2.1. Generating functions, MBPs, and ordering

Given an at most countable set X and a set Y, we consider a family of measures ${\boldsymbol{\mu}}=\{\mu_y\}_{y \in Y}$ defined on the (countable) measurable space $S_X\,:\!=\,\{f\colon X \to \mathbb{N}\colon |f|<+\infty\}$ equipped with the $\sigma$ -algebra $2^{S_X}$ where $|f|\,:\!=\,\sum_{x\in X}f(x)<+\infty$ ; throughout this paper we denote by $\mathbb{N}$ the set of natural numbers including 0. An interpretation is the following. Suppose that an individual marked with a label y has a random number of items to place in a space X; then $\mu_y(f)$ represents the probability that there are f(x) items placed at x (for all $x \in X$ ).

With the family $\{\mu_y\}_{y \in Y}$ , we associate the following generating function $G_{\boldsymbol{\mu}}\colon[0,1]^X \to [0,1]^Y$ :

(2.1) \begin{equation} G_{\boldsymbol{\mu}}({\mathbf{z}}|y) \, :\!= \, \sum_{f \in S_X} \mu_y(f) \prod_{x \in X} {\mathbf{z}}(x)^{f(x)},\end{equation}

where $G_{\boldsymbol{\mu}}({\mathbf{z}}|y)$ is the y coordinate of $G_{\boldsymbol{\mu}}({\mathbf{z}})$ . The family $\{\mu_y\}_{y \in Y}$ is uniquely determined by $G_{\boldsymbol{\mu}}$ (see, for instance, [Reference Bertacchi and Zucca6, Section 2.3] or [Reference Bertacchi and Zucca7, Section 2.2], and Lemma 2.1. Let $\mathbf{0}, \mathbf{1} \in [0,1]^X$ be such that $\mathbf{1}(x)\,:\!=\,1$ and $\mathbf{0}(x)\,:\!=\,0$ for all $x \in X$ . We define $\phi^{\boldsymbol{\mu}}_y(t)\,:\!=\,G_{\boldsymbol{\mu}}(t\mathbf{1}|y)$ for $t \in [0,1]$ and $y \in Y$ . Note that, if $\rho_y(n)\,:\!=\, \mu_y(\, f\colon |f|=n)$ , then $\phi_y$ is the one-dimensional generating function of $\rho_y$ . Henceforth, when it is not misleading, we drop the sub/superscripts and write G and $\phi$ . The topological properties of G are described in the following proposition, whose proof is left to the reader. In particular, we denote by $\|\mathbf{z}\|_\infty\,:\!=\,\sup_{x \in C} |\mathbf{z}(x)|$ the restriction of the norm of $l^\infty(C)$ to $[0,1]^C$ (for $C\in\{X,Y\}$ ).

A particular family of measures ${\boldsymbol{\mu}}$ is the multinomial one, which is defined given a family $(\rho_y)_{y \in Y}$ of measures on $\mathbb{N}$ and a nonnegative stochastic matrix $P=(p(y,x))_{y \in Y, x \in X}$ (where $\sum_{x \in X} p(y,x)=1$ for all $y \in Y$ ). We say that ${\boldsymbol{\mu}}$ is a multinomial family of measures if

(2.2) \begin{equation} \mu_y(f) = \rho_y\Bigg(\sum_{x\in X}f(x)\Bigg)\frac{\big(\sum_{x\in X}f(x)\big)!}{\prod_{x\in X}f(x)!} \prod_{x \in X} p(y,x)^{f(x)} \quad \text{for all}\ f \in S_X.\end{equation}

We can interpret multinomial families as measures where individuals marked with the label y draw a random number n of items (according to $\rho_y$ ) and place each of them independently in X according to the distribution $\{p(y, \cdot)\}$ . It is easy to prove that for a multinomial family ${\boldsymbol{\mu}}$ ,

(2.3) \begin{equation} G(\mathbf{z}|y)=\phi^{\boldsymbol{\mu}}_y (P\mathbf{z}(y)) \quad \text{for all}\ y \in Y, \, \mathbf{z} \in [0,1]^X,\end{equation}

where $P\mathbf{z}(y)=\sum_{x \in X} p(y,x)\mathbf{z}(x)$ . In fact, using the definitions of G and $\phi^{\boldsymbol{\mu}}$ , and (2.2), $G(\mathbf{z}|y)$ can be written as

\[ \sum_{n\in\mathbb{N}}\rho_y(n)\sum_{|f|=n}\frac{n!}{\prod_{x\in X}f(x)!}\prod_{x\in X}(p(y,x)\mathbf{z}(x))^{f(x)} = \sum_{n\in\mathbb{N}}\rho_y(n)\Bigg(\sum_{x\in X}p(y,x)\mathbf{z}(x)\Bigg)^n.\]

When $Y=X$ , the family ${\boldsymbol{\mu}}$ and its generating function $G_{\boldsymbol{\mu}}$ identify a discrete-time MBP on X, which we denote by $(X,{\boldsymbol{\mu}})$ . This is a process $\{\eta_n\}_{n \in \mathbb{N}}$ , where $\eta_n(x)$ is the number of particles alive at $x \in X$ at time n. The dynamics is as follows. A particle of generation n, at site $x\in X$ , lives one unit of time; after that, a function $f \in S_X$ is chosen at random according to the law $\mu_x$ . This function describes the number of children and their positions. The choice of f is independent for all breeding particles.

An explicit construction is the following. Given a family $\{ f_{i,n,x}\}_{i,n \in \mathbb{N}, x \in X}$ of independent $S_X$ -valued random variables such that, for every $x \in X$ , $\{f_{i,n,x}\}_{i,n \in \mathbb{N}}$ have the common law $\mu_x$ , then the discrete-time MBP $\{\eta_n\}_{n \in \mathbb{N}}$ is defined iteratively as

(2.4) \begin{equation} \eta_{n+1}(x) = \sum_{y\in X}\sum_{i=1}^{\eta_n(y)}f_{i,n,y}(x) = \sum_{y \in X} \sum_{j=0}^\infty 1_{\{\eta_n(y)=j\}} \sum_{i=1}^{j} f_{i,n,y}(x)\end{equation}

starting from an initial condition $\eta_0$ . The actual canonical construction of $\{\eta_n\}_{n \in \mathbb{N}}$ can be carried on, by using Kolmogorov’s theorem, in such a way that the probability space and the process $\{\eta_n\}_{n \in \mathbb{N}}$ are fixed, while the probability measure depends on the starting configuration and the family ${\boldsymbol{\mu}}$ .

When the initial configuration is $\eta\in\mathbb{N}^X$ , the corresponding law of $\{\eta_n\}_{n \in \mathbb{N}}$ is denoted by $\mathbb{P}^\eta_{\boldsymbol{\mu}}$ and the expectation by $\mathbb{E}^\eta_{\boldsymbol{\mu}}$ . In the particular case where the initial state is one particle at x, namely $\eta=\delta_x$ almost surely (a.s.), we write $\mathbb{P}^x_{\boldsymbol{\mu}}$ and $\mathbb{E}^x_{\boldsymbol{\mu}}$ . When ${\boldsymbol{\mu}}$ is fixed, we avoid the subscript ${\boldsymbol{\mu}}$ in the above notation. Clearly, $\{\eta_n\}_{n \in \mathbb{N}}$ is a Markov chain with absorbing state $\mathbf{0}$ , the configuration with no particles at all sites. We denote by $\{\mathcal{F}_n\}_{n \in \mathbb{N}}$ the filtration associated with the process, namely $\mathcal{F}_n\,:\!=\,\sigma(f_{i,j,x} \colon i,j \in \mathbb{N},\,j < n,\,x \in X)$ . Note that $\mathcal F_0$ is the trivial $\sigma$ -algebra. By using (2.4) it is easy to see that the MBP is adapted to $\{\mathcal{F}_n\}_{n \in \mathbb{N}}$ , i.e. $\eta_n$ is $\mathcal{F}_n$ -measurable for every $n \in \mathbb{N}$ .

Given an MBP $(X,{\boldsymbol{\mu}})$ , we denote by $m_{xy}\,:\!=\,\sum_{f\in S_X} f(y)\mu_x(f)$ the expected number of children that a particle living at x sends to y. It is easy to show that $\sum_{y \in X} m_{xy}=\bar \rho_x$ , where $\bar \rho_x$ is the expected value of a random variable with distribution $\rho_x$ . We can define the diffusion matrix P whose entries p(x, y) are the average fractions of children of a particle at x, placed at y, namely $p(x,y)=m_{xy}/\bar \rho_x$ .

It is important to note that, for a generic MBP, the locations of the offspring are not (necessarily) chosen independently, but are assigned according to the function $f\in S_X$ . When the children are dispersed independently, the process is completely characterised by $\{\rho_x\}_{x\in X}$ and P; its law ${\boldsymbol{\mu}}$ is a multinomial family (see (2.2)) and the process is called an MBP with independent diffusion.

We now recall the definition of stochastic order of measures and introduce the pgf and the germ orders.

We observe that ${\boldsymbol{\mu}} \succeq {\boldsymbol{\nu}} \Rightarrow \boldsymbol{\mu} \mathbin{\ge_{\mathrm{pgf}}} \boldsymbol{\nu} \Rightarrow \boldsymbol{\mu} \mathbin{\ge_{\mathrm{germ}}} \boldsymbol{\nu}$ , but the reverse implications do not hold. Clearly $G_{\boldsymbol{\mu}}(\mathbf{z}) \le G_{\boldsymbol{\nu}}(\mathbf{z})$ if and only if $G_{\boldsymbol{\mu}}(\mathbf{z}|y) \le G_{\boldsymbol{\nu}}(\mathbf{z}|y)$ for all $y \in Y$ ; thus, $\boldsymbol{\mu} \mathbin{\ge_{\mathrm{germ}}} \boldsymbol{\nu}$ (with a certain $\delta <1$ ) if and only if $\mu_y \mathbin{\ge_{\mathrm{germ}}} \nu_y$ for all $y \in Y$ (with $\delta_y$ such that $\sup_{y\in Y}\delta_y\le \delta<1$ ).

We recall that for real-valued measures (i.e. when Y is a singleton), $\mu \succeq \nu$ is equivalent to the existence of two random variables $\eta, \zeta$ , with laws $\mu$ and $\nu$ respectively, such that $\eta\ge\zeta$ a.s. (this construction is usually referred as an ordered coupling). This result can be extended to measures on partially ordered, compact metric spaces [Reference Liggett18, Theorem 2.4] and to measures on partially ordered Polish spaces (see, for instance, [Reference Kamae, Krengel and O’Brien17, Theorem 1]). It is not difficult to show that $\mathbb{R}^X$ , with a suitable finite metric, is a partially ordered Polish space.

Proposition 2.2 shows that $\ge_{\mathrm{germ}}$ is a partial order; the proof is based on the following technical lemma whose proof can be found in the Appendix.

Proof. The relation $\ge_{\mathrm{germ}}$ is clearly reflexive and transitive. Let us prove it is antisymmetric. Suppose that $\boldsymbol{\mu} \mathbin{\ge_{\mathrm{germ}}} \boldsymbol{\nu}$ and $\boldsymbol{\nu} \mathbin{\ge_{\mathrm{germ}}} \boldsymbol{\mu}$ , i.e. there exists $\delta<1$ such that, for all $\mathbf{z} \in [\delta,1]^X$ , $G_{\boldsymbol{\mu}}(\mathbf{z})=G_{\boldsymbol{\nu}}(\mathbf{z})$ ; we prove that ${\boldsymbol{\mu}}={\boldsymbol{\nu}}$ (this is equivalent to $G_{\boldsymbol{\mu}}=G_{\boldsymbol{\nu}}$ , as discussed in Section 2.3). It is enough to prove that $\mu_y=\nu_y$ (or equivalently that $G_{\boldsymbol{\mu}}(\! \cdot \! |y)=G_{\boldsymbol{\nu}}(\! \cdot \! |y)$ ) for every fixed $y \in Y$ . To this end, note that (2.1) defines a continuous function G on $D^X$ where $D\,:\!=\,\{z\in\mathbb{C}\colon|z|\le1\}$ is the closed disk of radius 1 in the complex plane. Hence, when X is finite, for every fixed $y \in X$ , the generating function $G(\cdot|y)$ can be seen as a holomorphic function of several variables. In this case the result follows from Lemma 2.1; indeed, since $G_{\boldsymbol{\mu}}(\cdot|x)-G_{\boldsymbol{\nu}}(\cdot|x)$ vanishes on $[\delta,1]^X$ , by Lemma 2.1 it vanishes on $D^X$ . Therefore, $\mu_y(f)-\nu_y(f)=0$ for every $f \in S_X$ . Now let X be infinite; given a subset $W \subseteq X$ , define $V(W)\,:\!=\,\{\mathbf{z} \in [0,1]^X \colon \mathbf{z}(x)=1$ for all $x \in X \setminus W\}$ and let $\pi\colon V(W) \mapsto [0,1]^W$ be the bijective map defined as $\pi(\mathbf{z})\,:\!=\,\mathbf{z}|_W$ (the restriction of $\mathbf{z}$ to W). Given $f \in S_X$ , define $\langle f \rangle_W\,:\!=\,\{g \in S_X \colon g|_W=f|_W\}$ , the set of functions extending the restriction of f to W; moreover, define $S_X(W)\,:\!=\,\{f \in S_X\colon \{f>0\} \subseteq W\}\}$ , the set of finitely supported functions on X whose support is in W. Clearly, since $x \to f(x) 1_{(x \in W)}$ is a map in $S_X(W)$ and $\langle f\rangle _W=\langle f(\! \cdot \!) 1_{(\cdot \in W)} \rangle _W$ , then the map $f \mapsto \langle f \rangle _W$ is a bijection from $S_X(W)$ onto $\{\langle f \rangle _W \colon f \in S_X\}$ . Roughly speaking, the $\langle f \rangle _W$ are equivalence classes containing exactly one function $g \in S_X(W)$ , and since every $g \in S_X(W)$ belongs to a class, there is a one-to-one correspondence between $S_X(W)$ and $\{\langle f \rangle _W \colon f \in S_X\}$ . We observe now that $G_{\boldsymbol{\mu}}(\cdot|x)|_{V(W)}$ and $G_{\boldsymbol{\nu}}(\cdot|x))|_{V(W)}$ can be seen as functions defined on $[0,1]^W$ ; indeed, $G_{\boldsymbol{\mu}}(\pi^{-1}(\! \cdot \!)|y)|_{V(W)}=G_{\boldsymbol{\mu}}(\pi^{-1}(\! \cdot \!)|y)$ and $\pi^{-1}$ is a bijection from $[0,1]^W$ onto V(W) (and the same holds for ${\boldsymbol{\nu}}$ ). More precisely,

\begin{align*} G_{\boldsymbol{\mu}}(\pi^{-1}(\mathbf{z})|y) = \sum_{f \in S_X(W)} \mu_x(\langle f \rangle _W)\prod_{w \in W} \mathbf{z}(w)^{f(w)}, \end{align*}

and an analogous expression holds for $G_{\boldsymbol{\nu}}$ . Suppose that W is finite; since $G_{\boldsymbol{\mu}}(\pi^{-1}(\! \cdot \!)|y)=G_{\boldsymbol{\nu}}(\pi^{-1}(\! \cdot \!)|y)$ on $[\delta,1]^W$ , the same equality holds on V(W) (by Lemma 2.1). This easily implies that $\mu_y(\langle f \rangle _W)=\nu_y(\langle f \rangle _W)$ for every $f \in S_X(W)$ or, equivalently, for every $f \in S_X$ . Consider now a fixed sequence of finite subsets of X, say $\{W_n\}_{n \in \mathbb{N}}$ , such that $W_n \subseteq W_{n+1}$ and $\bigcup_{n \in \mathbb{N}} W_n=X$ . Then, for all $f \in S_X$ we have $\langle f \rangle _{W_{n+1}}\subseteq \langle f \rangle _{W_n}$ and $\bigcap_{n \in \mathbb{N}}(\langle f \rangle _{W_n})=\{f\}$ , therefore

\begin{align*} \mu_y(f) = \lim_{n\to+\infty}\mu_y(\langle f\rangle_{W_n}) = \lim_{n\to+\infty}\nu_y(\langle f\rangle_{W_n}) = \nu_y(f). \\[-40pt] \nonumber \end{align*}

If $\# X=\# Y=1$ then, among the generating functions which admit a holomorphic extension in a neighbourhood of 1, $\ge_{\mathrm{germ}}$ defines a total order. Indeed in this case, if there is no $\delta\in[0,1)$ such that $G_{\boldsymbol{\mu}}(z) < G_{\boldsymbol{\nu}}(z)$ for all $z\in (\delta,1)$ or $G_{\boldsymbol{\mu}}(z) > G_{\boldsymbol{\nu}}(z)$ , then the two functions coincide by [Reference Rudin24, Theorem 10.18]. If X or Y has at least cardinality 2, then $\ge_{\mathrm{germ}}$ is not a total order, even if restricted to generating functions with holomorphic extensions. Indeed, if $\#Y \ge 2$ , then in general $\ge_{\mathrm{germ}}$ is not a total order. For instance, suppose that $\{a,b\}\subseteq Y$ with $\rho_{a}(t)>\rho_{b}(t)$ for all $t\in(\delta,1)$ . Let $G_{\boldsymbol{\mu}}(z)=\binom{\rho_a(z)}{\rho_b(z)}$ and $G_{\boldsymbol{\nu}}(z)=\binom{\rho_b(z)}{\rho_a(z)}$ : these two generating functions are not $\ge_{\mathrm{germ}}$ -comparable. For the case $\#Y=1$ and $\#X=2$ , see Example 2.1.

Note that the MBPs discussed in [Reference Hutchcroft15] are described by a multinomial family of measures, where $\rho_y \equiv \rho$ does not depend on y. The definition of germ order in [Reference Hutchcroft15] concerns the generating function of $\rho$ . We can extend this definition to general multinomial families by comparing, site by site, the corresponding generating function of $\rho_y$ . This is not our definition (which is based on the multidimensional generating function), but in the case of multinomial measures the two definitions are equivalent as the following proposition shows.

Proof of Proposition 2.3. For $(\mathrm{i}) \Rightarrow (\mathrm{ii})$ , using the hypothesis and the expression for $\phi_y$ , we get that for every $t \in [\delta,1]$ and for all $x \in X$ , since $t \mathbf{1} \in [\delta,1]^X$ ,

\begin{align*} \phi_y^{\boldsymbol{\mu}}(t) = G_{\boldsymbol{\mu}}(t\mathbf{1}|y) \le G_{\boldsymbol{\nu}}(t\mathbf{1}|y) = \phi_x^{\boldsymbol{\nu}}(t). \end{align*}

For $(\mathrm{ii}) \Rightarrow (\mathrm{i})$ , recall that, for multinomial families, the generating functions are $G_{\boldsymbol{\mu}}(\mathbf{z}|y)=\phi_y^{\boldsymbol{\mu}}(P\mathbf{z}(y))$ and $G_{\boldsymbol{\nu}}(\mathbf{z}|y)=\phi_y^{\boldsymbol{\nu}}(P\mathbf{z}(y))$ . We observe that the map $\mathbf{z} \mapsto P\mathbf{z}$ is nondecreasing and continuous from $[0,1]^X$ into $[0,1]^Y$ ; in particular, if $\mathbf{z} \in [\delta,1]^X$ , then $P\mathbf{z} \in [\delta,1]^Y$ . Indeed, $Pt\mathbf{1}=t\mathbf{1}$ , therefore $\delta\mathbf{1}=P\delta\mathbf{1} \le P \mathbf{z} \le P \mathbf{1}=\mathbf{1}$ . Take $\mathbf{z} \in [\delta,1]^X$ ; then, for all $y \in Y$ ,

\begin{align*} G_{\boldsymbol{\mu}}(\mathbf{z}|y)=\phi_y^{\boldsymbol{\mu}}(P\mathbf{z}(y)) \le \phi_y^{\boldsymbol{\nu}}(P\mathbf{z}(y))= G_{\boldsymbol{\nu}}(\mathbf{z}|y), \end{align*}

where we used the inequality $\phi_y^{\boldsymbol{\mu}}(t) \le \phi_y^{\boldsymbol{\nu}}(t)$ for $t=P\mathbf{z}(y) \in [\delta,1]$ (due to the monotonicity of P).

If the families are not multinomial, then Proposition 2.3(ii) does not imply that there is germ order, even when X and Y are finite, as the following example shows.

Example 2.1. Let $X=\{1,2\}$ , $Y=\{a\}$ , and

\begin{align*} G_{\boldsymbol{\mu}}(z_1,z_2|a) \, :\!= \, \frac56 z_1z_2 +\frac16, \qquad G_{\boldsymbol{\nu}}(z_1,z_2|a) \, :\!= \, \frac45\bigg(\frac{5z_1+z_2}6\bigg)^2 + \frac15. \end{align*}

Clearly $\phi_a^{\boldsymbol{\mu}}(t) \le \phi_a^{\boldsymbol{\nu}}(t)$ for all $t \in [0,1]$ . In fact, for all $t \in [0,1)$ we have

\begin{align*} G_{\boldsymbol{\mu}}(t,t|a) = \frac56 t^2 +\frac16 < \frac45 t^2 +\frac15 = G_{\boldsymbol{\nu}}(t,t|a), \end{align*}

while $G_{\boldsymbol{\mu}}(1,1|a)=G_{\boldsymbol{\nu}}(1,1|a)=1$ . Nevertheless,

\begin{align*} G_{\boldsymbol{\mu}}(t,1|a)=\frac{5t}6 +\frac16 > \frac{(5t+1)^2}{45}+\frac15=G_{\boldsymbol{\nu}}(t,1|a) \end{align*}

for all $t \in \big(\frac{1}/{10},1\big)$ ; thus, $G_{\boldsymbol{\mu}}$ and $G_{\boldsymbol{\nu}}$ are incomparable.

2.2. Survival and extinction of MBPs

With a generic discrete-time MBP we associate a directed graph $(X,E_{\boldsymbol{\mu}})$ , where $(x,y) \in E_{\boldsymbol{\mu}}$ if and only if $m_{xy}>0$ . Whenever x and y are in the same connected component of the graph, we write $x \rightleftharpoons y$ . If the graph $(X,E_{\boldsymbol{\mu}})$ is connected, then we say that the MBP is irreducible.

In order to avoid trivial situations where particles have exactly one offspring almost surely, we henceforth assume the following.

Assumption 2.1. For all $x \in X$ there is a vertex $y \rightleftharpoons x$ such that

$$\mu_y\Bigg(f\colon \sum_{w:w \rightleftharpoons y} f(w)=1\Bigg)<1.$$

$\ $

It is important to note that, in the canonical construction, the events $\{\mathcal{E}(A), \mathcal{S}(A)\}_{A \subseteq X}$ and the corresponding random variables $\{1_{\mathcal{E}(A)}, 1_{\mathcal{S}(A)}\}_{A \subseteq X}$ are fixed and depend neither on ${\boldsymbol{\mu}}$ nor on the initial configuration $\eta$ . The dependence on ${\boldsymbol{\mu}}$ and $\eta$ is in the probability measure $\mathbb{P}_{\boldsymbol{\mu}}^x$ .

$\ $

In the rest of the paper we use the notation ${\mathbf{q}}(x,y)$ instead of ${\mathbf{q}}(x, \{y\})$ for all $x,y \in X$ . It is worth noting that, in the irreducible case, for every $A \subseteq X$ the inequality ${\mathbf{q}}(x,A)<{1}$ holds for some $x \in X$ if and only if it holds for every $x \in X$ (although it may be that $\mathbf{q}(x,A) \neq \mathbf{q}(y,A)$ for some $x \neq y$ ); moreover, in the irreducible case, $\mathbf{q}(x,A)=\mathbf{q}(x,B)$ for all finite nonempty $A,B \subseteq X$ . For details and results on survival and extinction, see, for instance, [Reference Bertacchi and Zucca5, Reference Zucca26].

Note that in [Reference Hutchcroft15] the definition of a transient set corresponds to our definition of a set where there is extinction starting from every site $x\in X$ , while the definition of a recurrent set is equivalent to our definition of a set where there is strong survival starting from every site $x\in X$ . Strong survival has been studied by many authors in the last 15 years – see, for instance, [Reference Benjamini and Müller1, Reference Bertacchi and Zucca6, Reference Gantert, Müller, Yu and Vachkovskaia13, Reference Machado, Menshikov and Popov19–Reference Menshikov and Volkov22]. There are examples of MBPs where there is non-strong survival even in some finite sets (see [Reference Bertacchi and Zucca6, Example 4.2] or [Reference Bertacchi and Zucca7, Corollaries 4.3 and 4.4]).

2.3. Extinction probabilities

As in the case of the branching process, extinction probabilities are fixed points of the generating function. The smallest fixed point is ${\mathbf{q}(X)}$ . More generally, given a solution of $G(\mathbf{z}) \le \mathbf{z}$ , then $\mathbf{z} \ge {\mathbf{q}(X)}$ . Consider now the closed sets $F_G\,:\!=\,\{\mathbf{z} \in [0,1]^X \colon G(\mathbf{z})=\mathbf{z}\}$ , $U_G\,:\!=\,\{\mathbf{z} \in [0,1]^X \colon G(\mathbf{z}) \le \mathbf{z}\}$ , and $L_G\,:\!=\,\{\mathbf{z} \in [0,1]^X \colon G(\mathbf{z})\ge \mathbf{z}\}$ ; clearly $F_G = U_G \cap L_G$ . Moreover, by the monotonicity property, $G(U_G) \subseteq U_G$ and $G(L_G) \subseteq L_G$ . The iteration of G produces sequences converging to fixed points.

The proof is straightforward (see, for instance, [Reference Bertacchi and Zucca4]). The sequence $\{\mathbf{z}_n\}_{n \in \mathbb{N}}$ defined in Proposition 2.4 converges if $\mathbf{z}_0 \in L_G$ (resp. $\mathbf{z}_0 \in U_G$ ); in that case, $\mathbf{z}_n \uparrow \mathbf{z}$ (resp. $\mathbf{z}_n \downarrow \mathbf{z}$ ) for some $\mathbf{z} \in F_G$ . We note that $\mathbf{q}(X)$ is not only the smallest fixed point of G, but also of any of its iterates $G^{(n)}$ , where $G^{(1)}\,:\!=\,G$ and $G^{(n+1)}\,:\!=\,G \circ G^{(n)}$ , for every $n \ge 1$ . Indeed, for every $n \ge 1$ , $\mathbf{q}(X)=\lim_{i \to +\infty} G^{(i)}(\mathbf{0}) = \lim_{i \to +\infty} G^{(i\cdot n)}(\mathbf{0})$ (see, for instance, [Reference Zucca26]). By Proposition 2.4, since $\mathbf{0} \in L_G$ is the smallest point of $[0,1]^X$ , the above sequence converges to the smallest fixed point of $G^{(n)}$ for all $n \ge 1$ .

It is worth mentioning that if the MBP is irreducible and X is finite, the cardinality of the set of fixed points $F_G$ and of its subset $\operatorname{ext}(G)\,:\!=\,\{\mathbf{z} \in [0,1]^X\colon \mathbf{z}=\mathbf{q}(A),\, A \subseteq X\}$ , i.e. the set of extinction probability vectors, is at most 2. When X is infinite, there are examples where these cardinalities are any finite number, countable or uncountable, even in the irreducible case [Reference Bertacchi, Braunsteins, Hautphenne and Zucca2, Reference Bertacchi and Zucca7, Reference Bertacchi and Zucca8, Reference Braunsteins and Hautphenne11].

2.4. Super/submartingales related to extinction probabilities

Given $\mathbf{z}\in[0,1]^X$ and $\mathbf{w}\in[0,+\infty)^X$ , we define $\mathbf{z}^{\mathbf{w}}\in[0,1]$ as $\mathbf{z}^{\mathbf{w}}\,:\!=\,\prod_{x\in X} \mathbf{z}(x)^{\mathbf{w}(x)}$ . Note that this infinite product always converges, being the limit of a nonincreasing sequence (for any choice of ordering of the elements in X).

The first result gives an explicit expression for the conditional expectation of the above product in terms of the generating function of the process (the proof can be found in the Appendix).

Lemma 2.2. For every $\mathbf{z}\in[0,1]^X$ , $m\ge0$ , $k\ge1$ , and for every initial condition $\eta$ , we have

\begin{align*} \mathbb E^\eta[\mathbf{z}^{\eta_{m+k}}\mid\mathcal F_m] = (G^{(k)}(\mathbf{z}))^{\eta_m}, \quad \mathbb{P}^\eta\textit{-a.s.} \end{align*}

The previous lemma and Doob’s martingale convergence theorem imply the following.

We observe that, for every $\mathbf{z} \in [0,1]^X$ , $\mathbf{z}^{\eta_n} \to 1$ on $\mathcal{E}(X)$ ; hence $W_\mathbf{z}=1$ , $\mathbb{P}^\eta$ -a.s. on $\mathcal{E}(X)$ . Moreover, if $\mathbf{z} \in L_G$ , $\mathbb{E}^\eta[W_\mathbf{z}] \ge \mathbf{z}^\eta$ and $\mathbb{E}^x[W_\mathbf{z}] \ge \mathbf{z}(x)$ ; similarly, if $\mathbf{z} \in U_G$ , $\mathbb{E}^\eta[W_\mathbf{z}] \le \mathbf{z}^\eta$ and $\mathbb{E}^x[W_\mathbf{z}] \le \mathbf{z}(x)$ . Corollary 2.1 gives the limit of the martingale $\mathbf{z}^{\eta_n}$ when $\mathbf{z}=\mathbf{q}(A)$ . The submartingale plays a crucial role in the proof of Theorem 3.1.

Proof. Note that $\mathbb{E}^\eta[ 1_{\mathcal{E}(A)} \mid \mathcal{F}_n]=\mathbf{q}(A)^{\eta_n}$ . Indeed, it is enough to note that, for every sequence $\{f_i\}_{i=1}^n$ in $S_X$ , the Markov property implies

\begin{align*} \mathbb{P}^\eta(\mathcal{E}(A)|\eta_1=f_1, \ldots, \eta_n=f_n) = \mathbb{P}^x(\mathcal{E}(A)|\eta_n=f_n) = \mathbf{q}(A)^{\,f_n}. \end{align*}

By Proposition 2.5, $\{\mathbf{q}(A)^{\eta_n}\}_{n \in \mathbb{N}}$ is a martingale and, by [Reference Williams25, Theorem 14.2] or [Reference Chung12, Theorem 9.4.8], since $1_{\mathcal{E}(A)} \in L^p(\mathbb{P}^x)$ , then $\mathbb{E}^\eta[ 1_{\mathcal{E}(A)} \mid \mathcal{F}_n] \to \mathbb{E}^\eta[ 1_{\mathcal{E}(A)} \mid \mathcal{F}_\infty] = 1_{\mathcal{E}(A)}$ , $\mathbb{P}^\eta$ -a.s. and in $L^p(\mathbb{P}^\eta)$ for all $p \ge 1$ .