2. Irreducible kernels

In this section we give a summary of the basic definitions and results presented in [Reference Nummelin9], including Theorems 2.1, 5.1, 5.2 and Propositions 2.4, 2.8, 3.4.

Consider a measurable type space ${(E,\mathcal E)}$ assuming that the $\sigma$ -algebra $\mathcal E$ is countably generated. We denote by $\mathcal M_+$ the set of $\sigma$ -finite measures $\phi$ on ${(E,\mathcal E)}$ , and write $\phi\in\mathcal M^+$ if $\phi\in\mathcal M_+$ and $\phi(E)\in(0,\infty]$ .

Definition 1. A (non-negative) kernel on ${(E,\mathcal E)}$ is a map $M\,:\,E\times\mathcal E\to [0,\infty)$ such that, for any fixed $A\in \mathcal E$ , the function $M(\cdot,A)$ is measurable and, on the other hand, $M(x,\cdot)\in\mathcal M_+$ for any fixed $x\in E$ . For a pair ${(x,A)\in(E,\mathcal E)}$ , we write $x\to A$ if

\begin{equation*}M^n(x,A)>0 \text{ for some } n\ge1.\end{equation*}

The kernel M is called irreducible if there is a measure $\phi\in\mathcal M^+$ such that, for any $x\in E$ , we have $x\to A$ whenever $\phi(A)>0$ . The measure $\phi$ is then called an irreducibility measure for M.

If measure $\phi'\in\mathcal M^+$ is absolutely continuous with respect to an irreducibility measure $\phi$ , then $\phi'$ is itself an irreducibility measure. For an irreducible kernel M, there always exists a maximal irreducible measure $\psi$ such that any other irreducibility measure $\phi$ is absolutely continuous with respect to $\psi$ .

For an irreducible kernel M with a maximal irreducible measure $\psi$ , there is a decomposition of the form (see Theorem 2.1 in [Reference Nummelin9])

(3) \begin{equation} M^{n_0} (x,A)= m(x,A) + g(x)\gamma(A) \qquad \text{for all }x\in E,\ A\in \mathcal E, \end{equation}

where $\gamma$ is an irreducibility measure for M, g is a measurable non-negative function such that $\int g(x)\psi({\text{d}} x)>0$ , m is a another kernel on ${(E,\mathcal E)}$ , and $n_0$ is a positive integer number.

Definition 2. If (3) holds with $n_0=1$ , so that

(4) \begin{equation} M (x,A)= m(x,A) + g(x)\gamma(A),\qquad x\in E,\ A\in \mathcal E, \end{equation}

then the kernel M is said to have an atom ${(g,\gamma)}$ .

Given (3), put

(5) \begin{equation} F(s)= \sum_{n=1}^\infty F_ns^{n},\qquad F_n= \iint g(y)M^{n-1}(x,{\text{d}} y)\gamma({\text{d}} x).\end{equation}

Definition 3. Define the convergence parameter $R\in[0,\infty)$ of the irreducible kernel M by

\begin{equation*}F(s)<\infty \text{ for } s<R, \qquad F(s)=\infty \text{ for } s>R.\end{equation*}

If $F(R)<\infty$ , then the kernel M is called R-transient; if $F(R)=\infty$ , then the kernel M is called R-recurrent.

Definition 4. A non-negative measurable function h which is not identically infinite is called R-subinvariant for M if

\begin{equation*}h(x)\ge R\int h(y)M(x,{\text{d}} y), \qquad \text{for all } x\in E.\end{equation*}

An R-subinvariant function is called R-invariant if

\begin{equation*}h(x)= R\int h(y)M(x,{\text{d}} y) \qquad \text{for all } x\in E.\end{equation*}

A measure $\pi\in\mathcal M^+$ such that $\int g(y)\pi ({\text{d}} y)\in(0,\infty)$ is called R-subinvariant for M if

\begin{equation*}\pi(A)\ge R\int M(x,A)\pi({\text{d}} x) \qquad \text{for all } A\in \mathcal E.\end{equation*}

An R-subinvariant measure is called R-invariant if

\begin{equation*}\pi(A)= R\int M(x,A)\pi({\text{d}} x) \qquad \text{for all } A\in \mathcal E.\end{equation*}

Suppose M is R-recurrent. The function h and the measure $\pi$ defined by

(6) \begin{align}h(x)=\sum_{n=1}^\infty R^{nn_0}\int g(y)m^{n-1}(x,{\text{d}} y),\qquad \pi(A)=\sum_{n=1}^\infty R^{nn_0}\int m^{n-1}(x,A)\gamma({\text{d}} x) \end{align}

are R-invariant for M, scaled in such a way that

(7) \begin{align}\int h(x)\gamma({\text{d}} x)= \int g(y)\pi({\text{d}} y)=1.\end{align}

For any R-subinvariant function $\tilde h$ satisfying $\int \tilde h(x)\gamma({\text{d}} x)=1$ , we have

\begin{equation*}\tilde h=h\quad \psi\text{-everywhere}, \qquad \tilde h\ge h\quad \text{everywhere}.\end{equation*}

The measure $\pi$ is the unique R-subinvariant measure satisfying (7).

Definition 6. Kernel M has period d if d is the smallest positive integer such that there is a sequence of non-empty disjoint sets ${(D_0,D_1,\ldots D_{d-1})}$ having the following property:

\begin{equation*} \text{if } x\in D_i \text{ then } M(x,E\setminus D_j)=0 \ \text{for }j=i+1\ (\text{mod}\ d), \ i=0,\ldots, d-1.\end{equation*}

We call kernel M aperiodic if its period $d=1$ .

In the periodic case with $d\ge2$ , provided M is irreducible and satisfies (3), there is an index i, $0\le i\le d-1$ , such that $g=0$ over all $D_j$ except $D_i$ . Furthermore,

\begin{equation*}\gamma(E\setminus D_j)=0 \text{ for } j=i+n_0\ (\text{mod}\ d).\end{equation*}

3. GW processes with clusters

As will be explained later in this section, the following definition yields the abovementioned split chain construction in the particular case when ${\mathbb{P}}(\Xi^{(x)}(E)=1)=1$ .

Definition 7. Consider a multi-type GW process $\{Z_n\}_{n=0}^\infty $ whose reproduction measure can be decomposed into a sum of a random number of integer-valued random measures,

(8) \begin{equation}\Xi^{(x)}=\xi^{(x)}+\sum_{i=1}^{N^{(x)}}\tau_i. \end{equation}

Let each $\tau_i$ be independent of everything else and have a common distribution $ \tau_i\stackrel{{\text{d}}}=\tau$ .

1. (i) Such a multi-type GW process will be called a GW process with clusters.

2. (ii) Each group of particles behind a measure $\tau_i$ in (8) will be called a cluster, so that $N^{(x)}$ gives the number of clusters produced by a single particle of type x. Simple clusters correspond to the case ${\mathbb{P}}(\tau(E)=1)=1$ .

3. (iii) A multi-type GW process with the reproduction measure $\xi^{(x)}$ will be called a stem process.

Given (8) and

\begin{equation*} \mathbb{E} \xi^{(x)}(A)=m(x,A),\qquad \mathbb{E} N^{(x)}=g(x),\qquad \mathbb{E} \tau(A)=\gamma(A), \end{equation*}

by the total expectation formula we see that the kernel (1) satisfies (4). Note that we allow for dependence between $\xi^{(x)}$ and $N^{(x)}$ . Definition 7 puts no restrictions on the reproduction kernel m of the stem process. The example from Section 7 presents a case with $E=[0, \infty)$ , where the kernel m is reducible, in that for any ordered pair of types (x, y), where $x<y$ , type-x particles (within the stem process) may produce type-y particles but not otherwise.

Consider a GW process with simple clusters such that

\begin{align*}& {\mathbb{P}}\big(\xi^{(x)}(E)=0\big)=g(x),\qquad {\mathbb{P}}\big(\xi^{(x)}(E)=1\big)=1-g(x), \qquad g(x)\in[0,1],\\ &N^{(x)}=\mathbf{1}_{\{\xi^{(x)}(E)=0\}},\qquad x\in E,\\ &\tau=\delta_{Y},\qquad {\mathbb{P}}(Y\in A)=\gamma(A).\end{align*}

In this case, each particle produces exactly one offspring, and the GW process tracks the type of the regenerating particle. Using (4), we find that $M=P$ is a stochastic kernel satisfying (2) with

\begin{equation*}p(x,A)=(1-g(x)){\mathbb{P}}\big(\xi^{(x)}(A)=1 \mid \xi^{(x)}(E)=1\big).\end{equation*}

As a result we get a split chain corresponding to a stochastic kernel. Notice that the associated stem process is a pure death multi-type GW process.

An important family of GW processes with simple clusters is formed by linear-fractional multi-type GW processes; see [Reference Lindo and Sagitov7, Reference Sagitov11]. This family is framed by the following additional conditions:

\begin{align*}& {\mathbb{P}}\big(\xi^{(x)}(E)=0\big)+ {\mathbb{P}}\big(\xi^{(x)}(E)=1\big)=1,\\ &N^{(x)}=N\cdot \mathbf{1}_{\{\xi^{(x)}(E)=1\}},\qquad \text{where \textit{N} has a geometric distribution},\\ &\tau=\delta_{Y}.\end{align*}

In this case, the relation (4) holds with

\begin{equation*}m(x,A)={\mathbb{P}}\big(\xi^{(x)}(A)=1\big),\qquad g(x)=\mathbb{E} N\cdot {\mathbb{P}}\big(\xi^{(x)}(A)=1\big),\qquad \gamma(A)={\mathbb{P}}(Y\in A).\end{equation*}

Here again, the stem process is a pure death multi-type GW process.

4. Embedded CMJ process

The key assumption of Definition 7 guarantees that the procreation of particles constituting a cluster is independent of the other parts of the GW process with clusters. The main idea of this paper is to treat each cluster as a newborn CMJ individual, which recalls the construction of macro-individuals in the sibling dependence setting of [Reference Athreya and Ney10].

Consider the stem process starting from a single cluster at time 0, and denote by $L\in[1,\infty]$ its extinction time. Put $X_0=1$ and let $X_n$ stand for the number of new clusters generated at time n by the particles in the stem process born at time $n-1$ , $n\ge1$ . Observe that

\begin{align*}f_n&\,:\!= \mathbb{E} (X_n)=\iint g(y)m^{n-1}(x,{\text{d}} y)\gamma({\text{d}} x).\end{align*}

We treat the random vector ${(X_1,\ldots, X_{L})}$ as the life record of the initial individual in an embedded CMJ process, see Figure 1. A CMJ individual during its life of length L at different ages produces random numbers of offspring, cf. [Reference Jagers and Sagitov6]. Such independently reproducing CMJ individuals build a population with overlapping generations (in contrast to GW particles living one unit of time, so that there is no time overlap between generations).

Figure 1.

Embedding a CMJ individual into a multi-type GW process stemming from a single cluster of size $Z_0=3$ . Left: Solid lines represent the lineages of the stem process, which dies out by time $L=6$ . Dashed lines delineate the daughter clusters directly generated by the stem process. Dashed lines that terminate at times $n+\tfrac{1}{2}$ represent clusters with $\tau_i=0$ . We see that $X_1=3$ with $\tau_1(E)=0$ , $\tau_2(E)=1$ , $\tau_3(E)=3$ . Right: Summary of the individual life ${(X_1,\ldots,X_{L})=(3,2,2,0,2,1)}$ . The thicker line terminates at the time of extinction of the stem process. The dashed lines connect to the daughter clusters generated by the stem process.

Throughout this paper we assume that

(9) \begin{equation}f(s_0)\in( 0 , \infty) \text{ for some }s_0>0, \text{ where }f(s)= \sum_{n=1}^\infty f_ns^{n},\end{equation}

so that, on one hand, $ f_n>0$ for some $n\ge1$ , and on the other hand, the radius of convergence $r= \inf \{ s \ge 0 \colon f(s) = \infty \}$ is positive. The assumption $r>0$ prohibits very fast growing sequences of the type $f_n=\text{e}^{n^2}$ .

Definition 8. Given (9), define a parameter $R\in(0,\infty)$ as $R=r$ if $f(r)<1$ , and as the unique positive solution of the equation $f(R)=1$ if $f(r)\ge1$ .

Since $f(R)\le1$ , the sequence ${(f_nR^{n})}$ can be viewed as a (possibly defective) distribution on the lattice $\{1,2,\ldots\}$ . This is the distribution of the inter-arrival time for the renewal process naturally embedded into the CMJ process defined above. The renewal process is interpreted as the consecutive ages at childbearing as one tracks a single ancestral lineage backwards in time. Given $f(R)=1$ , the mean inter-arrival time for the embedded renewal process equals

\begin{equation*}\sum_{n = 1}^\infty nf_nR^{n}=Rf'(R),\end{equation*}

and is interpreted as the average age at childbearing or the mean generation length for the CMJ process; see [Reference Jagers5].

Focusing on the current waiting time of such a discrete renewal process, we get an irreducible Markov chain with the state space $\{0, 1,\ldots\}$ . The following observation concerning this Markov chain is straightforward.

Let $W_n$ be the number of newborn individuals at time n in the embedded CMJ process started from a single newborn individual, or in other words, the total number of clusters emerging at time n in the original GW process starting from a single cluster. Clearly, see (5),

\begin{align*}F_n&\,:\!= \mathbb{E} (W_n)= \iint g(y)M^{n-1}(x,{\text{d}} y)\gamma({\text{d}} x).\end{align*}

Theorem 1. Consider a kernel M with atom ${(g,\gamma)}$ . The parameter R from Definition 8 coincides with the convergence parameter of the kernel M. Moreover,

1. (i) if $f(r)<1$ , then $R=r$ , $f(R)<1$ , and $F(R)<\infty$ , so that M is R-transient;

2. (ii) if $f(r)\ge1$ , then $f(R)=1$ and $F(R)=\infty$ , so that M is R-recurrent;

3. (iii) if $f(R)=1$ , then either $f'(R)=\infty$ so that M is R-null recurrent, or $f'(R)\in(0,\infty)$ , so that M is R-positive recurrent.

Proof. Using the law of total expectation it is easy to justify the recursion

\begin{equation*}F_n=f_n+f_{n-1}F_1+\cdots+f_1F_{n-1}.\end{equation*}

This leads to the equality for the generating functions, see (5),

\begin{equation*}F(s)=f(s)+f(s)F(s),\end{equation*}

which yields

(10) \begin{equation} F(s)=\frac{f(s)}{1-f(s)}\qquad \text{for\ } \textit{s\ } \text{such that\ } f(s) \lt 1.\end{equation}

From here, and in view of Definition 3, it is obvious that the first statement is valid. Parts (i) and (ii) follow immediately. Part (iii) is proved in Section 5.

Remark 1. For a general starting configuration of particles $Z_0$ , putting $\mu_0=\mathbb{E} Z_0$ , we modify our formulas:

\begin{align*}\tilde f_n&\,:\!=\mathbb{E} (\tilde X_n)=\iint g(y)m^{n-1}(x,{\text{d}} y)\mu_0({\text{d}} x),\\\tilde F_n&\,:\!= \mathbb{E} (\tilde W_n)=\iint g(y)M^{n-1}(x,{\text{d}} y)\mu_0({\text{d}} x).\end{align*}

The corresponding generating functions,

\begin{equation*}\tilde f(s)= \sum_{n=1}^\infty \tilde f_ns^{n},\qquad \tilde F(s)= \sum_{n=1}^\infty \tilde F_ns^{n},\end{equation*}

are connected by

\begin{equation*} \tilde F(s) = \frac{\tilde f(s)}{1-f(s)},\qquad \text{for \textit{s} such that } f(s)<1.\end{equation*}

(To obtain this relation, observe that $\tilde F_n=\tilde f_n+\tilde f_{n-1}F_1+\cdots+\tilde f_1F_{n-1}$ , which gives $\tilde F(s)=\tilde f(s)(1+F(s))$ , and it remains to apply (10).)

As mentioned above, under the special initial condition $Z_0\stackrel{{\text{d}}}{=}\tau$ , the embedded CMJ process starts from a single newborn individual. For a general $Z_0$ , the embedded CMJ process has an immigration component characterised by the generating function $\tilde f(s)$ . By immigration we mean the inflow of new clusters generated by the stem process starting from $Z_0$ particles.

5. Null and positive recurrence of a kernel with atom

Consider a non-negative kernel M with atom ${(g,\gamma)}$ , and put

\begin{equation*}M_s(x,A)=\sum_{n=1}^\infty s^nM^{n-1}(x,A),\qquad m_s(x,A)=\sum_{n=1}^\infty s^nm^{n-1}(x,A),\quad s\ge0,\end{equation*}

so that the generating functions F and f earlier introduced can be presented as

\begin{align*}F(s)=\iint g(y)M_s(x,{\text{d}} y)\gamma({\text{d}} x),\qquad f(s)=\iint g(y)m_s(x,{\text{d}} y)\gamma({\text{d}} x).\end{align*}

Denote

(11) \begin{align}h_s(x)=\int g(y)m_s(x,{\text{d}} y),\qquad \pi_s(A)=\int m_s(x,A)\gamma({\text{d}} x),\end{align}

and observe that

\begin{align*}\int h_s(x)\gamma({\text{d}} x)= \int g(y)\pi_s({\text{d}} y)=f(s),\qquad \int h_s(y)\pi_s({\text{d}} y)=s^2f'(s).\end{align*}

The latter equality requires the following argument:

\begin{align*} \int h_s(x)\pi_s({\text{d}} x)&=\iiint g(y)m_s(x,{\text{d}} y)m_s(z,{\text{d}} x)\gamma({\text{d}} z) \\&=\iint g(y)m_s^2(z,{\text{d}} y)\gamma({\text{d}} z) =\sum_{n=1}^\infty ns^{n+1}f_{n}=s^2f'(s),\end{align*}

where we used the relation

\begin{align*}s^{-2}m_s^2(y,A)&= \int s^{-1}m_s(x,A)s^{-1}m_s(y,{\text{d}} x)=\sum_{n=0}^\infty\sum_{k=0}^\infty\int s^n m^{n}(x,A)s^km^{k}(y,{\text{d}} x)\\ &=\sum_{n=0}^\infty\sum_{k=0}^\infty s^{n+k}m^{n+k}(y,A)=\sum_{j=0}^\infty (j+1)s^{\,j}m^{\,j}(y,A).\end{align*}

Lemma 1. Consider a kernel with atom (4). If a positive s is such that $f(s)\le1$ , then the function $h_s$ and the measure $\pi_s$ defined by (11) satisfy

(12) \begin{equation} \int h_s(y)M(x,{\text{d}} y)=s^{-1}h_s(x)-(1-f(s))g(x), \end{equation}

(13) \begin{equation} \int M(y,A)\pi_s({\text{d}} y)=s^{-1}\pi_s(A)-(1-f(s))\gamma(A),\end{equation}

so that they are, respectively, an s-subinvariant function and measure for the kernel M.

Proof. By (4), we have

\begin{align*}\int m_s(y,A)M(x,{\text{d}} y)&=\sum_{n=1}^\infty s^{n}m^{n}(x,A)+g(x)\int m_s(y,A)\gamma({\text{d}} y)\\&=s^{-1}m_s(x,A)-\delta_x(A)+g(x)\pi_s(A),\end{align*}

which implies relation (12):

\begin{align*}\int h_s(y)M(x,{\text{d}} y)&=\iint g(w)m_s(y,{\text{d}} w)M(x,{\text{d}} y)=s^{-1}h_s(x)-g(x)+g(x)f(s).\end{align*}

Similarly, from

\begin{align*}\int M(y,A)m_s(x,{\text{d}} y)&=\sum_{n=1}^\infty s^{n}m^{n}(x,A)+\gamma(A)\int g(y)m_s(x,{\text{d}} y) \\&=s^{-1}m_s(x,A)-\delta_x(A)+\gamma(A)h_s(x),\end{align*}

we arrive at relation (13).

Lemma 1 yields the following statement, which in turn provides the proof of part (iii) of Theorem 1 (recall Definition 5).

Corollary 1. Consider an R-recurrent kernel M with atom ${(g,\gamma)}$ . If $f(R)=1$ , then $h=h_R$ and $\pi=\pi_R$ are an R-invariant function and a measure satisfying relation (6) with $n_0=1$ , relation (7), as well as

\begin{equation*}\int h(y)\pi({\text{d}} y)=R^2f'(R).\end{equation*}

Observe that

(14) \begin{equation} h(x)=\sum_{n=1}^\infty R^n\int g(y)m^{n-1}(x,{\text{d}} y)\end{equation}

is the expected R-discounted number of clusters ever produced by the stem process starting from a single particle of type x. From this angle, h(x) can be interpreted as the reproductive value of type x. On the other hand,

(15) \begin{equation}\pi(A)=\sum_{n=1}^\infty R^n\int m^{n-1}(x,A)\gamma({\text{d}} x).\end{equation}

is the expected R-discounted number of particles whose type belongs to A and which appear in the stem process starting from a single cluster of particles. As shown next, see Theorem 2, the measure $\pi$ can be viewed as an asymptotically stable distribution for the types of particles in the GW process with clusters.

6. Perron–Frobenius theorem for a kernel with atom

Theorem 2. Consider an aperiodic R-positive recurrent kernel M with atom ${(g,\gamma)}$ . Let h and $\pi$ be given by (14) and (15). If (x,A) are such that

(16) \begin{equation}R^{n}m^n(x,A)\to0,\qquad n\to\infty,\end{equation}

then

(17) \begin{equation} R^{n}M^n(x,A)\to \dfrac {h(x)\pi(A)}{R^2f'(R)} ,\qquad n\to\infty.\end{equation}

If $h(x)<\infty$ , then condition (16) holds for any A such that

(18) \begin{equation}A\subset\{y\,:\, g(y)\ge\varepsilon\} \text{ for some }\varepsilon>0.\end{equation}

To prove this result we need two lemmas. At the end of this section we give a remark addressing condition (16).

Lemma 2. Consider a kernel M with atom ${(g,\gamma)}$ . If $s>0$ is such that $f(s)<1$ , then

(19) \begin{align}M_s(x,A)&=m_s(x,A) + \frac{h_s(x)\pi_s(A)}{1-f(s)}\qquad \text{for all }x\in E,\ A\in\mathcal E. \end{align}

Proof. By (4), we have the recursion

\begin{align*}M^n(x,A)&=g(x)\int M^{n-1}(y,A)\gamma({\text{d}} y)+\int M^{n-1}(y,A)m(x,{\text{d}} y)\\&=g(x)\int M^{n-1}(y,A)\gamma({\text{d}} y)+\int g(y)m(x,{\text{d}} y)\int M^{n-2}(z,A)\gamma({\text{d}} z)\\&\quad +\int M^{n-2}(z,A)m^2(x,{\text{d}} z)\\ &=\sum_{i=1}^{n}\int g(y)m^{i-1}(x,{\text{d}} y)\int M^{n-i}(y,A)\gamma({\text{d}} y)+m^{n}(x,A),\end{align*}

which in terms of generating functions gives

\begin{align*}M_s(x,A)&=m_s(x,A)+h_s(x)\int M_s(y,A)\gamma({\text{d}} y) \end{align*}

and, after integration,

\begin{align*}\int M_s(x,A)\gamma({\text{d}} x)& = \frac{\pi_s(A)}{1-f(s)}.\end{align*}

Combining the last two relations we get (19). Observe also that the last formula yields (10).

Lemma 3. Let

\begin{equation*}a(s)=\sum_{n=0}^\infty a_ns^n,\qquad b(s)=\sum_{n=0}^\infty b_ns^n,\qquad c(s)=\sum_{n=0}^\infty c_ns^n\end{equation*}

be three generating functions for non-negative sequences connected by

\begin{equation*}c(s) = \frac{b(s)}{1-a(s)}.\end{equation*}

If sequence $\{a_n\}$ is aperiodic with $a(1)=1$ , $a'(1)\in(0,\infty)$ , then

\begin{equation*}c_n\to \dfrac {b(1)}{a'(1)},\qquad n\to\infty.\end{equation*}

Proof. This is a well-known result from Chapter XIII.4 in [Reference Feller3].

Proof of Theorem 2. R-positive recurrence implies $f(R)=1$ and $f'(R)\in(0,\infty)$ . Due to $f(R)=1$ , we can rewrite (19) as

\begin{equation*}M_{\hat s}(x,A)-m_{\hat s}(x,A) = \frac{b(s)}{1-a(s)},\end{equation*}

where $\hat s=sR$ and

\begin{equation*}a(s)=f(sR),\qquad b(s)=h_{\hat s}(x)\pi_{\hat s}(A),\end{equation*}

so that $a'(1)=Rf'(R)$ , $b(1)=h(x)\pi(A)$ . Applying Lemma 3, we find that, as $n\to\infty$ ,

\begin{equation*}R^{n}(M^{n}(x,A)-m^{n}(x,A))\to\dfrac {h(x)\pi(A)}{R^2f'(R)} , \end{equation*}

which, combined with condition (16), yields the main assertion. The stated sufficient condition for (16) is verified using

\begin{equation*}\sum_{n=1}^\infty R^nm^{n-1}(x,A)\le \sum_{n=1}^\infty R^n\int\mathbf{1}_{\{y:g(y)>\varepsilon\}}m^{n-1}(x,{\text{d}} y)\le\varepsilon^{-1}h(x)<\infty.\end{equation*}

Remark 2. To illustrate the role of the condition (16), consider the kernel (4) with $m(x,A)=g_1(x)\gamma_1(A)$ , assuming

\begin{equation*}\int g_1(x)\gamma_1({\text{d}} x)=a_1,\qquad \int g(x)\gamma({\text{d}} x)=a,\qquad \int g_1(x)\gamma({\text{d}} x)= \int g(x)\gamma_1({\text{d}} x)=0,\end{equation*}

where $a_1>a>0$ . In this particular case, we have

\begin{equation*}M^n(x,A)=m^n(x,A)+a^ng(x)\gamma(A),\qquad m^n(x,A)=a_1^ng_1(x)\gamma_1(A),\end{equation*}

and clearly,

\begin{equation*}M^n(x,A)\sim\left\{\begin{array}{ll} a_1^ng_1(x)\gamma_1(A), & \quad\text{if }g_1(x)\gamma_1(A)>0, \\[6pt] a^ng(x)\gamma(A), & \quad\text{if }g_1(x)\gamma_1(A)=0 \text{ and }g(x)\gamma(A)>0. \end{array}\right.\end{equation*}

Turning to the generating function defined by (5), we find that

\begin{equation*}F_n= \iint g(y)M^{n-1}(x,{\text{d}} y)\gamma({\text{d}} x)=a^{n+1},\qquad F(s) = \frac{a^2s}{1-as}.\end{equation*}

This yields $R=a^{-1}$ , and we see that condition (16) is not valid for (x, A) such that ${g_1(x)\gamma_1(A)>0}$ . On the other hand, if $g(x)<\infty$ and A satisfies (18), then

\begin{equation*}0=\int g(x)\gamma_1({\text{d}} x)\ge \int_A g(x)\gamma_1({\text{d}} x)\ge \varepsilon \gamma_1(A), \end{equation*}

so that $\gamma_1(A)=0$ and therefore $R^nM^n(x,A)\to g(x)\gamma(A)$ .

7. Three-parameter GW process with clusters

Here we construct a transparent example of a GW process with clusters having the type space $E = [0, \infty)$ . Its positive recurrent reproduction kernel is fully specified by just three parameters, $a,c\in(0,\infty)$ , and $b\in({-}1,\infty)$ :

\begin{equation*}M(x,{\text{d}} y)=ae^{x-y}1_{\{y\ge x\}}{\text{d}} y+ce^{-bx}\delta_0({\text{d}} y).\end{equation*}

This kernel satisfies (4) with

(20) \begin{equation} m(x,{\text{d}} y)=a\text{e}^{x-y}\mathbf{1}_{\{y\ge x\}}{\text{d}} y,\qquad g(x)=c\text{e}^{-bx},\qquad \gamma(A) = \delta_0(A),\end{equation}

implying that each cluster consists of a single particle of type 0.

The full specification of our example refers to a continuous-time Markov branching process modeling the size of a population of Markov particles having unit life-length mean and offspring mean a. The main idea is to count the Markov particles generation-wise, and to define the type of a GW particle as the birth-time of the corresponding Markov particle. The corresponding stem process $\{\xi_n\}_{n\ge0}$ is defined by

$\xi_n(A)=$ the number of n-generation Markov particles born in the time period A,

so that, conditionally on the parent’s birth time x,

\begin{equation*}m^n(x,[0,t])= a^n \mathbb{P}(x+T_1+\cdots+T_n\le t)= a^n \mathbb{P}(N_{t-x}\ge n), \qquad \text{for }t \gt x, \end{equation*}

where $T_i$ are independent exponentials with unit mean and $\{N_t\}_{t\ge0}$ is the standard Poisson process.

Proposition 2. Consider the multi-type GW process described above with clusters characterised by (20). Then we have

(21) \begin{align}f(s) = \frac{rcs}{r- s}, \qquad r = \frac{1+b}{a}, \qquad R = \frac{r}{1+cr}.\end{align}

The process is supercritical if $c \gt \frac{r-1}{r}$ , critical if $c = \frac{r-1}{r}$ , and subcritical if $c \lt \frac{r-1}{r}$ .

Convergence (17) holds for $A=[0,t]$ , $t\in [0,\infty)$ , with the right-hand side equal to $\text{e}^{-bx}(R\delta_0({\text{d}} y)+aR^2\text{e}^{(aR-1)y}{\text{d}} y)$ . If $Ra<1$ , then (17) holds even for $A=E$ with the right-hand side equal to $\frac{R\text{e}^{-bx}}{1-aR}$ .

Proof. Referring to the underlying Poisson process, we find that, for $s\ne1/a$ ,

\begin{align*}m_s(0,[0,t])&= s\sum _{n=0}^\infty s^n a^{n}\sum _{k=n}^\infty \mathbb{P}(N_{t}=k)= s \sum _{k=0}^\infty \mathbb{P}(N_{t}=k)\frac{ 1-(as)^{k+1}}{1-as}\\ & = \frac{s}{1-as}\big(1- as\mathbb{E}(as)^{N_t}\big)= \frac{s}{1-as}\big(1- as\text{e}^{t(as-1)}\big).\end{align*}

More generally, we have

\begin{equation*}m_s(x,[0,t])= m_s(0,[0,t-x])= \frac{s}{1-as}\big(1- as\text{e}^{(t-x)(as-1)}\big)\mathbf{1}_{\{t\ge x\}} ,\end{equation*}

so that

\begin{align*}m_s(x,{\text{d}} y)=s\delta_x({\text{d}} y)+as^2\text{e}^{(as-1)(y-x)}\mathbf{1}_{\{y\ge x\}}{\text{d}} y.\end{align*}

By (11),

\begin{align*}h_s(x)=\int g(y)m_s(x,{\text{d}} y)= sc\text{e}^{-bx}+ cas^2\int_{ 0}^\infty \text{e}^{u(as-1)}\text{e}^{-b(x+u)}{\text{d}} u=f(s)\text{e}^{-bx},\end{align*}

where f(s) satisfies (21). Since $f(r)=\infty$ , the stated value $R = \frac{r}{1+cr}$ is found from the equation $f(R)=1$ .

Applying (11) once again, we find

\begin{align*}\pi_s({\text{d}} y)=\int m_s(x,{\text{d}} y)\gamma({\text{d}} x)=m_s(0,{\text{d}} y)=s\delta_0({\text{d}} y)+as^2\text{e}^{(as-1)y}{\text{d}} y.\end{align*}

To check this and the previously obtained expressions, we verify the general formula for the integral

\begin{align*} \int h_s(x)\pi_s({\text{d}} x)&=sf(s)+ f(s)as^2\int \text{e}^{-(1+b)x}\text{e}^{asx}{\text{d}} x = \frac{rsf(s)}{r-s} = \frac{r^2s^2}{(r-s)^2}=s^2f'(s). \end{align*}

With

\begin{align*}h(x)&= \text{e}^{-bx},\qquad \pi({\text{d}} x)=R\delta_0({\text{d}} x)+aR^2\text{e}^{(aR-1)x}{\text{d}} x,\end{align*}

Theorem 2 specialised to the current example says that. for $t\in [0,\infty)$ ,

\begin{align*}R^{n}M^n(x,[0,t])&\to \text{e}^{-bx}\bigg(R+aR^2\int_0^t \text{e}^{(aR-1)y}{\text{d}} y\bigg)\\ &=\text{e}^{-bx}\bigg(R+ \frac{aR^2}{aR-1}\big(\text{e}^{(aR-1)t}-1\big)\bigg)=\text{e}^{-bx}\frac{aR^2\text{e}^{(aR-1)t}-R}{aR-1},\ n\to\infty. \end{align*}

If $aR<1$ and $A=E$ , then condition (16) holds since

\begin{equation*}\pi(E)=R + \frac{aR^2}{1-aR} = \frac{R}{1-aR}<\infty,\end{equation*}

and $R^{n}m^n(x,E)=(Ra)^n\to0$ .

Remark. If we further specialise this example by letting the stem process be the Yule process, then we have $a=2$ . If, furthermore, $b=2$ and $c \lt \frac{r-1}{r} = \frac{1}{3}$ , then the corresponding GW process with clusters is subcritical, although the total number of particles in the Yule process is infinite. The author is grateful for the constructive remarks of two anonymous reviewers addressing earlier versions of the paper.