z_{m-1}^{(1)}(q,B)} \!\!{\mathbb{E}}\bigg( \prod_{i=1}^2 \bigg[ 1- \exp \bigg\{ - \bigg(\sum_{j=1}^{U_{1,l}^{(q)}(Q,B)} \Delta^{(Q)}_{K-m, j,l}(B)\, f_i(x_j) - \epsilon_i \bigg)_+ \bigg\} \bigg] \bigg| Z_{m-1}^{(1)}(q,B) \bigg) \bigg] \nonumber \\ &\qquad = {\mathbb{E}}\bigg[ Z_{m-1}^{(1)}(q,B) {\mathbb{E}}\bigg( \prod_{i=1}^2 \bigg( 1 - \exp \bigg\{ - \bigg[ \sum_{j=1}^{U_1^{(q)}(B)} \Delta^{(Q)}_{K-m, j}(B)\, f_i(x_j) - \epsilon_i \bigg]_+\bigg\} \bigg) \bigg) \bigg] \nonumber \\ &\qquad = {\boldsymbol{e}}_1^t ( M(B) )^{m-1} {\boldsymbol{e}}_q {\mathbb{E}}\bigg[ \prod_{i=1}^2 \bigg( 1 - \exp \bigg\{ - \bigg( \sum_{j=1}^{U_1^{(q)}(Q, B)} \Delta^{(Q)}_{K-m, j}(B)\, f_i(x_j) - \epsilon_i \bigg)_+ \bigg\} \bigg) \bigg]. \end{align*}

Hence, the limiting expression in (5.14) becomes

(5.15) \begin{align} & \sum_{m=1}^K \sum_{q=1}^Q {\boldsymbol{e}}_1^t ( M(B) )^{m-1} {\boldsymbol{e}}_q \nonumber \\ &\qquad\times \int {\mathbb{E}}\bigg[ \prod_{i=1}^2 \bigg[ 1 - \exp \bigg\{ - \bigg( \sum_{j=1}^{U_1^{(q)}(Q,B)} \Delta_{K-m,j}^{(Q)}(B) f_i(x_j) - \epsilon_i \bigg)_+ \bigg\} \bigg] \bigg] \lambda({\text{d}}(x_1, \ldots, x_B)). \label{eqn27}\end{align}

In the last step we are using Wald’s identity. Note that by combining (5.14) and (5.15), we can write down the limit in the form of the right-hand side of (5.13) (with $p=1$ ), where

\begin{align*} m_1^*(\!\cdot\!) = \sum_{m=1}^K \sum_{q=1}^Q &{\boldsymbol{e}}_1^t (M(B) )^{m-1} {\boldsymbol{e}}_q \\ &\!\times{\mathbb{E}}\bigg[ \lambda^{(B)} \bigg\{ (x_1, \ldots, x_B) \in \mathbb{R}^B \colon \sum_{j=1}^{U_1^{(q)}(Q,B)} \Delta_{K-m,j}^{(Q)} (B) \delta_{x_j} \in \cdot \bigg\} \bigg]. \end{align*}

Similarly, for every $p=2, \ldots, Q$ ,

(5.16) \begin{align}% m^*_p(\!\cdot\!) = \sum_{m=1}^K \sum_{q=1}^Q &{\boldsymbol{e}}_p^t M^{m-1}(B) {\boldsymbol{e}}_q\nonumber \\ &\!\times {\mathbb{E}}\bigg[ \lambda^{(B)} \bigg\{ (x_1, \ldots, x_B) \in \mathbb{R}^B \colon \sum_{j=1}^{U_1^{(q)}(Q,B)} \Delta_{K-m,j}^{(Q)} (B) \delta_{x_j} \in \cdot \bigg\} \bigg]. \label{eqn28}\end{align}

Following the same steps discussed in [10, p.199 (last paragraph)], it can be verified that $m_p^* \in \mathbb{M}({{\char77}}_0)$ for every $p=1, \ldots, Q$ .

We are ready now to prove the main lemma of this section.

Proof of Lemma 4.2. Recall that in the regularization step, we replaced the displacements corresponding to the particles of type Q coming from the same parent by an independent copy of ${(X_1^{\,{(Q)}}, \ldots, X_B^{(Q)})}$ . The new displacements corresponding to the vertex $\boldsymbol{\varpi}$ is denoted by $X'_{\boldsymbol{\varpi}}$ . By regularization construction, to prove the first statement it suffices to show that the weak convergence ${N_n'^{ (K,B)} \Rightarrow N_*^{(K,B)}}$ as $n \to \infty$ for

\begin{align*} N_n'^{ (K,B)} = \sum_{p=1}^Q \sum_{j=1}^{|D_{n-K} ^{(p)}| } \sum_{{\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}_{pj}^{{(Q)}}(B)} A_{\boldsymbol{\varpi}}^{(B)} \delta_{b_n^{-1} {{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}^\prime}. \end{align*}

Now we will consider the Laplace functional of ${N_n'^{ (K,B)}}$ .

Consider a function $f \in C_c^+ (\bar{\mathbb{R}}_0)$ . We then have to find the limit of

(5.17) \begin{align} &{\mathbb{E}}[{\exp} \{ - N_n'^{ (K,B)}(\,f) \} ] \nonumber \\ &\qquad= {\mathbb{E}}\bigg[ \exp \bigg\{ - \sum_{p=1}^Q \sum_{j=1}^{Z_{n-K}(p)} \sum_{{\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}^{(Q)}_{pj}(B)} A_{\boldsymbol{\varpi}}^{(B)} f({b_n^{-1} X'_{\boldsymbol{\varpi}}}) \bigg\} \bigg] \nonumber \\ &\qquad= \prod_{p=1}^Q {\mathbb{E}}\bigg[{\mathbb{E}}\bigg( \exp \bigg\{ - \sum_{j=1}^{Z_{n-K}(p)} \sum_{{\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}_{pj}^{(Q)}(B)} A_{\boldsymbol{\varpi}}^{(B)} f({b_n^{-1} X'_{\boldsymbol{\varpi}}}) \bigg\} \bigg| Z_{n-K}^{(1)}, Z_{n-K}^{(2)}, \ldots, Z_{n-K}^{(Q)} \bigg) \bigg] \nonumber \\ &\qquad = \prod_{p=1}^{Q} {\mathbb{E}}\bigg(\bigg[{\mathbb{E}}\bigg( \exp \bigg\{ - \sum_{{\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}_{p1}^{(Q)} (B)} A_{\boldsymbol{\varpi}}^{(B)} f(b_n^{-1} X'_{\boldsymbol{\varpi}}) \bigg\} \bigg) \bigg]^{Z_{n-K}^{(p)}} \bigg), \label{eqn29}\end{align}

since the point processes associated to the trees ${{\mathbb{T}}_{pj}^{(Q)}(B)}$ are i.i.d. for every fixed $p \in \{1, \ldots,Q\}$ . Using the following convergence stated in Lemma 3 (see [10 (3.10, p.~192)] for details),

\begin{align*} \bigg[{\mathbb{E}}\bigg( \exp\bigg\{ - {{\boldsymbol{S}}}_{b_n^{-1}} N'^{(K,B,p,1)}(\,f) \bigg\} \bigg) \bigg]^{\rho^n} \to \exp \bigg\{-\int (1-\textrm{e}^{-\nu(\,f)}) m_{p}^*(\textrm{d} \nu) \bigg\}, \end{align*}

we can conclude that

\begin{align*} \bigg[{\mathbb{E}}\bigg( \exp \bigg\{ - \sum_{{\boldsymbol{\varpi}} \in \widetilde{\mathbb{T}}_{p1}^{(Q)}(B)} A_{\boldsymbol{\varpi}}^{(B)} f(b_n^{-1} {{\boldsymbol{X}}}_{{\boldsymbol{\varpi}}}') \bigg\} \bigg) \bigg]^{\rho^n} \end{align*}

converges to

\begin{align*} &\exp \bigg\{ - \sum_{m=1}^K \sum_{q=1}^Q {\boldsymbol{e}}_p^t M^{m-1}(B) {\boldsymbol{e}}_q \\ &\times\int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q,B)} \Delta^{(Q)}_{K-m,j}(B) f(x_j) \bigg\} \bigg) \lambda^{(B)}({\text{d}}(x_1, \ldots, x_B)) \bigg\} \end{align*}

as $n \to \infty$ . From the Kesten–Stigum theorem ([Reference Kesten and Stigum28], [7, Theorem 1, p.

192]), it follows that (5.17) converges to

\begin{align*} &\exp \bigg\{ - \frac{1}{\rho^K} W \varsigma_p \sum_{m=1}^K \sum_{q=1}^Q {\boldsymbol{e}}_p^t (M(B))^{m-1} {\boldsymbol{e}}_q \\ & \times \int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q,B)} \Delta^{(Q)}_{K-m,j}(B) f(x_j) \bigg\} \bigg) \lambda^{(B)}({\text{d}}(x_1, \ldots, x_B))\bigg\} \end{align*}

almost surely as $n \to \infty$ . Recall that $\varsigma_p$ denotes the pth entry of the right eigenvector $\varsigma$ of the matrix M. Finally, using the dominated convergence theorem, we obtain the convergence of the Laplace functional (5.17),

(5.18) \begin{align} &{\mathbb{E}}\bigg[ \exp\bigg\{- \frac{1}{\rho^K} W \sum_{p=1}^Q \varsigma_p \sum_{m=1}^K \sum_{q=1}^Q {\boldsymbol{e}}_p^t (M(B))^{m-1} {\boldsymbol{e}}_q \nonumber \\ &\times \int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q,B)} \Delta^{(Q)}_{K-m,j}(B) f(x_j) \bigg\} \bigg) \lambda^{(B)}({\text{d}}(x_1, \ldots, x_B))\bigg\} \bigg]. \label{eqn30}\end{align}

Here we shall construct a point process which has the same Laplace functional as obtained in (5.18). Consider a Poisson random measure

\begin{align*} {{\char80}}^{(B)} = \sum_{l=1}^\infty \delta_{\pmb{\varrho}_l} = \sum_{l=1}^\infty \delta_{(\varrho_{l1}, \ldots, \varrho_{lB})} \end{align*}

on $\mathbb{R}^B$ with intensity measure ${\lambda^{(B)}(\!\cdot\!)}$ and independent of the random variable W. Let ${(G^{(B)}}, {{{\boldsymbol{T}}}^{(B)})}$ be a $\Xi$ -valued random variable, where $\Xi = \cup_{i \in \mathbb{N}} \{i\} \times \mathbb{N}^i$ with probability mass function

\begin{align*} & \mathbb{P} (G^{(B)} = g, {{\boldsymbol{T}}}^{(B)} = (t_1, \ldots, t_g) ) \\ &\qquad= \frac{1}{s_B \rho^K} \sum_{p=1}^Q \varsigma_p \sum_{m=1}^K \sum_{q=1}^Q {\boldsymbol{e}}_p (M(B))^{m-1} {\boldsymbol{e}}_q \mathbb{P}( U_1^{(q)} (Q,B) = g) \prod_{j=1}^g \mathbb{P}(\Delta_{K-m, j}^{(Q)} = t_j) \end{align*}

for all $(t_1, \ldots, t_g) \in \mathbb{N}^g$ and $g \in \mathbb{N}$ , where $s_B$ is the normalizing constant that makes the above a probability measure on $\Xi$ . Now consider a collection ${\{(G_l^{(B)}, {{\boldsymbol{T}}}_l^{(B)})}\colon l \ge 1\}$ of independent copies of ${(G^{(B)}, {{\boldsymbol{T}}}^{(B)})}$ and also independent of W and ${{{\char80}}^{(B)}}$ . Now the Cox cluster process

\begin{align*} N_*^{(K,B)} = \sum_{l=1}^\infty \sum_{k=1}^{G_l^{(B)}} T_{lk} \delta_{(s_B W)^{1/\alpha}\varrho_{lk}} \end{align*}

can be shown to have the same Laplace functional as in (5.18) following the method discussed in [10, Equation (4.35), p. 201]. The construction is based on a Cox marked point process and then ${N_*^{(K,B)}}$ can be written in terms of the marked Cox point process.

Now, we let $B \to \infty$ and using the dominated convergence theorem once again, we obtain from the right-hand side of (5.18),

(5.19) \begin{align} &{\mathbb{E}}\bigg[ \exp\bigg\{- \frac{1}{\rho^K} W \sum_{p=1}^Q \varsigma_p \sum_{m=1}^K \sum_{q=1}^Q {\boldsymbol{e}}_p^t M^{m-1} {\boldsymbol{e}}_q \nonumber \\ & \times\int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{K-m,j} f(x_j) \bigg\} \bigg) \lambda(\textrm{d}{{\boldsymbol{x}}})\bigg\} \bigg]. \label{eqn31}\end{align}

Consider a $\Xi$ -valued random variable $(G', {{\boldsymbol{T}}}')$ with the probability mass function

(5.20) \begin{align} &\mathbb{P}(G'=g, {{\boldsymbol{T}}}'=(t_1, \ldots, t_g)) \nonumber \\ &\qquad= \frac{1}{s' \rho^K} \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{K-1} \sum_{q=1}^Q {\boldsymbol{e}}_p^t M^{K-m-1} {\boldsymbol{e}}_q \mathbb{P}(U_1^{(q)} (Q) =g) \prod_{j=1}^g \mathbb{P}(\Delta^{(Q)}_{m,j} = t_j) \label{eqn32}\end{align}

for every $(t_1, \ldots, t_g) \in \mathbb{N}^g$ and $g \in \mathbb{N}$ , where s’ is the normalizing constant. Now consider a collection $\{(G'_l, {{\boldsymbol{T}}}'_l) \colon l \ge 1 \}$ of independent copies of $(G', {{\boldsymbol{T}}}')$ which is also independent of ${{\char80}}$ and W. Now the point process

\begin{align*} N_*^{(K)} = \sum_{l=1}^\infty \sum_{k=1}^{G'_l} T'_{lk} \delta_{(s' W)^{1/\alpha} \xi_{lk}} \end{align*}

has the same Laplace functional as described in (5.19). It is easy to see that the right-hand side of (5.20) becomes

\begin{align*} & {\mathbb{E}}\bigg[ \exp\bigg\{- \frac{1}{\rho^K} W \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{K-1} \sum_{q=1}^Q {\boldsymbol{e}}_p^t M^{K-m-1} {\boldsymbol{e}}_q \\ &\times\int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda(\textrm{d}{{\boldsymbol{x}}})\bigg\} \bigg] \\ &\qquad= {\mathbb{E}}\bigg[ \exp\bigg\{- \frac{1}{\rho^K} W \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{K-1} \sum_{q=1}^Q {\boldsymbol{e}}_p^t (\rho^{K-m-1} P + R^{K-m-1}) {\boldsymbol{e}}_q \\ & \times\int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda(\textrm{d}{{\boldsymbol{x}}})\bigg\} \bigg] \end{align*}

using rearrangement of terms and Theorem 1 of [7, Chapter V, p.

185]. Here R is as in Assumption 1.

Note that Theorem 1 of [7, Chapter V, p.185] also implies that

\begin{align*} & \frac{1}{\rho^K} \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{K-1} \sum_{q=1}^Q |{\boldsymbol{e}}_p^t R^{K-m-1} {\boldsymbol{e}}_q | \int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda(\textrm{d}{{\boldsymbol{x}}}) \\ &\qquad \le \bigg( \frac{\rho_0}{\rho} \bigg)^K cQ \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{K-1} \frac{1}{\rho_0^{m+1}} \end{align*}

for some $1 &lt \rho_0 &lt \rho$ , which is negligible as $K \to \infty$ . Hence, we can let $K \to \infty$ and the Laplace functional then becomes

(5.21) \begin{align} & {\mathbb{E}}\bigg[ \!\exp\bigg\{\!- W \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{\infty} \frac{1}{\rho^{m+1}} \sum_{q=1}^Q {\boldsymbol{e}}_p^t P {\boldsymbol{e}}_q \int {\mathbb{E}}\bigg(\! 1 - \exp \bigg\{\! - \!\!\sum_{j=1}^{U_1^{(q)}(Q)} \!\!\Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda({\text{d}}{{\boldsymbol{x}}})\bigg\} \bigg] \nonumber \\ &\quad = {\mathbb{E}}\bigg[ \exp\bigg\{- W \sum_{p=1}^Q \varsigma_p \sum_{m=0}^{\infty} \frac{1}{\rho^{m+1}} \sum_{q=1}^Q {\boldsymbol{e}}_p^t \vartheta \varsigma^t {\boldsymbol{e}}_q \nonumber \\ &\times\int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda({\text{d}}{{\boldsymbol{x}}})\bigg\} \bigg] \nonumber \\ &\quad ={\mathbb{E}}\bigg[ \exp\bigg\{- W \sum_{p=1}^Q \varsigma_p \vartheta_p \sum_{m=0}^{\infty} \frac{1}{\rho^{m+1}} \nonumber \\ &\times \sum_{q=1}^Q u_q \int {\mathbb{E}}\bigg( 1 - \exp \bigg\{- \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda({\text{d}}{{\boldsymbol{x}}})\bigg\} \bigg] \nonumber \\ &\quad ={\mathbb{E}}\bigg[ \exp\bigg\{- W \sum_{m=0}^{\infty} \frac{1}{\rho^{m+1}} \sum_{q=1}^Q \varsigma_q \int {\mathbb{E}}\bigg( 1 - \exp \bigg\{ - \sum_{j=1}^{U_1^{(q)}(Q)} \Delta^{(Q)}_{m,j}\, f(x_j) \bigg\} \bigg) \lambda({\text{d}}{{\boldsymbol{x}}})\bigg\} \bigg], \label{eqn33}\end{align}

where we used the fact that $P = \vartheta. \varsigma^t$ for $\vartheta^t. \varsigma =1$ .

The last step will be to show that the expression obtained in (5.21) is the expression for the Laplace functional of the point process $N_*$ as described in Theorem 3. To compute the Laplace functional of $N_*$ , we will use the auxiliary marked Cox process

\begin{align*} P_* = \sum_{l=1}^\infty \delta_{(G_l, T_{l1}, \ldots, T_{l,G_l}, ((\rho-1)^{-1} W)^{1/\alpha} \pmb{\xi}_l )} \end{align*}

with the state space as $N \times \mathbb{R}^\mathbb{N}$ for $N = {\bigcup_{i=1}^\infty} (\{i\} \times \mathbb{N}^i)$ . We consider a positive, bounded, and continuous function f on the metric space $N \times \mathbb{R}^\mathbb{N}$ which vanishes outside a neighbourhood of $N \times \{\textbf{0}\}$ . Then we obtain

\begin{align*} &{\mathbb{E}}({\exp}\{ - P_*(\,f) \} ) \nonumber \\ &\qquad= {\mathbb{E}}\bigg[{\mathbb{E}}\bigg(\exp \bigg\{ - \sum_{l=1}^\infty f(G_l, T_{l1}, \ldots, T_{lG_l}, ((\rho-1)^{-1} W)^{1/\alpha} \pmb{\xi}_l ) \bigg\} \biggm| W \bigg) \bigg]. \nonumber \end{align*}

Now using the fact that $P_*$ conditioned on W is a marked Poisson point process with independent marks and following Proposition 3.8 of [Reference Resnick36], we can conclude that the right-hand side of the above equality becomes

\begin{align*} {\mathbb{E}}\left[ \exp \left\{ - \int_{\mathbb{R}_0^{\mathbb{N}}} {\mathbb{E}}( 1- \exp \{ - f(G, T_1, \ldots, T_G, ((\rho-1)^{-1} W)^{-1/\alpha} {{\boldsymbol{x}}}) \} \ \,\,|\, \ \, W ) \lambda({\text{d}}{{\boldsymbol{x}}})\right\} \right]. \end{align*}

Recall from (3.4) that $\lambda(a. {{\boldsymbol{x}}}) = a^{-\alpha} \lambda(a)$ for every $a > 0$ . Hence, the above expression equals

\begin{align*} {\mathbb{E}}\left[ \exp \left\{ - (\rho-1)^{-1} W\int_{\mathbb{R}_0^{\mathbb{N}}} {\mathbb{E}}( 1- \exp \{ - f (G, T_1, \ldots, T_G, {{\boldsymbol{x}}}) \} ) \lambda(\!\text{d} {{\boldsymbol{x}}})\right\} \right]. \end{align*}

Let $f' \colon \mathbb{R} \to \mathbb{R}_+$ be a bounded, continuous function vanishing on a neighbourhood of 0. Then

\begin{align*} f(g, t_1, \ldots, t_g, {{\boldsymbol{x}}}) = \sum_{m=1}^g t_m f'(x_m) \end{align*}

is a positive, bounded, and continuous function on $N \times \mathbb{R}^\mathbb{N}$ vanishing in a neighbourhood of $N \times \{\textbf{0}\}$ . Finally, we obtain

\begin{align*} & {\mathbb{E}}({\exp}\{ - P_*(f) \} ) \nonumber \\ &\qquad= {\mathbb{E}}({\exp}\{ - N_*(f') \} ) \nonumber \\ &\qquad = {\mathbb{E}}\bigg[{\exp}\bigg\{ - (\rho-1)^{-1} W \int_{\mathbb{R}_0^{\mathbb{N}}} {\mathbb{E}}\bigg( 1- \exp\bigg\{ - \sum_{j=1}^G T_j f'(x_j)\bigg\} \bigg) \lambda(\textrm{d} {{\boldsymbol{x}}}) \bigg\} \bigg] \nonumber \\ &\qquad = {\mathbb{E}}\bigg[{\exp} \bigg\{ - (\rho-1)^{-1} W \int_{\mathbb{R}_0^{\mathbb{N}}} \sum_{q=1}^Q \varsigma_q \sum_{g=1}^ \infty \mathbb{P}(U_1^{(q)}(Q) =g) \nonumber \\ &\times {\mathbb{E}}\bigg( 1- \exp\bigg\{ - \sum_{j=1}^g T_j f'(x_j)\bigg\} \bigg)\lambda(\textrm{d}{{\boldsymbol{x}}}) \bigg\} \bigg] \nonumber \\ &\qquad = {\mathbb{E}}\bigg[{\exp} \bigg\{ - (\rho-1)^{-1} W \int_{\mathbb{R}_0^{\mathbb{N}}} \sum_{q=1}^Q \varsigma_q \sum_{g=1}^ \infty \mathbb{P}(U_1^{(q)}(Q) =g) (\rho-1)\sum_{m=0}^ \infty \dfrac{1}{\rho^{m+1}} \nonumber \\ &\times \sum_{t_1, \ldots, t_g \in \mathbb{N}^g} \prod_{j=1}^g \mathbb{P}(\Delta_{m,j}^{(Q)} =t_j) \bigg( 1- \exp\bigg\{ - \sum_{j=1}^g t_j f'(x_j)\bigg\} \bigg) \lambda(\textrm{d} {{\boldsymbol{x}}}) \bigg\} \bigg] \nonumber \\ &\qquad= {\mathbb{E}}\bigg[{\exp} \bigg\{ - W \sum_{q=1}^Q \varsigma_q \sum_{m=0}^ \infty \dfrac{1}{\rho^{m+1}} \int_{\mathbb{R}_0^{\mathbb{N}}} {\mathbb{E}}\bigg( 1- \exp\bigg\{ - \!\!\sum_{j=1}^{U_1^{(q)}(Q)}\! \Delta_{m,j}^{(Q)} f'(x_j)\bigg\} \bigg) \lambda(\textrm{d} {{\boldsymbol{x}}}) \bigg\} \bigg], \end{align*}

which is the same as that obtained in (5.21) and, thus, $N_*$ is the limit point process.