2. Description of models

On a certain probability space $(\Omega ,\mathcal{A},\mathbb{P})$ , we consider the following three independent sets of random variables.

1. (i) Define the set $X=\{X_{n}(i),\, n,i=1,2,\ldots\}$ of non-negative integer-valued independent and identically distributed (i.i.d.) random variables with probability generating function (p.g.f.) $f(s)=\mathbb{E}[s^{X_{1}(1)}],\ 0\leq s\leq 1$ . Further, let $I_0$ be a positive integer-valued random variable independent of X with p.g.f. $\Delta (s)=\mathbb{E}[s^{I_{0}}],\ 0\leq s\leq 1$ . Recall that the classical BGW branching process with $I_{0}$ ancestors is defined as follows:

   (2.1) \begin{equation} Z_{0}=I_{0},\quad Z_{n}=\sum_{i=1}^{Z_{n-1}}X_n(i),\quad n=1,2,\ldots, \end{equation}
   where $\sum_{i=1}^0=0$ . Taking into account the independence of the individual evolutions, (2.1) implies $\mathbb{E}[s^{Z_{n}}\mid Z_{0}=I_0]=\Delta(f_{n}(s))$ , where $f_{n}(s)=\mathbb{E}[s^{Z_{n}}\mid Z_{0}=1]$ and $f_{n}(s)=f_{n-1}(f(s))$ , $ n=1,2,\ldots.$ Clearly zero is an absorbing state.

2. (ii) Define the set $\phi =\{\phi _{n}(k),\, n=1,2,\ldots;\, k=0,1,\ldots\}$ of non-negative integer-valued random variables, independent of X, where for every fixed k the subset $\phi(k)=\{\phi_{n}(k),\, n=1,2,\ldots\}$ consists of i.i.d. random variables with p.g.f. $g_{k}(s)=\mathbb{E}[s^{\phi_{1}(k)}],$ $0\leq s\leq 1$ . The CBP is defined as follows:

   (2.2) \begin{equation} Z_{0}=I_{0},\quad Z_n=\sum_{i=1}^{\phi_{n}(Z_{n-1})} X_{n}(i),\quad n=1,2,\ldots, \end{equation}
   with $\phi$ referred to as the set of random control functions. Note that if $\phi_{1}(k) \equiv k $ a.s. for every k, then $\{Z_{n},\, n=0,1,\ldots\}$ is a BGW branching process defined by (2.1). If this is not the case, let us point out that since $h_{n}(s)=\mathbb{E}[s^{Z_{n}}\mid Z_{0}=I_{0}]\neq \Delta (\psi _{n}(s))$ where $\psi _{n}(s)=\mathbb{E}[s^{Z_{n}}\mid Z_{0}=1]$ , we have that the evolutions of the individuals are not independent. It follows from (2.2) that the state zero will be absorbing if and only if $\phi_1(0)=0$ a.s. For more details see [Reference González, del Puerto and Yanev11].

   Definition (2.2) can be generalized by introducing the set of random control functions $\phi_{D}=\{\phi_{n,d}(k),\, n=1,2,\ldots;\, k=0,1,\ldots;\, d\in D\}$ and the set of random variables $X_{D}=\{X_{n,d}(i),\, n,i=1,2,\ldots;\, d\in D\}$ , where D is an index set. Then the CBP with multitype control functions is defined as follows:

   (2.3) \begin{equation} Z_{0}=I_{0},\quad Z_{n}=\bigg(\sum_{d\in D}\sum_{i=1}^{\phi_{n,d}(Z_{n-1})} X_{n,d}(i)\bigg)^+,\quad n=1,2,\ldots, \end{equation}
   where, as usual, $a^{+}=\max \{0,a\}$ . In some branching models it is assumed that for every fixed d the random variables $\{X_{n,d}(i),\, n,i=1,2,\ldots\}$ are integer-valued i.i.d. and, for every fixed k and d, the subset $\phi_{d}(k)=\{\phi _{n,d}(k),\, n=1,2,\ldots\}$ consists of non-negative integer-valued i.i.d. random variables. Note that the random variable $X_{n,d}(i)$ can be negative, allowing individual emigration in the model. We will consider a particular case of (2.3) in Section 5, which admits a random migration component.

3. (iii) Finally, define the set $J=\{J_{n},\, n=1,2,\ldots\}$ of positive i.i.d. random variables, independent of X and $\phi$ , with cumulative distribution function (c.d.f.) $G(x)=\mathbb{P}(J_{1}\leq x)$ , $ x>0$ ; $G(0)=0$ . Define the renewal process $\{N(t),\, t\geq 0\}$ by

   (2.4) \begin{equation} N(t)=\max \{n\geq 0\colon S_{n}\leq t\},\quad t\geq 0, \end{equation}
   where $S_{0}=0$ , $S_{n}=\sum_{i=1}^{n}J_{i}$ , $n=1,2,\ldots.$ This yields the renewal function
   \begin{equation*}\mathbb{E}[N(t)]=\sum_{n=0}^{\infty}G^{\ast n}(t),\quad t\geq 0,\end{equation*}
   where $G^{\ast n}$ is the n-fold convolution of G with $G^{\ast 0}=1$ .

Alternatively, the process $\{Y(t),\, t\geq 0\}$ can be called a randomly indexed CBP with CT or CBP subordinated by a renewal process.

3. Preliminaries

We now provide auxiliary results for investigating limiting distributions of the CBPs with CT $\{Y(t),\, t\geq 0\}$ defined in the previous section. The main tools for further investigations are the p.g.f.

(3.1) \begin{equation} h(t;s)=\mathbb{E}[s^{Y(t)} \mid Y(0)=I_{0}]=\sum_{n=0}^{\infty }\mathbb{P}(N(t)=n)h_{n}(s),\quad t\geq 0,\quad 0\leq s\leq 1, \end{equation}

and the conditional distribution function $\mathbb{P}(Y(t)\leq x \mid Y(t)>0)$ , which satisfies

(3.2) \begin{equation} \mathbb{P}(Y(t)\leq x \mid Y(t)>0)=\int_{0}^{\infty}\mathbb{P}(Z_{\lfloor y\rfloor}\leq x \mid Z_{\lfloor y\rfloor}>0)\,{\textrm{d}}_y\mathbb{P}(N(t)\leq y \mid Z_{N(t)}>0), \end{equation}

where $\lfloor y\rfloor$ denotes the integer part of y and $\{N(t),\, t\geq 0\}$ is the renewal process (2.4).

Let

\begin{equation*} \mu=\mathbb{E}[J_{1}]=\int_0^\infty(1-G(y))\,{\textrm{d}} y, \end{equation*}

the mean of the renewal periods. When $\mu<\infty$ , it is known (see [Reference Bingham, Goldie and Teugels1, Ch. 8.6.1] or [Reference Feller4, Ch. XI.1]) that

\begin{equation*} \mathbb{P}\biggl(\lim_{t\rightarrow\infty} \dfrac{N(t)}{t/\mu}=1\biggr)=1. \end{equation*}

Let $\{Z_{n},\, n=0,1,\ldots\}$ be a non-negative discrete-time process with absorbing state zero, and denote $Q_{n}=\mathbb{P}(Z_{n}>0)$ , $n=0,1,\ldots.$ It is shown using equation (7.17) of [Reference Mitov, Mitov and Yanev21] that if $Q_{n}\downarrow 0$ and $Q_{n}\sim L(n)n^{-{{\gamma}}}$ as $n\rightarrow\infty$ , where $0<{{\gamma}}<1$ and L(x) is a slowly varying function (s.v.f.), then

(3.3) \begin{equation} \lim_{t\rightarrow\infty} \mathbb{P}\biggl(\dfrac{N(t)}{t/\mu}\leq x \Bigm| Z_{N(t)}>0\biggr)=V_{1}(x),\quad x\geq 0, \end{equation}

where $V_{1}(x)=\textbf{1}_{\{x\geq 1\}}$ , with $\textbf{1}_A$ the indicator function of the set A.

We will also consider the case when $\mu =\infty $ and

(3.4) \begin{equation} 1-G(t)\sim\dfrac{t^{-\rho }\mathcal{L}(t)}{\Gamma(1-\rho)}\quad \text{as $ t\rightarrow \infty$,}\quad 0<\rho<1, \end{equation}

where $\mathcal{L}(t)$ is an s.v.f. Let $a(t)=(\Gamma(1-\rho)(1-G(t)))^{-1}$ . It is known (see [Reference Bingham, Goldie and Teugels1, Ch. 8.6.2] or [Reference Feller4, Ch. XIV.3]) that

\begin{equation*} \lim_{t\to\infty}\mathbb{P}\biggl(\dfrac{N(t)}{a(t)}\leq x\biggr)=G_\rho(x),\quad x\geq 0,\end{equation*}

where $G_\rho(x)= \mathbb{P}(\Lambda_{\rho }\leq x)$ , with $\Lambda_{\rho }\overset{d}{=}\xi_{\rho}^{-\rho}$ and $\xi_{\rho }$ , has a one-sided $\rho$ -stable distribution, i.e. $\mathbb{E} [{\textrm{e}}^{-\lambda\xi_{\rho }}]={\textrm{e}}^{-\lambda^{\rho }}$ , $ \lambda>0$ . Note that (3.4) yields $a(t)\sim t^{\rho }/\mathcal{L}(t)$ as $t\to\infty$ . Hence, referring to equation (7.18) in [Reference Mitov, Mitov and Yanev21] and the conditions on $\{Z_{n}\colon n=0,1,\ldots\}$ , we have

(3.5) \begin{equation} \lim_{t\rightarrow\infty}\mathbb{P}\biggl(\dfrac{N(t)}{a(t)} \leq x \Bigm| Z_{N(t)}>0\biggr)=V_{\rho,{{\gamma}} }(x),\quad x\geq 0, \end{equation}

where $V_{\rho,{{\gamma}}}(x)=\mathbb{E}[\Lambda_{\rho }^{-{{\gamma}}}{\textbf{1}}_{\{\Lambda_{\rho}\leq x\}}]/\mathbb{E}[\Lambda _{\rho}^{-{{\gamma}}}]$ and $\mathbb{E}[\Lambda_{\rho}^{-{{\gamma}}}]=\Gamma(1-{{\gamma}})/\Gamma(1-{{\gamma}}\rho)$ . Note that $G_\rho(x)$ is known as the c.d.f. of the Mittag–Leffler distribution of order $\rho$ , with Laplace transform $\varphi _{\rho }(\lambda )=(1+\lambda ^{\rho })^{-1}$ (see [Reference Huillet12] for more details).

Further on we will need some results from weighted renewal theory as follows. For any sequence of real numbers $\{w_{n},\, n=0,1,\ldots\}$ , consider the function ${W(x)=\sum_{k=0}^{\lfloor x\rfloor}w_{k}}$ , $x\geq 0$ . The weighted renewal function $H_w(t)$ , $t\geq 0$ , is defined as follows:

\begin{equation*} H_{w}(t)= \sum_{n=0}^{\infty}w_{n}G^{\ast n}(t) =\sum_{k=0}^{\infty}\mathbb{P}(N(t)=k)W(k)=\mathbb{E}[W(N(t))]. \end{equation*}

Lemma 3.2. Let $\mu=\infty$ . Assume (3.4) and $W({t})\sim L_{w}({t}){t}^{\beta }$ as ${t}\rightarrow \infty$ , where $0<\beta<1$ and $L_{w}(x)$ is an s.v.f. Then ${H_{w}(t)\sim CW(t/m(t))}$ as $t\rightarrow \infty$ , where $C=\Gamma (1+\beta)/(\Gamma(1+\rho\beta)\Gamma^{\beta}(2-\rho))$ and

\begin{equation*} m(t)=\int_{0}^{t}(1-G(x)) \,{\textrm{d}} x. \end{equation*}

Lemma 3.1 is proved in [Reference Mallor and Omey13, Ch. 2, Theorem 38(a) and Theorem 46(i)]. Lemma 3.2 follows from [Reference Mallor and Omey13, Theorem 43].

4. Critical controlled branching processes subordinated by a renewal process

In this section we will consider the process $\{Y(t)=Z_{N(t)}, t\geq 0\}$ , where $ \{Z_{n}, n=0,1,\ldots\} $ is the CBP (2.2) with zero as an absorbing state and $ \{N(t),\, t\geq 0\}$ is the renewal process (2.4). Let us introduce the following notation for $k=0,1,\ldots$ :

\begin{align*} \varepsilon (k)&=\mathbb{E}[\phi _{1}(k)],\quad \nu ^{2}(k)=\operatorname{Var} [\phi _{1}(k)],\\[3pt] \tau _{m}(k)&= k^{-1} \mathbb{E}[Z_{n+1}|Z_{n}=k]=k^{-1}\varepsilon (k)m, \quad \text{where } m=\mathbb{E}[X_1(1)],\\[3pt] l_{2}(k)&=\operatorname{Var} [Z_{n+1}|Z_{n}=k]=m^{2}\nu ^{2}(k)+\sigma ^{2}\varepsilon (k), \quad \text{where } \sigma^2=\operatorname{Var} [X_1(1)]. \end{align*}

Later on the following condition is assumed to hold.

It is worth pointing out that for a CBP, the threshold parameter determining the criticality of the process is the asymptotic mean growth rate, i.e. $\lim_{k\to\infty}\tau_m(k)$ , when it exists (note that the asymptotic mean growth rate for a BGW branching process coincides with m). Thus, under Condition A we consider a critical CBP with additional hypotheses.

Finally, denote $Q_n=\mathbb{P}(Z_n>0)$ , $n= 0,1,\ldots,$ and $Q(x)=\sum_{k=0}^{{\lfloor x \rfloor}-1}Q_{k}$ , $x\geq 1$ , and $Q(x)=0$ for $0\leq x<1$ .

Proof. Assertions (i) and (ii) follow from Theorems 3 and 4 in [Reference González, Molina and del Puerto10] by using Condition A. We will prove (iii). Let $m_n=\mathbb{E}[Z_n]$ . Using Condition A(a), we obtain

(4.1) \begin{align} m_{n+1}&=\sum_{k=1}^{\infty }\mathbb{P}(Z_{n}=k)\mathbb{E}[Z_{n+1}|Z_{n}=k]\notag \\[2pt] &=\sum_{k=1}^{\infty }\mathbb{P}(Z_{n}=k)(k+c)\notag \\[2pt] &=\mathbb{E}[Z_{n}]+cQ_{n}\notag \\[2pt] &=m_{0}+c\sum_{k=0}^{n}Q_{k},\end{align}

where $\{Q_{n}\colon n=0,1,\ldots\}$ is monotone decreasing. Applying Theorem 5 from [Reference Feller4, Ch. XIII.5] and using (i) of this theorem, we obtain $ \sum_{k=0}^{n}Q_{k}\sim (K/\delta )n^{\delta }$ and the first statement in (iii) follows.

Similarly, by Condition A(b), we have

$$\eqalign{ & {\mathop{\rm Var}\nolimits} [{Z_{n + 1}}] = {\mathbb E}[{\mathop{\rm Var}\nolimits} [{Z_{n + 1}}\mid {Z_n}]] + {\mathop{\rm Var}\nolimits} [{\mathbb E}[{Z_{n + 1}}\mid {Z_n}]] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sum\limits_{k = 1}^\infty {\mathbb P} ({Z_n} = k){\mathop{\rm Var}\nolimits} [{Z_{n + 1}}|{Z_n} = k] + {\mathop{\rm Var}\nolimits} [m\varepsilon ({Z_n})] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sum\limits_{k = 1}^\infty {\mathbb P} ({Z_n} = k)(\nu k + {\rm{O}}(1)) + {\mathop{\rm Var}\nolimits} [{Z_n} + c] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\mathop{\rm Var}\nolimits} [{Z_0}] + \nu \sum\limits_{k = 0}^n {\mathbb E} [{Z_k}] + {\rm{O}}(n). \cr} $$

Since $m_{k}=\mathbb{E}[Z_{k}]\sim (K\nu /2)k^{\delta }$ as $k\rightarrow \infty$ , using well-known properties of the regular varying functions (r.v.f.s) (e.g. [Reference Feller4, Ch. VIII.9]), we obtain

\begin{equation*}\sum_{k=0}^{n}\mathbb{E}[Z_{k}]\sim \dfrac{K\nu}{ 2(\delta +1)}n^{\delta+1}\quad \text{as $n\rightarrow \infty$.}\end{equation*}

Consequently

\begin{align*} \operatorname{Var} [Z_{n}]\sim {{\dfrac{ K\nu ^{2}}{2(\delta +1)}}}n^{\delta +1}\quad \text{as $ n\rightarrow \infty$.}\\[-40pt] \end{align*}

□

We will investigate the continuous-time process $\{Y(t)=Z_{N(t)}, t\geq 0\}$ in Definition 2.1 with (2.2).

Proof. (i) Note that from (3.1) we have that $\mathbb{P}(Y(t)=0)=h(t;0)$ and therefore

(4.2) \begin{equation}\mathbb{P}(Y(t)=0)=\sum_{n=0}^{\infty }\mathbb{P}(N(t)=n)P_{n}=\mathbb{E}[P_{N(t)}], \end{equation}

where $P_{n}=\mathbb{P}(Z_{n}=0)=1-Q_n\uparrow 1$ as $n\rightarrow \infty$ .

Introduce $w_{0}=P_{0}=0$ , $w_{k}=P_{k}-P_{k-1}$ , $k=1,2,\ldots.$ Note that $ w_{k}\geq 0$ and $W(x)=\sum_{k=0}^{\lfloor x\rfloor}w_{k}=P_{n}$ for $n\leq x < n+1$ , $n=0,1,\ldots.$ Hence we can rewrite (4.2) as

(4.3) \begin{equation} \mathbb{P}(Y(t)=0)=\sum_{n=0}^{\infty }\mathbb{P}(N(t)=n)W(n)= \mathbb{E}[W(N(t))]. \end{equation}

Note that in this case $W(t)\uparrow 1$ as $t\to\infty$ , and by Lemma 4.1(i) we have $1-W(t)=Q_{\lfloor t \rfloor}\sim Kt^{-(1-\delta )}$ as $t\to\infty$ , where $0<K<\infty$ . Therefore, applying Lemma 3.1(ii), we obtain that $\mathbb{P}(Y(t)>0)\sim \mu ^{1-\delta }(1- W(t))$ as $t\rightarrow \infty$ , which completes the proof of (i).

(ii) Using (4.1) we obtain

\begin{equation*} \mathbb{E}[Y(t)]=\sum_{n=0}^{\infty }\mathbb{P}(N(t)=n)m_{n}=m_{0}+c\sum_{n=1}^{\infty }\mathbb{P}(N(t)=n) \sum_{k=0}^{n-1}Q_{k}, \end{equation*}

where we recall that $m_n=\mathbb{E}[Z_n]$ . Then

(4.4) \begin{equation} \mathbb{E}[Y(t)]= m_0+c\mathbb{E}[Q(N(t))]. \end{equation}

As it was proved in Lemma 4.1 that $ Q(n)=\sum_{k=0}^{n-1}Q_{k}\sim Kn^{\delta }/\delta $ as $n\rightarrow \infty$ , Lemma 3.1(i) implies $\mathbb{E}[Q(N(t))]\sim \mu ^{-\delta }Kt^{\delta }/\delta$ as $t\rightarrow \infty$ . Since $c/\delta =\nu /2$ , the first statement in (ii) follows from (4.4). Note that

\begin{equation*}\operatorname{Var} [Y(t)]=\mathbb{E}[\mathbb{E}[Y(t)^2\mid N(t)]]-\mathbb{E}[Y(t)]^2 \\ = \sum_{n=0}^\infty \mathbb{E}[Z_n^2]\mathbb{P}(N(t)=n)-\mathbb{E}[Y(t)]^2. \end{equation*}

Now denote $w_0=\mathbb{E}[Z_0^2]$ and $w_n= \mathbb{E}[Z_n^2]-\mathbb{E}[Z_{n-1}^2]$ , $n=1,2,\ldots.$ Note that $w_n\geq 0$ and let us define $W(x)=\sum_{k=0}^{{\lfloor x \rfloor}} w_k=\mathbb{E}[Z_n^2]$ for $n\leq x<n+1$ , $n=1,2,\ldots,$ and $W(x)=0$ for $0\leq x<1$ . Consequently $\operatorname{Var} [Y(t)]= \mathbb{E}[W(N(t))]-\mathbb{E}[Y(t)]^2$ . Applying Lemma 4.1(iii) with $0<\delta <1$ , we have $W(t)\sim (K\nu^2/2(\delta+1))t^{\delta +1}$ as $t\to\infty$ . Moreover, since $W(t)\uparrow\infty$ as $t\to\infty$ , using Lemma 3.1(i) and the first statement in part (ii) of this theorem, we obtain $\operatorname{Var} [Y(t)]\sim (K\nu^2/2(\delta+1))\nu^{-(\delta+1)}t^{\delta +1}$ as $t\to\infty$ . This completes the proof of (ii).

(iii) Let

$${\Phi _t}(x) = \mathbb P({{Y(t)} \over {t/\mu }} \le x|Y(t) > 0),\quad x \ge 0.$$

Now by equation (3.2) we have

\begin{equation*} \Phi _{t}(x)= \int_{0}^{\infty }\mathbb{P}\biggl( Z_{\lfloor{y}\rfloor}\leq \dfrac{x t}{\mu} \Bigm|Z_{\lfloor{y}\rfloor}>0\biggr) \,{\textrm{d}}_{y}\mathbb{P}(N(t)\leq y\mid Z_{N(t)}>0). \end{equation*}

Then, after the substitution $y=ut/\mu $ , we obtain

(4.5) \begin{equation} \Phi _{t}(x)=\int_{0}^{\infty }\mathbb{P}\biggl( \dfrac{Z_{\lfloor{ut/\mu}\rfloor}}{\lfloor{ut/\mu}\rfloor} \leq \dfrac{xt}{\mu \lfloor{ut/\mu }\rfloor}\Bigm|Z_{\lfloor{ut/\mu }\rfloor}>0\biggr)\, {\textrm{d}}_{u} \mathbb{P}\biggl(\dfrac{\mu N(t)}{t }\leq u\Bigm|Z_{N(t)}>0\biggr). \end{equation}

Hence, from this latter equality, Lemma 4.1(ii), (3.3), and the generalized Lebesgue dominated convergence theorem (see Theorem 2.4 in [Reference Serfozo23]), we obtain

\begin{equation*}\lim_{t\rightarrow \infty }\Phi _{t}(x)=\int_{0}^{\infty }\Gamma _{\nu /2,1}\biggl(\dfrac{x}{u}\biggr)\,{\textrm{d}} V_{1}(u)=\Gamma _{\nu /2,1}(x), \quad x\geq 0,\end{equation*}

which proves (iii).□

Now we consider the case $\mu=\infty$ . The appropriate normalization factor of $\{Y(t),\, t\geq 0\}$ is $a(t)=(\Gamma(1-\rho)(1-G(t)))^{-1}$ introduced in Section 3. Let $\mathcal{L}$ be the s.v.f. introduced in (3.4).

Proof. (i) From (4.2) we have

(4.6) \begin{equation} \mathbb{P}(Y(t)>0)=\sum_{n=0}^{\infty }\mathbb{P}(N(t)=n)Q_{n}= \mathbb{E}[Q_{N(t)}]. \end{equation}

Moreover, since (3.4) implies ${N(t)}/{a(t)}\overset{d}{\rightarrow }\Lambda _{\rho }$ , with $\overset{d}{\rightarrow }$ denoting the convergence in distribution (see Section 3), we can apply equation (7.10) from [Reference Mitov, Mitov and Yanev21] to obtain that, as $t\rightarrow \infty$ ,

(4.7) \begin{equation} \mathbb{P}(Y(t)>0)\sim Q_{\lfloor a(t)\rfloor }\mathbb{E}[\Lambda _{\rho }^{-(1-\delta)}]. \end{equation}

Note that from (3.4) and Lemma 4.1(i) we have, as $t\rightarrow \infty$ ,

\begin{equation*}Q_{\lfloor a(t)\rfloor }\sim Kt^{-\rho (1-{\delta})}\mathcal{L}^{1-{\delta}}(t).\end{equation*}

On the other hand (3.5) yields

(4.8) \begin{equation} \mathbb{E}[\Lambda _{\rho }^{-(1-\delta)}]=\dfrac{\Gamma (\delta)}{\Gamma (1-\rho +\delta\rho )}, \end{equation}

which proves (i).

(ii) Under assumption (3.4) we obtain (see [Reference Feller4, Ch. VIII.9]) that, as $t\to \infty$ ,

(4.9) \begin{equation} m(t)=\int_0^t (1-G(x))\,{\textrm{d}} x \sim \dfrac{\mathcal{L}(t)t^{1-\rho }}{\Gamma (2-\rho )}. \end{equation}

From (4.9) we have $t/m(t)\sim \Gamma (2-\rho )t^{\rho }/\mathcal{L}(t)$ as $t\rightarrow \infty$ . Moreover, it was proved in Lemma 4.1 that $Q(n)= \sum_{k=0}^{n-1}Q_{k}\sim Kn^{\delta }/\delta$ as $n\rightarrow \infty$ . Therefore

\begin{equation*} Q(t/m(t))\sim \dfrac{K\Gamma ^{\delta }(2-\rho )}{\delta \mathcal{L}^{\delta }(t)}t^{\rho \delta }\quad \text{as $ t\rightarrow \infty$.}\end{equation*}

Then, from the latter and by Lemma 3.2, we obtain, as $t\rightarrow \infty$ ,

\begin{equation*}\mathbb{E}[Q(N(t))]\sim C_{1}Q(t/m(t)), \quad \text{with }C_{1}=\dfrac{\Gamma (1+\delta )}{\Gamma (1+\rho \delta )\Gamma ^{\delta }(2-\rho )},\end{equation*}

which proves (ii).

(iii) Similarly to the proof of Theorem 4.1, by using

\begin{equation*} \Psi _{t}(x)= \mathbb{P}\biggl(\dfrac{Y(t)}{a(t)}\leq x\Bigm| Y(t)>0\biggr), \end{equation*}

Lemma 4.1(ii), and (3.5), we obtain

(4.10) \begin{equation}\lim_{t\rightarrow \infty }\Psi _{t}(x)=\int_{0}^{\infty }\Gamma _{\nu /2,1}\biggl(\dfrac{x}{u}\biggr)\,{\textrm{d}} V_{\rho ,1-\delta }(u), \end{equation}

where

\begin{equation*} V_{\rho,1-\delta}(x)=\dfrac{ \mathbb{E}[ \Lambda _{\rho }^{-(1-\delta)}I_{\{\Lambda _{\rho }\leq x\}}]}{ \mathbb{E}[\Lambda _{\rho }^{-(1-\delta)}]}. \end{equation*}

Since

\begin{equation*} {\textrm{d}} \mathbb{E}[ \Lambda _{\rho }^{-(1-\delta )}\textbf{1}_{\{\Lambda _{\rho}\leq x\}}] ={\textrm{d}} \int_{0}^{x}u^{-(1-\delta )}\,{\textrm{d}} G_{\rho }(u)= x^{-(1-\delta )}\,{\textrm{d}} G_{\rho }(x), \end{equation*}

and from (4.8), we obtain (iii).□

5. Critical branching processes with random migration and continuous time

In this section we will consider a particular case of the CBP with multitype control functions (2.3). Let $X=\{X_n(i),\, n, i=1,2,\ldots\}$ be non-negative integer-valued i.i.d. random variables and let $\eta=\{(\eta _{n,1},\eta _{n,2}),\, n=1,2,\ldots\}$ and $I=\{I_{n},\, n=1,2,\ldots\}$ be two independent sets of non-negative integer-valued i.i.d. random variables, which are independent from X. Let $\{\xi_n,\, n=1,2,\}$ be i.i.d. random variables with $\mathbb{P}(\xi _{n}=-1)=p$ , $\mathbb{P}(\xi _{n}=0)=q$ , $\mathbb{P}(\xi _{n}=1)=r$ , being $p+q+r=1$ , and independent from X, $\eta$ , and I.

Consider $D=\{1,2,3\}$ , $X_{n,1}(i)= X_{n}(i)$ , $X_{n,2}(i)= -\eta_{n,2}$ , $X_{n,3}(i)= I_{n}$ , $\phi _{n,1}(k)=\min\{k,k+\xi _{n}\cdot \eta_{n,1}\}^{+}$ , $\phi _{n,2}(k)=\xi _{n}^{-}\textbf{1}_{\{k>0\}}$ , $\phi _{n,3}(k)=\xi _{n}^{+}\textbf{1}_{\{k>0\}}$ , where we recall that $a^{+}=\max \{0,a\}$ and $a^{-}=\max \{0, -a\}$ . The CBP (2.3) with the previous specifications of the offspring distributions and the control functions can be rewritten as

(5.1) \begin{equation} Z_{0}>0,\quad Z_{n}=\biggl(\sum_{i=1}^{Z_{n-1}}X_{n}(i)+M_{n} \textbf{1} _{\{Z_{n-1}>0\}}\biggr)^+,\quad n=1, 2, \ldots, \end{equation}

where

\begin{equation*} M_n= \begin{cases} -\sum_{j=1}^{\eta_{n,1}}X_{n}(j)-\eta_{n,2} & \text{with probability p,} \\ \\[-7pt] 0 & \text{with probability q,} \\ \\[-7pt] I_{n} & \text{with probability r,} \end{cases} \end{equation*}

with $\sum_{k=1}^x=0$ , if $x\leq 0$ . This model can be interpreted as follows: in each generation a random number of families and individuals can emigrate with probability p, or new immigrants appear with probability r, or there is no migration with probability q. Zero is an absorbing state. The particular case with $\eta_{n,1}\equiv 1$ a.s. and $\eta_{n,2}\equiv 0$ a.s., for every n, is considered in detail in the monograph [Reference González, del Puerto and Yanev11]. The CBP defined by (5.1) is studied in [Reference Yanev and Yanev24] and [Reference Yanev and Yanev25].

Denote $m=\mathbb{E}[X_{1}(1)]$ , $2b=\operatorname{Var} [X_{1}(1)]$ , $\epsilon _{1}=\mathbb{E}[\eta _{1,1}]$ , $\epsilon _{2}=\mathbb{E}[\eta _{1,2}]$ , $a=\mathbb{E}[I_{1}]$ , and

\begin{equation*}{{\theta}}=\dfrac{2\mathbb{E}[M_{1}]}{\operatorname{Var} [ X_{1}(1)]}=\dfrac{ra-p(m\epsilon _{1}+\epsilon _{2})}{b}.\end{equation*}

Note that for this model the asymptotic mean growth rate, namely

\begin{align*} \lim_{k\to\infty}k^{-1}\mathbb{E}[Z_n\mid Z_{n-1}=k]&= \lim_{k\to\infty}k^{-1} \sum_{i=1}^3\mathbb{E}[X_{1,i}(1)]\mathbb{E}[\phi_{1,i}(k)]\\&= \lim_{k\to\infty}k^{-1}[ (km-\epsilon _{1}m-\epsilon _{2})p+kmq+ (km+a)r]\\&= m. \end{align*}

Thus the criticality parameter is the offspring mean, as in a BGW branching process. In this section we consider the model (5.1) in the critical case under the following condition.

Recall that in Section 2 we introduced $\Delta (s)=\mathbb{E}[s^{Z_{0}}]$ . The following results take place (see [Reference Yanev and Yanev24, Theorem 2.2] and [Reference Yanev and Yanev25, Theorem 2.1, Theorem 2.2, and Section 5]).

As in the previous section, let us denote $Q_n=\mathbb{P}(Z_n>0)$ , $n=0,1,\ldots.$

We will investigate the continuous-time process $Y(t)=Z_{N(t)}$ in Definition 2.1 by considering (5.1). Note that in this case $\{Y(t),\, t\geq 0\}$ is a CBP with CT which admits two types of emigration (family and individual) and an immigration component in the non-zero states.

Proof. (i)-(a) We will use (4.2) and (4.3) from the proof of Theorem 4.1, where now $W(t)\uparrow 1$ as $t\to\infty$ , and by Lemma 5.1(i)-(a) we have $1-W(t)=Q_{\lfloor t \rfloor}\sim K_1(t)t^{-(1-\theta)}$ as $t\to\infty$ . Therefore, applying Lemma 3.1(ii), we obtain that $\mathbb{P}(Y(t)>0)\sim \mu ^{1-\theta }(1- W(t))$ as $t\rightarrow \infty$ , which completes the proof of (i)-(a).

(i)-(b) Now introduce the function $W_{1}(x)=\mathbb{E}[Z_n]=m_{n}$ for $n\leq x<n+1$ , $n=0,1,\ldots.$ Note that in this case $W_{1}(n)=$ $\sum_{k=0}^{n}w_{k,1}$ , where $ w_{0,1}=m_{0}$ , $ w_{k,1}=m_{k}-m_{k-1}$ , $ k=1,2,\ldots.$ Therefore $\mathbb{E}[Y(t)]=\mathbb{E}[W_{1}(N(t))]$ . From Lemma 5.1(i)-(c) we have $W_{1}(t)\sim b{{\theta}}K_{1}(t)t^{{{\theta}}}$ as $t\rightarrow \infty$ , and the conditions of Lemma 3.1(i) are satisfied; then we obtain that $\mathbb{E}[Y(t)]\sim \mu ^{-{{\theta}}}W_{1}(t)$ as $ t\rightarrow \infty $ , and the first relation of (i)-(b) follows.

Similarly we can introduce the function $W_{2}(x)=\sum_{k=0}^{{\lfloor x\rfloor}} w_{k,2}=b_{n}$ for $n\leq x<n+1$ , $n=0,1,\ldots,$ where $w_{0,2}=b_{0}$ and $w_{k,2}=b_{k}-b_{k-1}$ , $k=1,2,\ldots$ ; recall that $b_n=\mathbb{E}[Z_n^2]$ . Therefore $\mathbb{E}[Y^{2}(t)] =\mathbb{E}[W_{2}(N(t))]$ . Again from Lemma 5.1(i)-(c) we have $W_{2}(t)\sim b^{2}{{\theta}}(\theta+1)K_{1}(t)t^{1+{{\theta}}}$ as $ t\rightarrow \infty $ , and by Lemma 3.1(i) we obtain $\mathbb{E}[Y^{2}(t)]\sim \mu ^{-1-\theta} W_{2}(t)$ as $ t\rightarrow \infty $ , which proves the second relation of (i)-(b).

(i)-(c) The proof is similar to that of Theorem 4.1(iii) by considering

\begin{equation*} \Phi _{t}(x)=\mathbb{P}\biggl(\dfrac{\mu Y(t)}{bt}\leq x\Bigm| Y(t)>0\biggr),\quad x\geq 0, \end{equation*}

and applying Lemma 5.1(i)-(b) and (3.3). Thus

\begin{equation*} \lim_{t\rightarrow \infty }\Phi _{t}(x)=\int_{0}^\infty {{{\mathcal{E}\biggl(\dfrac{x}{u}\biggr)}}}\,{\textrm{d}} V_{1}(u)={{\mathcal{E}(x)}},\quad x\geq 0, \end{equation*}

which proves (i)-(c).

(ii)-(a) In this case we will use (4.2) and Lemma 5.1(ii)-(b). Then we have that, as $t\to \infty$ , both $W(t)\uparrow 1$ and $1-W(t)=Q_{\lfloor t \rfloor}\sim K_{2}(t)t^{-{{\kappa}}}$ . Hence, by Lemma 3.1(ii), we obtain that $\mathbb{P}(Y(t)>0)\sim \mu ^{{{\kappa}}}(1-W(t))$ as $t\rightarrow \infty$ , which proves (ii)-(a).

(ii)-(b) It follows from equation (4.5), applying Lemma 5.1(ii)-(b) and (3.3) that

\begin{equation*} \lim_{t\rightarrow \infty }\Phi _{t}(x)=\int_{0}^{\infty }F\biggl(\dfrac{x}{u}\biggr)\,{\textrm{d}} V_{1}(u)=F(x),\quad x\geq 0,\end{equation*}

which proves (ii)-(b) in this theorem.□

We now consider the case $\mu=\infty$ . Recall that the appropriate normalization factor of $\{Y(t),\, t\geq 0\}$ is defined by $a(t)=(\Gamma(1-\rho)(1-G(t)))^{-1}$ introduced in Section 3. Recall also that $\mathcal{L}$ is the s.v.f. introduced in (3.4).

Proof. (i)-(a) As in the proof of Theorem 4.2(i), we obtain

\begin{equation*} \mathbb{P}(Y(t)>0)\sim Q_{\lfloor a(t)\rfloor}\mathbb{E}[\Lambda_{\rho }^{-(1-\theta)}]\quad\text{as $t\rightarrow \infty$.} \end{equation*}

Now, from Lemma 5.1(i)-(a) and (3.4), we have

\begin{equation*} Q_{\lfloor a(t)\rfloor }\sim K_{1}(t^{\rho }/\mathcal{L}(t))t^{-\rho (1-{{\theta}})}\mathcal{L}^{1-{{\theta}}}(t) \quad\text{as $t\rightarrow \infty$.} \end{equation*}

Therefore, by the line after (3.5), we obtain

\begin{equation*} \mathbb{E}[\Lambda _{\rho }^{-(1-\theta)}]=\dfrac{\Gamma (\theta)}{\Gamma (1-\rho +\theta\rho )}, \end{equation*}

which completes the proof of (i)-(a).

(i)-(b) From (4.9) we have that $t/m(t)\sim \Gamma (2-\rho )t^{\rho} \mathcal{L}(t)$ as $t\to \infty$ . Moreover, $W_{1}(t)\sim b{{\theta}}K_{1}(t)t^{\theta}$ as $t\to\infty$ (see the proof of Theorem 5.1(i)-(b)). Hence, as $t\rightarrow \infty$ ,

(5.3) \begin{equation} W_{1}(t/m(t))\sim b{{\theta}}K_{1}(t^{\rho }/\mathcal{L} (t))\Gamma ^{{{\theta}}}(2-\rho )t^{\rho {{\theta}}}/\mathcal{L}^{\theta}(t). \end{equation}

Since all assumptions of Lemma 3.2 are fulfilled, then, as $t\rightarrow\infty$ ,

(5.4) \begin{equation} \mathbb{E}[Y(t)]\sim C_{2}W_{1}(t/m(t)), \quad \text{with } C_{2}=\dfrac{\Gamma (1+{{\theta}})}{\Gamma (1+\rho {{\theta}})\Gamma ^{{{\theta}}}(2-\rho )}. \end{equation}

Therefore, by (5.3) and (5.4), we obtain (i)-(b).

(i)-(c) As in the proof of Theorem 4.2, denoting

\begin{equation*}\Phi _{t}(x)=\mathbb{P}\biggl(\dfrac{Y(t)}{ba(t)}\leq x\big|Y(t)>0\biggr),\quad x\geq 0,\end{equation*}

and taking into account Lemma 5.1(i)-(b) and (3.5), we have

(5.5) \begin{equation} \lim_{t\rightarrow \infty }\Phi _{t}(x)=\int_{0}^\infty{{\mathcal{E}\biggl(\dfrac{x}{u}\biggr)}}\,{\textrm{d}} V_{\rho ,1-\theta}(u) \end{equation}

and consequently (i)-(c).

(ii)-(a) The proof is similar to that in (i)-(a), applying (4.6), (4.7), and Lemma 5.1(ii)-(a).

(ii)-(b) The proof is similar to that in (i)-(c), by using Lemma 5.1(ii)-(b) and (3.5).

It is also interesting to point out that for the limiting c.d.f. $\Phi(x)$ in (5.5) we have $\Phi (x)=\mathbb{P}(\xi \eta \leq x)$ , where $\xi $ and $\eta $ are independent random variables with c.d.f. $\mathcal{E}(x)$ and $V_{\rho ,1-\theta}(x)$ , respectively. Theorem 5.2 shows that for $0<\theta<1$ , $\mu=\infty$ , and (3.4), the asymptotic behaviour of the continuous-time CBP $ \{Y(t),\, t\geq 0\}$ is quite different from that of the embedded discrete-time process $\{Z_{n},\, n=0,1,\ldots\}$ . One explanation is that the c.d.f. of the individual lifespan G(x) has a heavy tail with $\mu =\infty$ . Investigating the process $\{Y(t),\, t\geq 0\}$ in the cases ${{\theta}}\leq 0$ and ${{\theta}}\geq 1$ is an open problem.

6. Regenerative controlled branching processes with continuous time

So far we have studied models of branching processes absorbed at zero. In this section we will extend these models allowing an immigration component at zero.

Let $Y=\{Y(t),\, t\geq 0\}$ be the CBP with CT investigated in Sections 4 or 5, where $\tau =\inf\{t\colon Y(t)=0\}$ is the life period with $\mathbb{P}(\tau<\infty)=1$ and c.d.f. $B(t)=\mathbb{P}(\tau \leq t)=\mathbb{P}(Y(t)=0)$ . Assume also that $\zeta=\{\zeta_{i},\, i=1,2,\ldots\}$ are non-negative i.i.d. random variables with c.d.f. $A(x)=\mathbb{P}(\zeta_{1}\leq x)$ . The sets $\zeta$ and Y are assumed independent.

Let $Y_{k}=\{Y_{k}(t),\, t\geq 0\},\ k=1,2,\ldots$ , be the i.i.d. copies of $Y=\{Y(t),\, t\geq 0\}$ with corresponding life periods $\tau_{k}$ and c.d.f. B(x).

We will use the sequence of the random vectors $\{(\zeta_{i},\tau_{i}),\, i=1,2,\ldots\}$ to define the renewal epochs $S_{0}=0$ , $ S_{n}=S_{n-1}+\eta_{n}$ , where $\eta_{n}=\zeta_{n}+\tau_{n}$ , $n=1,2,\ldots.$ Let $\varkappa (t)=\max\{n\colon S_{n}\leq t\}$ be the corresponding renewal process. Also consider the alternating renewal epochs

\begin{equation*}\{(S_{n},\, S_{n+1}^{\ast})\colon S_{n+1}^{\ast }=S_{n}+\zeta _{n+1},\, n=0,1,\ldots\}\end{equation*}

and introduce the process $\{\sigma(t),\, t\geq 0\}$ , $ \sigma(t)=t-S_{\varkappa (t)+1}^{\ast}$ . Then the regenerative branching process $U=\{U(t),\, t\geq 0\}$ is defined as follows:

(6.1) \begin{equation} U(t)=Y_{\varkappa(t)+1}(\sigma(t)){1}_{\{\sigma(t)\geq 0\}},\quad t\geq 0. \end{equation}

Note that $\zeta$ is interpreted as a set of waiting periods. If $\sigma(t)\geq 0$ then it is called a spent lifetime, and if $\sigma (t)<0$ then $|\sigma(t)|$ is called a rest waiting time.

The process $\{U(t),\, t\geq 0\}$ develops as follows: U(t) is defined as zero during the waiting periods $S_{n-1}\leq t< S_{n}^{\ast}$ , and U(t) coincides with the process $Y_{n}(t-S_{n}^{\ast})$ during the life periods $S_{n}^{\ast}\leq t< S_{n}$ , $ n=1,2,\ldots.$

Recall that $\Delta(s)=\mathbb{E}[s^{I_{0}}]=\mathbb{E}[s^{Y(0)}]$ . Then $Y_{k}(0)$ can be interpreted as immigration components at state zero. If $Y(n)=Z_{n}$ , $ n=0,1,\ldots,$ and $\{Z_{n},\, n=0,1,\ldots\}$ is a BGW branching process, then $\{U(n),\, n=0,1,\ldots\}$ is a branching process with state-dependent immigration (i.e. immigration at zero only), well known as the Foster–Pakes model (see [Reference Mitov and Yanev19] for more details and further generalizations).

Further on we will consider the case when A(x) and B(x) are non-lattice c.d.f.s, $A(0)=B(0)=0$ , and there exists

(6.2) \begin{equation} \lim_{t\rightarrow\infty}\dfrac{1-A(t)}{1-B(t)}=C,\quad 0\leq C\leq\infty. \end{equation}

For the distribution of the waiting periods, A(x) say, we will assume one of the following conditions:

(6.3) \begin{align} m_{A}&= \mathbb{E}[\zeta_1]<\infty, \end{align}

(6.4) \begin{align} m_{A}&= \infty,\quad 1-A(x)\sim x^{-\alpha}L_{A}(x) \quad \text{as $ x\rightarrow \infty$,} \end{align}

where $1/2<\alpha\leq 1$ , $L_{A}(x)$ is an s.v.f., and for every $ h >0$ fixed, $A(t)-A(t-h)={\textrm{O}} (1/t)$ as $t\rightarrow\infty$ .

The asymptotic behaviour of U(t) is related to the asymptotic behaviour of the regeneration period. Assume that the latter is described by the following condition:

(6.5) \begin{equation} \lim_{t\rightarrow \infty }\mathbb{P}\biggl(\dfrac{Y_{k}(t)}{M(t)}\leq x\Bigm|\tau _{k}>t\biggr)=D(x),\quad x\geq 0, \end{equation}

where M(t) is a positive r.v.f. with exponent $\varsigma \geq 0$ and D(x) is a proper c.d.f.

Note that Lemma 6.1 follows by Theorem 2.1 (Basic Regeneration Theorem) in [Reference Mitov and Yanev19].

Referring to the limiting results in Theorems 4.1–4.2, we are now in a position to apply Lemma 6.1. Let $\{U(t),\, t\geq 0\}$ be the regenerative branching process defined by (6.1), where $\{Y(t),\, t\geq 0\}$ is the CBP with CT investigated in Section 4. The following result holds.

Proof. Note that by Theorem 4.1(i), by denoting $L_\delta(t)=K\mu^{1-\delta}$ , and (6.4), we have in (6.2) that

\begin{equation*}\dfrac{1-A(t)}{1-B(t)}=(L_{A}(t)/L_{{{\delta}}}(t))t^{1-{{\delta}}-\alpha}. \end{equation*}

Therefore $C=0$ if $1-\delta<\alpha$ and $C=\infty $ if $1-\delta>\alpha$ . In the case $1-\delta=\alpha$ we have that $C=0$ if ${L_{A}(t)}/{L_\delta(t)}\rightarrow 0$ , $C=\infty$ if ${L_{A}(t)}/{L_\delta(t)}\rightarrow \infty$ , and $0<C<\infty$ if ${L_{A}(t)}/{L_\delta(t)}$ converges to a positive constant. One has $1-B(t)={\textrm{o}} (1-A(t))$ and hence $C=\infty$ in this case. Note that under the conditions of the theorem, the limiting results (i) and (iii) in Theorem 4.1 are valid. Therefore we can use the limiting distributions (6.6) and (6.7) of Lemma 6.1, where upon substituting $\beta=1-\delta$ , $M(t)=t/\mu$ , $D(x)=\Gamma _{\nu /2,1}(x)$ , and $\varsigma=1$ , we establish (6.8) and (6.9).□

Proof. The proof is similar to that of Theorem 6.1. Indeed, in the limiting distributions (6.6) and (6.7) of Lemma 6.1, we have to make the substitutions $\beta=(1-\delta)\rho$ , $M(t)=a(t)$ , ${{\varsigma}}=\rho$ , and $D(x)=\Psi(x)$ .

We can also consider the process $\{U(t),\, t\geq 0\}$ defined by (6.1), where $\{Y(t),\, t\geq 0\}$ is the CBP with CT given by Definition 2.1 with (5.1), investigated in Section 5. Then, applying limiting distributions (6.6) and (6.7) of Lemma 6.1 together with Theorems 5.1 and 5.2, we can obtain counterparts of Theorems 6.1 and 6.2.

More precisely, Theorem 5.1 has to be combined with Lemma 6.1, where we have to set $\beta=1-\theta $ under the condition $0<\theta<1/2$ , $M(t)=bt/\mu$ , $D(x)=\mathcal{E}(x)$ , and $\varsigma=1$ . Similarly, Theorem 5.2 has to be combined with Lemma 6.1, where we have to set $\beta=(1-\delta)\rho$ under the condition $1/2<(1-\delta )\rho <1$ , $M(t)=a(t)$ , $D(x)=\Phi(x)$ , and $\varsigma=\rho$ .

Finally, note that some other applications of the Basic Regeneration Theorem from [Reference Mitov and Yanev19] in the theory of branching processes are given in [Reference Mitov, Mitov and Yanev21] and references therein.

7. Concluding remarks

We have introduced a new class of CBPs with CT and investigated its asymptotic behaviour in some critical cases, using the limiting distributions of the CBP with discrete time and certain transfer-type limit theorems for renewal and regenerative processes. If the mean $\mu $ of the renewal periods is finite, then the limiting behaviour of the processes with discrete and continuous time is similar. However, if $\mu $ is infinite, then the normalization of the processes is different as well as their limiting distributions.

As mentioned in the Introduction, randomly indexed BGW branching processes were successfully applied as stock price models. The new extensions introduced here open possibilities for more diverse applications. Although the randomly indexed branching processes appeared in financial mathematics, it seems that they could also be applied in cell biology studies, especially in the analysis of clonal data, PCR processes, and cell proliferation models.

The investigation of the non-critical processes as well as the critical processes under different conditions is an open problem. It would also be interesting to obtain more detailed results for some particular CBP classes. Another open problem is to define a CBP with CT along the lines of the construction in [Reference Molina and Yanev22] for two-sex branching processes.