# Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types

Mathilde André, Jean-Jil Duchamps

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Mathilde André, Jean-Jil Duchamps. Sharp $L \log L$ condition for supercritical Galton-Watson processes with countable types. 2025. hal-04984074

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Distributed under a Creative Commons Attribution 4.0 International License
Sharp $L \log L$ condition for supercritical Galton–Watson
processes with countable types

Mathilde ANDRÉ¹,² and Jean-Jil DUCHAMPS³

¹Institut de Biologie de l'ENS (IBENS), École Normale Supérieure, PSL Université, CNRS UMR 8197, INSERM U1024, Paris, France

²Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

³Marie et Louis Pasteur University, CNRS, LmB (UMR 6623), F-25000 Besançon, France

March 10, 2025

Abstract

We investigate Kesten–Stigum-like results for multi-type Galton–Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp $L \log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

Contents

<table><tr><td>1 Introduction and main results</td><td>1</td></tr><tr><td>2 Examples and counter-examples</td><td>6</td></tr><tr><td>3 Proof of Theorem 2</td><td>8</td></tr><tr><td>4 Proof of Theorem 5</td><td>13</td></tr><tr><td>References</td><td>18</td></tr></table>

1 Introduction and main results

The large-time behavior of Galton–Watson processes, both in the single-type and multi-type settings, has been thoroughly explored in various works (e.g. Harris, 1963; Athreya and Ney, 1972). However, more general settings remain less investigated. The Galton–Watson process with countably many types arises naturally when studying discretized structured population, such as ageing processes or spatially structured populations (e.g. Bansaye and Camanes, 2018; André and Duchamps, 2025). Notably, although some Kesten–Stigum limit theorems have been established for such processes, these results often assume additional conditions such as compactness of the mean reproduction operator or some kind of uniform control of the types. These assumptions fall short in most examples relevant to the study we made in André and Duchamps (in prep.), which motivated this work. To the best of our knowledge, no sharp Kesten–Stigum-like result had been provided yet for the objects we consider, namely the general supercritical countable-type recurrent Galton–Watson processes.
**Preliminary notation.** In the following, we consider a countable set $\mathcal{X}$. We will write $\mathbb{R}^{\mathcal{X}}$ and $\mathbb{N}^{\mathcal{X}}$ for the column vectors indexed by $\mathcal{X}$ with entries in respectively $\mathbb{R}$ and $\mathbb{N}$ (in this article we denote by $\mathbb{N}$ the set of nonnegative integers). The set of row vectors with entries in $\mathbb{R}$ is denoted $\mathbb{R}^{\mathcal{X}*}$. We make the distinction between row and column vectors in order to use the notation for matrix multiplication: for $f \in \mathbb{R}^{\mathcal{X}*}$ and $g \in \mathbb{R}^{\mathcal{X}}$, the inner product (when it makes sense) is $fg := \sum_{x \in \mathcal{X}} g_x f_x \in \mathbb{R}$ and $gf \in \mathbb{R}^{\mathcal{X} \times \mathcal{X}}$ is the matrix with entries $(g_x f_y)_{x,y \in \mathcal{X}}$. Note that we will only deal with vectors and matrices with nonnegative entries, so that matrix multiplication is always well-defined and associative. If $\mathcal{X}' \subset \mathcal{X}$ we write $g|_{\mathcal{X}'}$ for the vector $(g_x 1_{x \in \mathcal{X}'})_x$, so that in particular $fg|_{\mathcal{X}'} := \sum_{x \in \mathcal{X}'} f_x g_x$. We will also use the notation $e_y = (1_{x=y})_x \in \mathbb{N}^{\mathcal{X}*}$.

**The model.** We consider a multi-type Galton-Watson model, with a countable set of types $\mathcal{X}$. At each generation, each particle reproduces independently and generates an offspring vector (a random variable in $\mathbb{N}^{\mathcal{X}*}$, whose distribution depends on the type of the reproducing particle) – the sum of all these vectors make up the next generation of particles.

We write $L^{(x)} = (L_y^{(x)})_{y \in \mathcal{X}}$ for a random row vector with the type-x offspring distribution. We assume that $M_{xy} := \mathbb{E}[L_y^{(x)}]$ is finite for all $x, y \in \mathcal{X}$, and we define the mean offspring matrix as $M = (M_{xy})_{x,y \in \mathcal{X}}$. We will write $Z(n) = (Z_x(n))_{x \in \mathcal{X}}$ for the (row) population vector at time $n$, and without loss of generality, we will always assume that $Z(0)$ is deterministic. We point out the fact that the progeny of an individual is almost surely finite coordinate-wise, however their total number of children need not be. Whence the total number of individuals in the population at generation $n$, given by $\sum_{x \in \mathcal{X}} Z_x(n)$, is allowed to be infinite.

Since $M$ is a matrix with only non-negative entries, its $n$th power is well defined for all $n \ge 0$, and we assume that $M^n < \infty$ (any equality or inequality we write concerning matrices or vectors means it holds coordinate-wise) for all $n \ge 0$, so that for any $x, y \in \mathcal{X}$, starting from $Z(0) = e_x$, we have $\mathbb{E}[Z_y(n)] = M_{xy}^n < \infty$.

We summarize here some known definitions and results on the extension of Perron-Frobenius theory to countable nonnegative matrices, for which we refer to Kitchens (1998, Chapter 7, Section 1); see also Seneta (1981, Chapter 10) or Vere-Jones (1967). The matrix $M$ is said to be irreducible if the adjacency graph of $M$ (i.e. the graph with vertex set $\mathcal{X}$ and where $x \to y$ is a directed edge iff $M_{xy} > 0$) is strongly connected. It is said to be aperiodic if for all $x \in \mathcal{X}$, $\gcd\{n, M_{xx}^{(n)} > 0\} = 1$. If $M$ is irreducible the period is common to all indices and “for all” can be replaced by “for some”. Let $M$ be irreducible and aperiodic. Its Perron value, defined as $\rho := \lim_{n \to \infty} (M_{xx}^n)^{1/n} \in (0, \infty]$, is independent of $x \in \mathcal{X}$ and $\rho^{-1}$ is the radius of convergence of the power series $f^{(x,y)}: s \mapsto \sum_{n \in \mathbb{N}} M_{xy}^n s^n$, for all pairs $(x, y)$. Then $M$ is said to be recurrent if $f^{(x,x)}(\rho^{-1}) = \infty$ for some $x$ (hence all $x$), otherwise transient.

In the finite type setting and under assumptions of irreducibility and aperiodicity, the Perron-Frobenius theorem gives the mean behavior of the process, whose exponential growth is driven by the greatest eigenvalue of the mean matrix. When it comes to infinite non-negative matrices, the following proposition (Kitchens, 1998, Theorem 7.1.3) gives an analogous result.

**Proposition 1 (Extension of Perron-Frobenius theory to infinite matrices).** Assume $M$ is irreducible, aperiodic and recurrent, with a finite Perron value $\rho$. Then $\rho$ is the largest eigenvalue of $M$ and is simple. Moreover, there exists associated left and right eigenvectors $\tilde{h} \in \mathbb{R}^{\mathcal{X}*}$ and $h \in \mathbb{R}^{\mathcal{X}}$, which are positive and unique up to multiplicative constants. When $\tilde{h}h < \infty$, $M$ is said to be positive recurrent and normalizing the eigenvectors such that $\tilde{h}h = 1$, we have

$$ \lim_{n \to \infty} \rho^{-n} M^n = h\tilde{h}. $$

Otherwise (i.e. when $\tilde{h}h = \infty$), $M$ is said to be null recurrent and we have $\lim_{n \to \infty} \rho^{-n} M^n = 0$.

From now on, we work under the following assumption.

**Assumption A.** The mean offspring matrix $M$ is irreducible, aperiodic and recurrent, with a Perron value $1 < \rho < \infty$. In the case where $M$ is positive recurrent, we will always assume that the normalization $hh = 1$ holds.
Under this assumption, the mean behavior of the process is an exponential growth driven by the Malthusian parameter $\rho$.

**Related works.** Kesten-Stigum-like results typically consist in finding conditions for the non-degeneracy of the a.s. limit $W$ of the so-called additive martingale of the process (defined as $W_n := Z(n)h$), relating it to the extinction event $\text{Ext} := \{\exists n \ge 0: Z(n) = 0\}$ of the process, and studying the asymptotic type-distribution (i.e. the behavior of $\rho^{-n}Z(n)$). For branching processes with a finite set of types, under Assumption A and with $\tilde{h} = \tilde{h}_1 = 1$, it is well-known that

$$ \mathbb{E} \left[ \sum_{x,y \in \mathcal{X}} L_y^{(x)} \log_+ (L_y^{(x)}) \right] < \infty \iff \begin{cases} \mathbb{E}[W] = Z(0)h, \\ \mathbb{P}(W > 0) > 0, \\ \{W = 0\} = \text{Ext} \text{ up to a null-measure set,} \end{cases} $$

the three items on the right-hand side being equivalent. Moreover, the normalized distribution of the types in the population converges almost surely to $Wh$ as $n \to \infty$. A short proof of this result is the main result of Kurtz et al. (1997), building on the spinal change of measure developed by the same authors for single type branching processes (Lyons et al., 1995).

When it comes to broader sets of types, several studies investigated the limit behavior of the large time type-population vector in the supercritical Galton-Watson process. Limit theorems for the countable-type setting already appear in the seminal works of Harris (1963, Chapter III) and Moy (1967). Both derive extinction criterion and almost-sure convergence of $\rho^{-n}Z(n)$ under strong additional assumptions. Moy (1967) works with a second moment assumption, and Harris (1963, Chapter III) assumes the coefficients of $M$ are uniformly bounded. Some more recent work (de Raphélis, 2017; Vatutin et al., 2021) focused on the critical case. They derived Yaglom-type results, that is, convergence in distribution of the type vector conditional upon non-extinction.

Several works have focused on $L \log L$ criteria, in the countable setting or in more general multi-type settings, however, these works generally require some kind of compactness assumption or uniform bound on moments of the offspring distribution. Asmussen and Hering (1976) give a necessary and sufficient $L \log L$ condition and almost-sure convergence of the types assuming a uniformly bounded mean number of offspring over the type space, and a uniform convergence assumption of the renormalized first moment semigroup. Very recently, Bansaye et al. (2025) manage to relax these uniformity assumptions under a slightly stronger $L \log L$ condition. Kesten (1989) work under polynomial bounds for the fitness (that is, the contribution of the progeny of an individual to the demography of the population in the long run) of an individual outside a set $A_t$ that does not grow too quickly. Biggins et al. (1999) tackle $L^p$ convergence of the type distribution in a random environment, under ergodicity conditions and uniform integrability conditions. Athreya (2000) and Biggins and Kyprianou (2004) give some criteria for the uniform integrability of the additive martingale in general type spaces, involving uniform $L \log L$ moment assumptions. More recently Ligonnière (2025), in the setting of varying environment and countable types, also obtains sufficient conditions for convergence of the type distribution, under some uniform control of the second moments of the offspring vectors.

In a more specific setting, discretized Crump-Mode-Jagers processes studied in Jagers and Sagitov (2008) can also be interpreted as branching processes with a countable set of types. For these specific processes independence between the progeny produced by a same individual at distinct time allows to obtain Kesten-Stigum results and limit theorems under minimal assumptions, as is well documented in the work of Nerman (1981).

**Main results.** We seek to generalize Kesten-Stigum uniform integrability and ergodicity results to the countable type setting under minimal assumptions (namely Assumption A). We start from a population satisfying $Z(0)h < \infty$. Because $h$ is a right eigenvector for $M$, for every $n \ge 0$ we have $\mathbb{E}[Z(n)h] = \rho^n Z(0)h < \infty$, and in particular $Z(n)h < \infty$ holds almost surely. This way, the population vector at generation $n$ can be seen as a random element of the Banach space $\mathbb{B}_h = \{v \in \mathbb{R}^{\mathcal{X}^*}: \|v\|_h := \sum_x |v_x|h_x < \infty\}$. In the case where $M$ is positive recurrent, $\tilde{h}$ is a unit
vector in $\mathbb{B}_h$. In the same spirit as the additive martingale in finite multi-type branching theory, it is not hard to see that for all $n \in \mathbb{N}$,

$$W_n := \varrho^{-n}Z(n)h$$

defines a (scalar) nonnegative martingale. We denote by $W$ its almost-sure limit as $n \to \infty$. We will see that in the general setting of recurrence for $M$, a sufficient condition for $\mathbb{E}[W] = Z(0)h$ – or equivalently, for the uniform integrability of $(W_n)_n$ – is

$$\sum_{x \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ (L^{(x)}h) \log_+(L^{(x)}h) \right] < \infty. \quad (1)$$

This relates to previous works, in particular to Asmussen and Hering (1976) where the analogue of this condition is necessary and sufficient to have $P(W > 0) > 0$ in their setting. Note that (1) is not necessary in general, even when $M$ is further assumed to be positive recurrent – in this case we can express a necessary and sufficient condition that is less easily checked.

In order to state the latter condition, we need some more definition. Let us define the spinal chain, that is a Markov chain $(X_n)_{n \ge 0}$ on $\mathcal{X}$ with transition probabilities

$$p_{xy} = \frac{M_{xy}h_y}{\varrho h_x}.$$

Note that $X$ is a recurrent Markov chain when $M$ is recurrent, and $x \mapsto \tilde{h}_x h_x$ is a stationary distribution for $X$, so that positive recurrence of $M$ is equivalent to positive recurrence of $X$.

Let us define the $\sigma$-field $\mathcal{G} = \sigma(X_n, n \ge 0)$ and on the same probability space, define the sequence of nonnegative random variables $(Y_n)_{n \ge 1}$ such that:

* the $(Y_n)$ are independent conditional on $\mathcal{G}$.

* the distribution of $Y_n$ conditional on $\mathcal{G}$ and on $(X_{n-1}, X_n) = (x, y)$ is the distribution of $L^{(x)}h$ biased by $L_y^{(x)}$, that is:

$$\mathbb{E}[F(Y_n) | \mathcal{G}] \mathbb{1}_{X_{n-1}=x, X_n=y} = \frac{1}{M_{xy}} \mathbb{E}[L_y^{(x)} F(L^{(x)}h)] \mathbb{1}_{X_{n-1}=x, X_n=y}, \quad (2)$$

for any nonnegative measurable function $F$.

Note that throughout the paper we will use the phrase “A biased by B”, always in the same usual sense, referring to a random variable $A'$ with distribution given by $\mathbb{E}[F(A')] = \mathbb{E}[BF(A)]/\mathbb{E}[B]$ when this makes sense. We will also use the term test function for a nonnegative measurable function, whenever needed.

We will write $\mathbb{E}_x$ to denote expectation conditional on $\{X_0 = x\}$. The random sequence $(X_n, Y_n)$ allows us to express the sharp $L \log L$ condition in the positive recurrent case, namely:

**Theorem 2 (Uniform integrability of the additive martingale).** Fix any state $x$, and write $T_1$ for the first return time to $x$ for the spinal Markov chain $X$. Assume that the initial population satisfies $Z(0)h < \infty$ and consider the condition:

$$\mathbb{E}_x \left[ \log_+ \left( \sum_{k=1}^{T_1} \varrho^{-k} Y_k \right) \right] < \infty. \quad (3)$$

If $M$ is positive recurrent, then (3) is equivalent to any of the following (equivalent) points:

(i) $\mathbb{E}[W] = Z(0)h$;

(ii) $\mathbb{P}(W > 0) > 0$;

(iii) $\{W = 0\} = \text{LocExt up to a null-measure set, where } \text{LocExt} = \{\forall x \in \mathcal{X}, \lim_{n \to \infty} Z_x(n) = 0\}$ is the event of local extinction.
Condition (1) is sufficient, but not necessary. If M is null recurrent, then both conditions are sufficient.

**Remark 3 (Extinction of the branching process).** As highlighted before, the Kesten-Stigum theorem on branching processes with a finite set of types includes that $P(W > 0) > 0$ is equivalent to $\text{Ext} = \{W = 0\}$ up to a null-measure set, where $\text{Ext}$ is the event of extinction. This is usually checked through the immediate inclusion $\{W = 0\} \subset \text{Ext}$ and the equality $P(W = 0) \in \{\mathbb{P}(\text{Ext}), 1\}$, which is derived using a fixed point argument on the generating function of the offspring distribution (Harris, 1963).

However when the set of types is infinite the question is much more involved. As a matter of fact, the number of fixed points of the offspring generating function is not 2 but is at least countably infinite, as shown in Tan and Zhang (2024). In fact, every type could become extinct eventually while the whole population does not, and may even grow exponentially. This phenomenon of local extinction have been studied in many recent works (Braunsteins et al., 2019; Ligonnière, 2025; Kyprianou and Palau, 2018). Examples of such processes can be found in Braunsteins et al. (2019), and we give in Section 2 an instance where even under positive recurrence of the mean matrix we have $P(W > 0) > 0$ yet $P(\text{Ext}^c \cap \{W = 0\}) > 0$. Nevertheless in our countable-type setting there is a dichotomy between the events $\{W > 0\}$ and local extinction of the process.

**Remark 4.** Using the fact that $\log_+(\sum_i a_i) \le \sum_i \log(1 + a_i)$ for any sequence of nonnegative numbers $(a_i)$, we get

$$ \mathbb{E}_x \left[ \log_+ \left( \sum_{k=1}^{T_1} \varrho^{-k} Y_k \right) \right] \le \mathbb{E}_x \left[ \log_+ \left( \sum_{k=1}^{T_1} Y_k \right) \right] \le \mathbb{E}_x \left[ \sum_{k=1}^{T_1} \log(1 + Y_k) \right], $$

and writing $\pi(x) = \tilde{h}_x h_x$ for the stationary distribution of $X$, the last term above can be rewritten

$$ \begin{aligned} & \frac{1}{\pi(x)} \sum_{y \in \mathcal{X}} \pi(y) \mathbb{E}_y[\log(1 + Y_1)] \\ &= \frac{1}{\pi(x)} \sum_{y,z \in \mathcal{X}} \pi(y) \frac{M_{yz} h_z}{\varrho h_y} \cdot \frac{1}{M_{yz}} \mathbb{E}[L_z^{(y)} \log(1 + L^{(y)}h)] \\ &= \frac{1}{\pi(x)} \sum_{y \in \mathcal{X}} \tilde{h}_y \mathbb{E}[(L^{(y)}h) \log(1 + L^{(y)}h)], \end{aligned} $$

which, in the positive recurrent case, is finite only if (1) holds, showing that (1) is stronger than (3). In the null recurrent case it is not clear with this argument, but we will prove directly that (1) is sufficient for the uniform integrability of $(W_n)$.

We now turn to the large population asymptotic of the type distribution $Z(n)/\varrho^n$. Let $L^1(\Omega, \mathcal{A}, \mathbb{P}, \mathbb{B}_h)$ be the subspace of random variables $f: \Omega \to \mathbb{B}_h$ such that $\mathbb{E}[\|f\|_h] < \infty$. For reasons of brevity, we will use the notations of Pisier (2016, Chapter 1) and write $L^1(\mathbb{B}_h)$ instead of $L^1(\Omega, \mathcal{A}, \mathbb{P}, \mathbb{B}_h)$.

**Theorem 5 (Convergence of the type distribution).**

(i) In the null recurrent case, we have $\varrho^{-n}\mathbb{E}[Z_x(n)] \to 0$ as $n \to \infty$, for all $x \in \mathcal{X}$.

(ii) In the positive recurrent case, when $P(W > 0) > 0$ we have

$$ \mathbb{E} \left[ \| \varrho^{-n} Z(n) - W \tilde{h} \|_h \right] = \sum_{x \in \mathcal{X}} \mathbb{E} \left[ \left| \varrho^{-n} Z_x(n) - W \tilde{h}_x \right| \right] h_x \underset{n \to \infty}{\longrightarrow} 0, \quad (4) $$

in other words, $\varrho^{-n}Z(n) \to W\tilde{h}$ in $L^1(\mathbb{B}_h)$, and so in probability.

Otherwise, $\|\varrho^{-n}Z(n)\|_h = W_n \to 0$ almost surely as $n \to \infty$, but we do not have $\varrho^{-n}Z(n) \to 0$ in $L^1(\mathbb{B}_h)$.
(iii) In the positive recurrent case, when $\mathbb{P}(W > 0) > 0$, the convergence $\varrho^{-n}Z(n) \to W\tilde{h}$ also holds almost surely in $\mathbb{B}_h$, i.e. we have

$$ \|\varrho^{-n}Z(n) - W\tilde{h}\|_h \underset{n\to\infty}{\longrightarrow} 0 \quad a.s., \qquad (5) $$

if one of the following conditions is satisfied:

(a) We have $\sum_{x,y\in\mathcal{X}} \tilde{h}_x \mathrm{Var}(L_y^{(x)})\tilde{h}_y^{-2} < \infty$;

(b) The sufficient $L \log L$ condition (1) holds and the distribution $x \mapsto \tilde{h}_x h_x$ has finite entropy. That is,

$$ \sum_{x \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ (L^{(x)} h) \log_+ (L^{(x)} h) \right] < \infty \quad \text{and} \quad - \sum_{x \in \mathcal{X}} \tilde{h}_x h_x \log (\tilde{h}_x h_x) < \infty. $$

**Remark 6.** We will see in the proof that we actually recover a weaker condition for point (iii).(b), although it doesn't offer a clear probabilistic interpretation. As a matter of fact, in the positive recurrent case with $\mathbb{P}(W > 0) > 0$, the almost-sure type distribution convergence is ensured if

$$ \sum_{x,y \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ L_y^{(x)} \log_+ (L_y^{(x)} h_y^{-1} \tilde{h}_y^{-2}) \right] h_y < \infty. \qquad (6) $$

**Remark 7.** Condition (iii).(a) is reminiscent of the condition that can be read in Moy (1967, Theorem 1). Translated to our notation, Moy's condition reads $\sum_x \tilde{h}_x \mathbb{E}[(L^{(x)}h)^2]$, and she shows almost-sure and $L^2$ convergence of the type distribution in the positive recurrent case.

## 2 Examples and counter-examples

A convenient way to build some examples of countable-type Galton-Watson processes with specific properties consists in first fixing a spinal Markov chain, then building the rest of the branching process around this. Specifically, let us consider an irreducible, aperiodic, recurrent Markov chain $(X_n)$ with values on some countable space $\mathcal{X}$, with transition matrix denoted by $(p_{xy})$ and a stationary distribution $(\pi_x)$, assumed to be a probability distribution in the positive recurrent case. Fix any $\varrho > 1$ and any entry-wise positive $h = (h_x) \in \mathbb{R}^\mathcal{X}$, and define the matrix $M$ and the row vector $\tilde{h} \in \mathbb{R}^{\mathcal{X}*}$ via

$$ M_{xy} = \varrho \frac{p_{xy}h_x}{h_y}, \quad \text{and} \quad \tilde{h}_x = \frac{\pi_x}{h_x}. $$

Then by construction $M$ is a matrix with nonnegative entries and satisfies $Mh = \varrho h$ and $\tilde{h}M = \varrho \tilde{h}$. Also, it is not hard to see that for all $x \in \mathcal{X}$ and $n \ge 1$, we have

$$ \varrho^{-n} M_{xy}^n = \frac{h_x}{h_y} \mathbb{P}_x(X_n = y) \underset{n \to \infty}{\longrightarrow} \begin{cases} h_x \tilde{h}_y & \text{if } X \text{ is positive recurrent,} \\ 0 & \text{otherwise,} \end{cases} $$

in other words, $\varrho, h$ and $\tilde{h}$ are exactly as in Assumption A for the matrix $M$, and $X$ is the associated spinal chain. Now it remains to choose, for all $x \in \mathcal{X}$, any random vector $L^{(x)} = (L_y^{(x)})_{y \in \mathcal{X}}$ to fix our type-$x$ offspring distribution, and as long as $\mathbb{E}[L_y^{(x)}] = M_{xy}$, we have built a countable-type Galton-Watson process with a given spinal chain. Note that there is a large freedom in the last step which consist in fixing the offspring distribution, as everything that comes before only depends on the mean reproduction matrix $M$. Using this construction, we now illustrate some noteworthy facts about our results.

**Condition (1) may fail while (3) holds, even in the positive recurrent case.** Looking at Remark 4, it seems clear that the two conditions are not equivalent, but we will still provide
a simple example to show this. Consider a positive probability vector $(q_n)_{n \in \mathbb{N}}$ and define $X$ a Markov chain on $\mathcal{X} := \{0\} \cup \{(n, m) \in \mathbb{N}^2 : 1 \le m \le n\}$ with the following transition probabilities:

$$ 
\begin{array}{lll}
p_{0,(n,1)} = q_n, & \text{and } p_{(n,n),0} = 1 & \forall n \ge 1, \\
& \text{and } p_{(n,m),(n,m+1)} = 1 & \forall n > m.
\end{array}
$$

We assume $\sum_n nq_n < \infty$, so that (considering the return time at 0) $X$ is positive recurrent. It is easily seen that $\pi_0 = 1$ and $\pi_{(n,m)} = q_n$ also defines a stationary distribution for the Markov chain. We now define $h$ to be equal to (the transpose of) $\pi$, so that $\tilde{h}$ is uniformly equal to 1. Choosing any $\varrho > 1$, we see that the above construction implies $M_{(n,m),(n,m+1)} = \varrho$, and we may take $L_{(n,m+1)}^{(n,m)}$ to be Poisson distributed with mean $\varrho$ (let us write $V$ for such a random variable). This way, the sum in (1) is greater than

$$ \sum_{n \ge 1} (n-1) \times \mathbb{E}[V \log_+ V], $$

which obviously diverges. We still need to choose some other distributions for $L^{(0)}$ and $L^{(n,n)}$ to define completely the branching process, but if one chooses them in a very simple way (for instance vectors of independent Poisson random variables), it is not hard to see that the sum in (3) (taking of course 0 as the starting point) is finite if

$$ \sum_{n \ge 1} q_n \mathbb{E} \left[ \log_+ \left( \sum_{k=1}^{n-1} \varrho^{-k} (1+V_k) \right) \right] < \infty, $$

where $(V_k)$ are i.i.d. Poisson($\varrho$) random variables, and where we have used the fact that the size-biasing of $V$ (i.e. $V$ biased by $V$) is distributed as $1 + V$. Now we can crudely bound this by $\sum_n q_n \mathbb{E}[\sum_k \varrho^{-k}(1+V_k)]$ which is finite. Therefore we see that there exist examples in which (1) is far from true, while (3) easily holds.

Note that there does not seem to be a reason for (3) to be necessary in the null-recurrent case to obtain the uniform integrability of $(W_n)$, but we did not find a counter-example for this situation.

**Extinction is not characterized by $W = 0$, even in the positive recurrent case.** As pointed out in Remark 3, we do not have $\mathbb{P}(W > 0) > 0 \Rightarrow \{W > 0\} = \text{Ext}^c$ up to a null-measure set, as for Galton-Watson processes with a finite set of types. This paragraph gives a counter-example in the positive recurrent supercritical framework. This example already appears in Braunsteins et al. (2019, Section 4.1), but for the sake of completeness we still use our spinal chain construction to describe it.

Take $\varrho = 4/3$ and let $X$ be the Markov chain with values in $\mathbb{N}$ whose transitions probabilities are given by $p_{n,n+1} = p_{n,0} = 1/2$, for every $n \in \mathbb{N}$. $X$ is a positive recurrent Markov chain, $(\pi_n) = (2^{-(n+1)})$ being its stationary probability distribution. Define the vector $h$ as $h_n = (2/3)^n$, such that $\tilde{h}_n = (3/4)^n/2$ and $M_{n,n+1} = 1$, $M_{n,0} = 2 \cdot (2/3)^{n+1}$. We can now take the following offspring distributions:

$$ L^{(n)} = \delta_{n+1} + \text{Poi}(2 \cdot (2/3)^{n+1}) \delta_0 \quad \forall n \ge 1, $$

$$ L^{(0)} = \text{Poi}(1)\delta_1 + \text{Poi}(4/3)\delta_0 \quad n = 0, $$

where the Poisson random variables are independent.

It is easy to check that (1) holds, so that $\mathbb{P}(W > 0) > 0$. Theorem 2 then implies that up to a null-measure set, we have

$$ \text{Ext} \subset \text{LocExt} = \{W = 0\}, $$

and we will see that in this example the first inclusion is not an equality. Indeed, the event of seeing a type-1 offspring in the first generation whose descendants (note that the population descending from it survives forever, with exactly one descendant whose type goes to $\infty$) never produce any
type-0 individuals is included in {$W = 0$} ∩ Ext$^c$, so we have

$$
\begin{align*}
\mathbb{P}(\{W=0\} \cap \mathrm{Ext}^c) &\ge \mathbb{P}(L_1^{(0)} = 1) \prod_{n \ge 0} \mathbb{P}(L_0^{(n)} = 0) \\
&= \exp\left(-1 - 2 \sum_{n \ge 0} \left(\frac{2}{3}\right)^{n+1}\right) = e^{-5} > 0.
\end{align*}
$$

**Motivation and applications to population biology.** In this last paragraph we further illus-
trate how our result may prove relevant to some natural settings inspired from biology. In André and Duchamps (in prep)
we study a constant-sized population in which individuals make uniform infectious contacts over
time, propagating an epidemic through the population. Individuals may remain infectious for an
indefinite period of time, hence considering the entire process the cumulative number of contacts
of a given individual over time is a priori infinite.

More specifically, in order to study the spread of the disease in this general epidemic model
we study the typical lengths of geodesics in a directed graph where each edge is endowed with a
positive integer corresponding to the number of time steps from the infection of the head node
before contact with the tail node. Here, the so-called infection graph is distributed as a directed
configuration model whose set of edge lengths is countably infinite. We address this first-passage
percolation problem with a breadth-first exploration of the graph. A common approach to do so is
to construct a coupling with a branching process on the set of edges, where a progeny distributed
as the size-biased degree distribution is associated to each edge as they are traversed.

In the framework of André and Duchamps (in prep.) we recover a multitype Galton-Watson
process with an ageing dynamic and whose set of types is the subset of $\mathbb{N}^2$ given by

$$
\mathcal{X} = \{(x, a), x \in \mathbb{N}, a \le x\}.
$$

For $(x, a) \in \mathcal{X}$, $x$ is the length of the edge and $a$ their “age” – that is, the number of generations since addition to the process. The next generation is obtained from the following dynamics.

• Every edge with type $(x, a)$, $a < x$ is replaced by a child with type $(x, a + 1)$;

* Every edge with type $(x, x)$ produces a multitype juvenile progeny – whose types are in $\{(y, 1), y \in \mathbb{N}\} \subset \mathcal{X}$.

Translating in the notations of this article we have $L^{(x,a)} = \delta_{(x,a+1)}$ for $a < x$, and the type-$ (x, x)$ offspring distribution is $(L^{(x,x)}_{(y,1)})_{y \in \mathbb{N}}$, where $L^{(x,x)}_{(y,1)}$ has the distribution of the out-degree of edges with length $y$ biased by the in-degree of edges with length $x$. Kesten-Stigum criteria and ergodicity estimates for the asymptotic distribution of types enable us to prove convergence in distribution of the geodesics' lengths to the atoms of a Cox point process with a well-characterized intensity. As natural example of discrete-time epidemics where the assumptions in previous studies fail is the following: once an individual gets infected, they remain infectious for an independent, geometrically distributed period of time. While infectious, assume the number of contacts per unit of time follows a fat-tailed distribution – e.g. a power law. This allows for the presence of rare “super-spreaders” who are highly connected within the graph. Here, the mean matrix may be positive recurrent, however due to the skewness of the degree distribution the size-biased distribution lacks finite variance. Our results could find application in other contexts, such as seed bank models or delayed branching processes where countable sets of types typically appear.

3 Proof of Theorem 2

Theorem 2 puts together several classical ideas, the first of which is the so-called spinal change of
measure. In order to state this change of measure, up to enlarging our probability space, let us
assume that our branching process keeps track of the whole genealogy of individuals. Specifically,
it will be convenient to use a Neveu-Ulam-Harris notation and identify individuals with elements
of $\mathcal{N} := (\mathcal{X} \times \mathbb{N})^{\mathcal{N}}$, the set of finite sequences of elements of $\mathcal{X} \times \mathbb{N}$. Individuals of type $x$ in generation 0 are labelled $(x, 1), (x, 2), \dots, (x, Z_x(0))$; the type-$y$ children of individual $(x, i)$ are then labelled $(x, i, y, 1), (x, i, y, 2), \dots$, and so on.

With these labelling rules, we may view our branching process as a random forest of genealogical typed trees (each tree rooted at an individual in generation 0). Let us write $\mathcal{T}_n$ for the genealogy of our branching process consisting of all individuals up to generation $n$. This genealogy $\mathcal{T}_n$ can formally be seen as a random subset of $\mathcal{N}$, i.e. a random variable with values in $\mathcal{T} := \{0, 1\}^{\mathcal{N}}$ endowed with the product $\sigma$-field – a value of 1 at $v \in \mathcal{N}$ indicating the presence of individual $v$ in the genealogy, which we will write $v \in \mathcal{T}_n$.

To build $\mathcal{T}_n$ rigorously, first let $(\hat{L}^v)_{v \in \mathcal{N}}$ be independent random variables, where $\hat{L}^v \stackrel{d}{=} L^{(x)}$ if $v$ is of type $x$ – in what follows, we will use $x^v$ to denote the type of $v$. Then, we define $\mathcal{T}_n$ as the set of individuals $v \in \mathcal{N}$ satisfying that $v = (x_0, i_0, \dots, x_m, i_m)$ for some $m \le n$, and writing $v_k = (x_0, i_0, \dots, x_k, i_k)$ for the ancestor of $v$ at generation $k$:

$$i_0 \le Z_x(0) \quad \text{and} \quad i_k \le \hat{L}_{x_k}^{v_{k-1}}, \quad 1 \le k \le n.$$

With this construction, we can define $Z(n) \in \mathbb{N}^{\mathcal{X}}$ as the counts of individuals of each type at generation $n$ (i.e. the leaves of $\mathcal{T}_n$), and $(Z(n))_n$ is indeed the countable-type Galton-Watson process associated with the offspring distributions $(L^{(x)})_{x \in \mathcal{X}}$.

The spinal change of measure relies on the next lemma, which deals with random genealogies with a marked individual – note that a marked forest can be viewed as a random variable $(\mathcal{T}, v)$ with values in $\mathcal{T}^* := \mathcal{T} \times \mathcal{N}$. Before stating the result, let us recursively build the “Kesten tree” (which is technically a forest here) associated with our branching process:

* Define $(\mathcal{T}_0^*, v_0)$ as the population at time 0 (so $\mathcal{T}_0^*$ is deterministic and equal to the $\mathcal{T}_0$ defined above) with a marked vertex $v_0$, which is chosen at random proportional to $h(x^{v_0})$.

* For all $n \ge 0$, conditional on $(\mathcal{T}_n^*, v_n)$, we build $(\mathcal{T}_{n+1}^*, v_{n+1})$ as follows (each step is independent from everything else):

- First choose a random type $X \in \mathcal{X}$ by taking a step of the spinal Markov chain starting from $x^{v_n}$.

- Let $Z$ be a random vector distributed as $L^{(x^{v_n})}$ biased by $L_X^{(x^{v_n})}$.

- $\mathcal{T}_{n+1}^*$ is build from $\mathcal{T}_n^*$ by letting $v_n$ reproduce according to $Z$ (i.e. $v_n$ has $Z_x$ children of type $x$), and all other individuals $v$ at generation $n$ reproduce independently according to a copy of $L^{(x^v)}$.

- $v_{n+1}$ is chosen uniformly among the $Z_X \ge 1$ type-$X$ children of $v_n$.

**Lemma 8.** Let $(\tilde{\mathcal{T}}_n, \tilde{v}_n)$ be the random marked forest whose distribution is given by

$$\mathbb{E}[F(\tilde{\mathcal{T}}_n, \tilde{v}_n)] = \mathbb{E} \left[ \frac{1}{\rho^n Z(0)h} \sum_{v \in G_n} h(x^v) F(\mathcal{T}_n, v) \right],$$

for any test function $F: \mathcal{T}^* \to \mathbb{R}$, and where $G_n$ denotes the vertices of $\mathcal{T}_n$ living at generation exactly $n$. Then $(\tilde{\mathcal{T}}_n, \tilde{v}_n)$ has the distribution of $(\mathcal{T}_n^*, v_n)$.

*Proof.* We prove this by induction. The equality in distribution is clear for $n = 0$ by construction. We now assume it holds for some $n \ge 0$, and fix a test function $F$. For any individual $u \in \mathcal{T}_{n+1}$, we will denote by $\text{Ch}(u)$ (resp. $\text{Ch}_y(u)$) the set of children (resp. type-$y$ children) of $u$, and recall
that $L_y^u$ is the cardinal of $\mathrm{Ch}_y(u)$. We compute:

$$
\begin{align*}
& \mathbb{E} \left[ \frac{1}{\varrho^{n+1} Z(0) h} \sum_{v \in G_{n+1}} h(x^v) F(\mathcal{T}_{n+1}, v) \right] \\
&= \mathbb{E} \left[ \frac{1}{\varrho^n Z(0) h} \sum_{u \in G_n} \frac{1}{\varrho} \sum_{v \in \mathrm{Ch}(u)} h(x^v) F(\mathcal{T}_{n+1}, v) \right] \\
&= \mathbb{E} \left[ \frac{1}{\varrho^n Z(0) h} \sum_{u \in G_n} h(x^u) \sum_{y \in \mathcal{X}} \underbrace{\frac{M_{x^u y} h(y)}{\varrho h(x^u)}}_{=p_{x^u y}} \cdot \frac{L_y^u}{M_{x^u y}} \cdot \frac{1}{L_y^u} \sum_{v \in \mathrm{Ch}_y(u)} F(\mathcal{T}_{n+1}, v) \right].
\end{align*}
$$

It suffices to note that by the induction hypothesis and the recursive construction of $(\mathcal{T}_{n+1}^*, v_{n+1})$ from $(\mathcal{T}_n^*, v_n)$, the last expression above is exactly $\mathbb{E}[F(\mathcal{T}_{n+1}^*, v_{n+1})]$, which concludes the proof. ■

This result can now be translated in terms of the martingale $(W_n)_{n \ge 0}$. Here and in the following, we let $(\mathcal{F}_n)$ denote the natural filtration associated to $(Z(n))$. It is a standard result that there exists a probability measure $\mathbb{Q}$ on $\mathcal{F}_\infty = \sigma(\cup_n \mathcal{F}_n)$ satisfying that (writing respectively $\mathbb{P}_n$ and $\mathbb{Q}_n$ for the restriction of $\mathbb{P}$ and $\mathbb{Q}$ to $\mathcal{F}_n$) $\mathbb{Q}_n \ll \mathbb{P}_n$, with Radon–Nikodym derivative proportional to $W_n = \varrho^{-n} Z(n) h$. The fact that $Z(n) h = \sum_{v \in G_n} h(x^v)$ allows us to describe easily the distribution of $(W_n)$ under $\mathbb{Q}$ (in the upcoming Corollary 9). Note that we will never need to use expectation under $\mathbb{Q}$, so that $\mathbb{E}$ always denotes expectation under $\mathbb{P}$.

Let $(X_n)_{n \ge 0}$ be the spinal chain, started from an initial condition $X_0$ drawn at random from $(1/Z(0)h) \sum_x Z_x(0)h(x)\delta_x$, and let us expand on the construction of $(Y_n)_{n \ge 1}$ in (2) by defining a sequence of $\mathcal{X}^\mathbb{N}$-valued random processes $(Z^{(n)}(m))_{n,m \ge 0}$, such that:

* The $(Z^{(n)}(0))_{n \ge 0}$ are independent conditional on $\mathcal{G} = \sigma(X_n, n \ge 0)$.

* For $n=0$, set $Z^{(n)}(0) = Z(0) - e_{X_0}$, where we recall the notation $e_y = (1_{x=y})_x \in \mathbb{N}^\mathcal{X}$.

* For $n \ge 1$, the distribution of $Z^{(n)}(0)$ conditional on $\mathcal{G}$ and on $(X_{n-1}, X_n) = (x, y)$ is the distribution of $(L^{(x)} - e_y)$ biased by $L_y^{(x)}$.

* Finally, conditional on $(Z^{(n)}(0))_{n \ge 0}$ and independently of everything else, for each $n$ independently let $(Z^{(n)}(m))_{m \ge 0}$ be a branching process started from $Z^{(n)}(0)$, with the same Markov kernel as the original Galton–Watson process $(Z(m))_{m \ge 0}$.

With this definition, note that a version of $(Y_n)_{n}$ defined in (2) is given by $Y_n = h(X_n) + Z^{(n)}(0)h$. Now let $(\tilde{W}_n)_{n \ge 0}$ denote the process defined by

$$
\tilde{W}_n = \varrho^{-n} \left( h(X_n) + \sum_{k=0}^{n} Z^{(k)}(n-k)h \right) = \varrho^{-n} h(X_n) + \sum_{k=0}^{n} \varrho^{-k} \left( \frac{Z^{(k)}(n-k)h}{\varrho^{n-k}} \right). \quad (7)
$$

Lemma 8 has the following immediate consequence.

**Corollary 9.** The distribution of $(W_n)$ under $\mathbb{Q}$ is that of $(\tilde{W}_n)$.

As in Lyons et al. (1995), The two following classic results are key to proving a Kesten–Stigum theorem through the spinal change of measures. We state them without proof: the first one is deduced for instance from Theorem 5.3.3 in Durrett (2010), with $\mu = \mathbb{Q}$, $\nu = \mathbb{P}$ and $A = \{W > 0\}$; the second one is a simple consequence of the Borel–Cantelli lemmas.

**Lemma 10.**

(i) If $\limsup_n \tilde{W}_n < \infty$ a.s, then $\mathbb{E}[W] = \mathbb{E}[W_0]$.

(ii) If $\limsup_n \tilde{W}_n = \infty$ a.s, then $\mathbb{E}[W] = 0$.
**Lemma 11.** Let $(X_n)_{n \ge 1}$ be i.i.d. nonnegative random variables. Then almost surely:

$$ \limsup_{n \to \infty} \frac{X_n}{n} = \begin{cases} 0 & \text{if } \mathbb{E}[X_1] < \infty, \\ \infty & \text{if } \mathbb{E}[X_1] = \infty. \end{cases} $$

If $(X_n)_{n \ge 1}$ are only identically distributed, then $\mathbb{E}[X_1] < \infty$ still implies that $\limsup_{n \to \infty} \frac{X_n}{n} = 0$.

We are now ready to prove the different parts of Theorem 2.

**Proposition 12.** We have $\limsup_n \tilde{W}_n < \infty$ a.s. if (1) or (3) is satisfied.

*Proof.* Let us first show that it is sufficient to study $(\tilde{W}_n)$ for $Z(0) = e_{x_0}$, for a fixed $x_0 \in \mathcal{X}$. For any other $Z(0)$, defining $T_{x_0} = \min\{n \ge 0 : X_n = x_0\}$, which is finite a.s. since the spinal chain is recurrent, by (7) we can write

$$ \tilde{W}_n = \sum_{k=0}^{T_{x_0}-1} \varrho^{-k} \left( \frac{Z^{(k)}(n-k)h}{\varrho^{n-k}} \right) + \tilde{W}_{n-T_{x_0}}^0, \quad n \ge T_{x_0}, $$

where by the strong Markov property $(\tilde{W}_m^0)_{m \ge 0}$ is distributed as $\tilde{W}_m$ with initial condition $Z(0) = e_{x_0}$. Now the sum in the display above is an almost surely finite sum of processes that converge almost surely, since conditional on $Z^{(k)}(0)$, the process $(\varrho^{-m}Z^{(k)}(m)h)_{m \ge 0}$ is a nonnegative martingale. This shows that $\limsup_n \tilde{W}_n = \infty$ if and only if $\limsup_n \tilde{W}_n^0 = \infty$; thus for the rest of the proof, we may assume that $Z(0) = e_{x_0}$.

The description of $(\tilde{W}_n)$ given in (7) is convenient for this problem since conditional on $\mathcal{G}' := \sigma(X_n, Z^{(n)}(0), n \ge 0)$, we see that

$$ W'_n := \tilde{W}_n - \varrho^{-n}h(X_n) $$

is a nonnegative submartingale. Specifically, we have

$$ \mathbb{E}[W'_{n+1} | \mathcal{G}' \lor \mathcal{F}_n] = W'_n + \varrho^{-n} Z^{(n)}(0)h. $$

Therefore, if we can show that

$$ \sum_{n \ge 1} \varrho^{-n} Z^{(n)}(0)h < \infty \quad \text{a.s.,} \qquad (8) $$

we deduce from standard martingale arguments that $W' = \lim_{n \to \infty} W'_n$ exists and is finite.

First, we assume that (1) is satisfied. Since $\pi(x) = \tilde{h}_x h_x$ is a stationary distribution for the spinal Markov chain, $Y_n$ is identically distributed under the (non-necessarily finite in the recurrent case) measure $\mu = \sum_{x \in \mathcal{X}} \pi(x) \mathbb{P}_x(\cdot)$. We also compute

$$ \begin{aligned} \sum_{x \in \mathcal{X}} \pi(x) \mathbb{E}_x[\log_+(Y_1)] &= \sum_{x \in \mathcal{X}} \tilde{h}_x h_x \sum_{y \in \mathcal{X}} \frac{M_{xy} h_y}{\varrho h_x} \cdot \frac{1}{M_{xy}} \mathbb{E}[L_y^{(x)} \log_+(L^{(x)}h)] \\ &= \frac{1}{\varrho} \sum_{x \in \mathcal{X}} \tilde{h}_x \mathbb{E}[(L^{(x)}h) \log_+(L^{(x)}h)] < \infty, \end{aligned} $$

so by Lemma 11 we have $\limsup_n \log(Y_n)/n = 0$, in other words $Y_n$ grows subexponentially: for any $\gamma > 0$, we have $\gamma^{-n}Y_n \to 0$. Note that this holds $\mu$-a.e, which implies that this is true $\mathbb{P}_{x_0}$-a.s. In the construction of $(\tilde{W}_n)$, we can write $Y_n = h(X_n) + Z^{(n)}(0)h$, so that both $Z^{(n)}(0)h$ and $h(X_n)$ grow subexponentially, implying (8) and also $|\tilde{W}_n - W'_n| = \varrho^{-n}h(X_n) \to 0$. Putting everything together, we get that $\lim_{n \to \infty} \tilde{W}_n$ exists and is finite.

We now assume that (3) is satisfied for $x = x_0$. Let us define $T_0 = 0$ and $T_1, T_2, \dots$ the successive return times to $x_0$ for the spinal chain $(X_n)$, and for $n \ge 0$, let us define $k_n$ such that
$T_{k_n-1} \le n < T_{k_n}$. Grouping terms, the sum in (8) is bounded by

$$ \sum_{k=1}^{k_n} \sum_{j=T_{k-1}+1}^{T_k} \varrho^{-j} Y_j = \sum_{k=1}^{k_n} \varrho^{-T_{k-1}} \left( \sum_{j=1}^{T_k-T_{k-1}} \varrho^{-j} Y_{T_{k-1}+j} \right) = \sum_{k=1}^{k_n} \varrho^{-T_{k-1}} Y'_k, \quad (9) $$

where the $Y'_k$ are i.i.d, with $Y'_1 = \sum_{j=1}^{T_1} \varrho^{-j} Y_j$. Therefore, by Lemma 11, Condition (3) implies that $(Y'_k)$ grows subexponentially, and since necessarily $T_k \ge k$, this shows that (8) holds. Therefore $\lim_n W'_n$ exists and is finite, and it remains to show that $\varrho^{-n}h(X_n) \to 0$. But by definition of $k_n$, we have $\varrho^{-n}h(X_n) \le \varrho^{-n}Y_n \le \varrho^{-T_{k_n-1}}Y'_k$, so we conclude that $\lim_n \tilde{W}_n$ exists and is finite. ■

**Proposition 13.** If M is positive recurrent and (3) does not hold, we have

$$ \limsup_{n \to \infty} \tilde{W}_n = \infty, $$

up to a null measure set.

*Proof.* Here we take the same notation as in the proof above, and assume that (3) does not hold. Since the $(Y'_k)$ are i.i.d, by Lemma 11 we get that almost surely, for any $\gamma > 1$, we have $Y'_k \ge \gamma^k$ for infinitely many $k$. Let us fix $\gamma > \varrho^c$, where $c = \mathbb{E}[T_1]$, which is finite because $(X_n)$ is assumed to be positive recurrent. Note that by the definition of $Y'_k$ in (9), if $Y'_k \ge \gamma^k$ holds, then necessarily we have

$$ \max_{T_{k-1}+1 \le j \le T_k} (\varrho^{-j} Y_j) \ge \frac{\gamma^k \varrho^{-T_{k-1}}}{2(T_k - T_{k-1})}. $$

Because $\mathbb{E}[T_1] = c < \infty$, we have $T_k \sim ck$, and furthermore (again by Lemma 11), for $k$ large enough, we will have $T_k - T_{k-1} \le k$, so with our choice of $\gamma$, the right-hand-side above tends to $\infty$ as $k \to \infty$. Summing up, we have shown that almost surely, there exists a sequence $j_m$ such that $\varrho^{-j_m} Y_{j_m} \to \infty$, and the fact that by definition $\tilde{W}_n \ge \varrho^{-n}Y_n$ concludes the proof. ■

Putting together Proposition 12 and Proposition 13, we have shown that (3) is equivalent to $\mathbb{E}[W] = Z(0)h$ in the positive recurrent setting, and that (1) is in general only sufficient – see Section 2 for an example satisfying (3) but not (1).

To complete the proof of Theorem 2, it remains to tackle the equivalence of the three items (i)-(iii). First, we have already seen that $\mathbb{E}[W]$ is either 0 or $Z(0)h$, which shows that (i) and (ii) are equivalent. Let us see that (iii) implies (ii). We assume (iii), fix $x \in \mathcal{X}$ and note that since we are in the supercritical regime, we have $M_{x,x}^d > 1$ for some $d$. Then started from $Z(0) = e_x$, the process $(Z_x(nd))_{n \ge 0}$ stochastically dominates a supercritical (single-type) branching process, which therefore has a nonzero probability of survival. But on the event of survival, there is no local extinction, so $W > 0$ by assumption – (ii) is therefore satisfied.

We now only need to show that (ii) implies (iii). Let us assume (ii) holds. First, we use Theorem 5.(ii), which shows in particular that for a fixed $x \in \mathcal{X}$, we have $\varrho^{-n}Z_x(n) \to W\tilde{h}_x$ in probability. Note that we are not using a circular argument here since the proof of Theorem 5 in Section 4 only uses the fact that $\mathbb{P}(W > 0) \iff \mathbb{E}[W] = Z(0)h$. We can define an increasing subsequence $(n_k)_{k \ge 1}$ such that $\varrho^{-n_k}Z_x(n_k) \to W\tilde{h}_x$ almost surely, and this show that on the event $\{W > 0\}$, there cannot be local extinction. In other words we have

$$ \{W > 0\} \subset \text{LocExt}^c. $$

It remains to derive the converse inclusion. By definition of the local extinction event we have $\text{LocExt}^c = \{\exists x_0, \limsup_n Z_{x_0}(n) > 0\}$. We fix $x_0 \in \mathcal{X}$ and focus on the event $\{\limsup_n Z_{x_0}(n) > 0\}$. Let us fix some integer $K > 0$. Under Assumption A one can find $d \in \mathbb{N}$ such that $M_{x_0,x_0}^d > K$, and so in particular, started from $Z(0) = e_{x_0}$, we have $\mathbb{P}(Z_{x_0}(d) \ge K) = c > 0$.

Define the recursive sequence of stopping times $\tau_0 = \inf\{n > 0, Z_{x_0}(n) \ge 1\}$ and for $k \ge 0$, $\tau_{k+1} = \inf\{n > \tau_k + d, Z_{x_0}(n) \ge 1\}$. The $(\tau_k)_k$ are all almost surely finite on $\{\limsup_n Z_{x_0}(n) > 0\}$
and by the strong Markov property, the $(Z_{x_0}(\tau_n + d))_n$ are independent conditional on $\mathcal{G} := \sigma(Z_{x_0}(\tau_n), n \ge 0)$ and all satisfy

$$ \mathbb{P}(Z_{x_0}(\tau_n + d) > K | \mathcal{G}) \ge c. $$

The conditional Borel-Cantelli lemma then yields that on $\{\limsup_n Z_{x_0}(n) > 0\}$, $\limsup_n Z_x(n) > K$ holds almost surely, and the stopping time $t_K = \inf\{n > 0, Z_{x_0}(n) \ge K\}$ is almost surely finite.

By the branching property at time $t_K$, each of the type-$x_0$ individuals at generation $n$ (name them $u$) is the ancestor to an independent multi-type branching process with same offspring distribution to which we associate an additive martingale $W_n^{(u)}$ converging pointwise to a copy $W^{(u)}$ of $W$ (independent of the other $W^{(v)}$, $v \ne u$). The initial martingale being smaller than the sum of the $W_n^{(u)}$, we have $W = 0$ only if for all $u$, $W^{(u)} = 0$. Since at time $t_K$ there are at least $K$ type-$x_0$ individuals, we then have

$$ \mathbb{P}\left(\limsup_{n \to \infty} Z_{x_0}(n) > 0 \text{ and } W = 0\right) \le \mathbb{P}(t_K < \infty \text{ and } W = 0) \\ \le \mathbb{P}(t_K < \infty) \mathbb{P}(W = 0 | Z(0) = e_{x_0})^K. $$

The upper bound tends to 0 as $K \to \infty$, which yields $\{\limsup_n Z_{x_0}(n) > 0\} \subset \{W > 0\}$. Taking the union over $x_0$, we finally derive

$$ \mathrm{LocExt}^c \subset \{W > 0\}. $$

This concludes the proof of Theorem 2. Note that survival of a given type almost surely upon non extinction of the process is stated as a sufficient criterion stated in Ligonnière (2025) (yet within a different setting with varying environment and stronger moment assumptions) to recover the dichotomy between $\{W = 0\}$ and the event of survival.

# 4 Proof of Theorem 5

In this part, we will use several times a version of Scheffé's lemma for nonnegative random variables, which we state in our specific setting, for the sake of clarity.

**Lemma 14.** For $n \ge 1$ and $x \in \mathcal{X}$, let $U_{n,x}$ and $V_x$ be nonnegative random variables. If

$$ \forall x \in \mathcal{X}, \varepsilon > 0, \quad \mathbb{P}(U_{n,x} \ge V_x - \varepsilon) \to 1, \quad \text{and} \quad \sum_{x \in \mathcal{X}} \mathbb{E}[U_{n,x}] \to \sum_{x \in \mathcal{X}} \mathbb{E}[V_x] < \infty, $$

then we have $\mathbb{E}[\sum_x |U_{n,x} - V_x|] \to 0$. Note that the first of the two conditions above is implied by the condition that $\liminf_{n \to \infty} U_{n,x} \ge V_x$ for all $x \in \mathcal{X}$.

*Proof.* First let us fix $x \in \mathcal{X}$ and show that $\mathbb{E}[U_{n,x} \wedge V_x] \to \mathbb{E}[V_x]$. We compute, for any $\varepsilon > 0$:

$$ \mathbb{E}[V_x] \ge \mathbb{E}[U_{n,x} \wedge V_x] \ge \mathbb{E}[\mathbb{1}_{\{U_{n,x} \ge V_x - \varepsilon\}}(V_x - \varepsilon)] = \mathbb{E}[V_x] - \varepsilon - \mathbb{E}[\mathbb{1}_{\{U_{n,x} < V_x - \varepsilon\}}(V_x - \varepsilon)]. $$

Since $V_x - \varepsilon$ is integrable and $\mathbb{P}(U_{n,x} < V_x - \varepsilon) \to 0$, the last term above tends to 0, and so $\liminf_n \mathbb{E}[U_{n,x} \wedge V_x] \ge \mathbb{E}[V_x] - \varepsilon$. Since $\varepsilon$ is arbitrary, we get $\mathbb{E}[U_{n,x} \wedge V_x] \to \mathbb{E}[V_x]$. Now, by dominated convergence we have $\sum_x \mathbb{E}[U_{n,x} \wedge V_x] \to C := \sum_{x \in \mathcal{X}} \mathbb{E}[V_x]$, and we now just use the fact that $|U_{n,x} - V_x| = U_{n,x} + V_x - 2(U_{n,x} \wedge V_x)$ to get

$$ \mathbb{E}\left[\sum_{x \in \mathcal{X}} |U_{n,x} - V_x|\right] \to C + C - 2C = 0, $$

concluding the proof.

<figure><img src="image_2.png" /></figure>
## Proof of (i) and (ii)

Proposition 1 directly gives the first part of Theorem 5 in the null recurrent setting. From now on we can assume $M$ is positive recurrent with $\tilde{h}h = 1$.

**Ergodicity in mean.** This paragraph is dedicated to proving convergence of the type vector in $L^1(\mathbb{B}_h)$, which corresponds to the second part (ii) in Theorem 5. The proof relies on two intermediate lemmas. First and foremost, we recall that

$$ \mathbb{E}[\|\varrho^{-n}Z(n) - W\tilde{h}\|_h] = \mathbb{E}\left[\sum_{x \in \mathcal{X}} |\varrho^{-n}Z_x(n) - W\tilde{h}_x|h_x\right]. $$

**Lemma 15 (Restriction to a finite set of types).** For all $\varepsilon, \eta > 0$, there exists a finite subset $\mathcal{X}_0 \subset \mathcal{X}$ and $n_0 \ge 1$ such that $\forall n \ge n_0$,

$$ \mathbb{P}(\varrho^{-n}Z(n)h|_{\mathcal{X}_0^c} \le \varepsilon) \ge 1 - \eta. $$

*Proof.* Recall that under Assumption A, by Proposition 1, for all $x, y \in \mathcal{X}$, we have $\varrho^{-n}(M^n)_{xy} \to h_x\tilde{h}_y$. Therefore, using Fatou's lemma, for all $y \in \mathcal{X}$,

$$ \liminf_{n \to \infty} \mathbb{E}[\varrho^{-n}Z_y(n)] = \liminf_{n \to \infty} \sum_{x \in \mathcal{X}} Z_x(0)\varrho^{-n}(M^n)_{xy} \ge \sum_{x \in \mathcal{X}} Z_x(0)h_x\tilde{h}_y = Z(0)h \cdot \tilde{h}_y. $$

By the martingale property and the condition $\tilde{h}h = 1$, we also have

$$ \sum_{x \in \mathcal{X}} \mathbb{E}[\varrho^{-n}Z_x(n)] h_x = \mathbb{E}[W_n] = Z(0)h \cdot \tilde{h}h = \|Z(0)h \cdot \tilde{h}\|_h. \quad (10) $$

Therefore, applying Lemma 14 with the deterministic sequence $U_{n,x} = \mathbb{E}[\varrho^{-n}Z_y(n)]h_x$ and $V_x = Z(0)h \cdot \tilde{h}_xh_x$, we get

$$ \sum_{x \in \mathcal{X}} \left| \mathbb{E}[\varrho^{-n}Z_x(n)] - Z(0)h \cdot \tilde{h}_x \right| h_x \to 0, \quad (11) $$

in other words we have the convergence $\mathbb{E}[\varrho^{-n}Z(n)] \to Z(0)h \cdot \tilde{h}$ in $\mathbb{B}_h$. Now let $\varepsilon, \eta > 0$ and $\mathcal{X}_0$ a finite subset of $\mathcal{X}$ such that $Z(0)h|_{\mathcal{X}_0^c} \le \varepsilon\eta/2$. Using (11) and the fact that $u \in \mathbb{B}_h \mapsto uh|_{\mathcal{X}_0^c} \in \mathbb{R}$ is continuous, we can find $n_0$ such that

$$ \forall n \ge n_0, \mathbb{E}[\varrho^{-n}Z(n)h|_{\mathcal{X}_0^c}] \le \varepsilon\eta. $$

We finally get with Markov's inequality that

$$ \mathbb{P}(\varrho^{-n}Z(n)h|_{\mathcal{X}_0^c} \ge \varepsilon) \le \eta, $$

which concludes the lemma.

**Lemma 16 (Truncating the offspring distribution).** Let $\varrho_1, \varrho_2$ verifying

$$ \varrho_1 \in (\sqrt{\varrho}, \varrho), \quad 1 < \varrho_2 < \varrho \quad \text{and} \quad \varrho_1^2/\varrho_2 > \varrho. $$

*Note that since $\varrho > 1$, such $\varrho_1$ and $\varrho_2$ can indeed be found. For $n \ge 1$, introduce the capped mean matrix $\bar{M}(n)$ such that for every $x, y \in \mathcal{X}$,

$$ \bar{M}_{xy}(n) = \mathbb{E}\left[L_y^{(x)} \wedge (\varrho_2^n h_y \tilde{h}_y^2)\right]. \quad (12) $$

Then almost surely, there exists $n_0 \in \mathbb{N}$ such that for all $n \ge n_0$ and $y \in \mathcal{X}$,

$$ Z_y(n+1) \ge (Z\bar{M}(n))_y - \varrho_1^n \tilde{h}_y. $$
Proof. From the definition of the capped matrix, $\forall x, y, \bar{M}_{xy}(n) \le M_{xy}$ and converges to $M_{xy}$ as $n \to \infty$. Let $y \in \mathcal{X}$ and $n \ge 0$ be fixed. By the branching property on $Z$ and $\bar{M}$, we can write

$$ 
\begin{aligned}
Z_y(n+1) &= \sum_{x \in \mathcal{X}} \sum_{i=1}^{Z_x(n)} L_y^{(x,i)} \\
&\ge \sum_{x \in \mathcal{X}} \sum_{i=1}^{Z_x(n)} \bar{L}_y^{(x,i)}, \quad \text{with } \bar{L}_y^{(x,i)} := L_y^{(x,i)} \wedge (\varrho_2^n h_y \tilde{h}_y^2),
\end{aligned}
$$

where the $(L^{(x,i)})_{i \ge 1, x \in \mathcal{X}}$ are independent, with each $L^{(x,i)}$ distributed as $L^{(x)}$, and the $(\bar{L}_y^{(x,i)})_{i \ge 1}$ have finite variance, such that $\mathrm{Var}(\bar{L}_y^{(x,i)}) \le \mathbb{E}[(\bar{L}_y^{(x,i)})^2] \le \mathbb{E}[L_y^{(x,i)}] \varrho_2^n h_y \tilde{h}_y^2 = M_{xy} \varrho_2^n h_y \tilde{h}_y^2$. Now we can apply the Bienaymé-Chebyshev inequality to get

$$ 
\begin{aligned}
\mathbb{P} (Z_y(n+1) \le (Z(n)\bar{M}(n))_y - \varrho_1^n \tilde{h}(y)) &\le \mathbb{P} \left( \sum_{x \in \mathcal{X}} \sum_{i=1}^{Z_x(n)} \bar{L}_y^{(x,i)} \le (Z(n)\bar{M}(n))_y - \varrho_1^n \tilde{h}(y) \right) \\
&\le \mathbb{E} \left[ \frac{1}{\varrho_1^{2n} \tilde{h}_y^2} \mathrm{Var} \left( \sum_{x \in \mathcal{X}} \sum_{i=1}^{Z_x(n)} \bar{L}_y^{(x,i)} \mid \mathcal{F}_n \right) \right] \\
&\le \sum_{x \in \mathcal{X}} \mathbb{E} [Z_x(n)] \frac{\mathrm{Var}(\bar{L}_y^{(x,i)})}{\varrho_1^{2n} \tilde{h}_y^2} \\
&\le \sum_{x \in \mathcal{X}} \mathbb{E} [Z_x(n)] M_{xy} h_y \left( \frac{\varrho_2}{\varrho_1^2} \right)^n.
\end{aligned}
$$

By the union bound and harmonicity of $h$,

$$ 
\begin{aligned}
\mathbb{P} (\exists y \in \mathcal{X}, Z_y(n+1) \le (Z(n)\bar{M}(n))_y - \varrho_1^n \tilde{h}_y) &\le \left(\frac{\varrho_2}{\varrho_1^2}\right)^n \sum_{x,y \in \mathcal{X}} \mathbb{E}[Z_x(n)] M_{xy} h_y \\
&\le \left(\frac{\varrho_2 \varrho}{\varrho_1^2}\right)^n \varrho \cdot Z(0)h.
\end{aligned}
$$

Since we picked $\varrho_1$ and $\varrho_2$ to have $\varrho_2\varrho/\varrho_1^2 < 1$, the previous bound is summable over $n$ and the result holds by Borel-Cantelli's lemma. ■

Now let us prove point (ii) in Theorem 5. Take $\mathcal{X}_0$ as in Lemma 15 and fix $y \in \mathcal{X}$ and $\varepsilon > 0$ for good. Using Proposition 1, one can choose some $k \in \mathbb{N}^*$ such that the following holds:

$$ \forall x \in \mathcal{X}_0, M_{xy}^k \ge (1 - \varepsilon) \varrho^k h_x \tilde{h}_y. \quad (13) $$

Using that $\tilde{h}$ is the left eigenvector associated with $\varrho$, Lemma 16 implies that almost surely, for all $n$ large enough, coordinate-wise we have

$$ 
\begin{aligned}
Z(n+k) &\ge Z(n)\bar{M}(n,k) - (\varrho^{k-1} + \varrho^{k-2}\varrho_1 + \cdots + \varrho\varrho_1^{k-2} + \varrho_1^{k-1}) \varrho_1^n \tilde{h} \\
&= Z(n)\bar{M}(n,k) - (\varrho - \varrho_1)^{-1}(\varrho^k - \varrho_1^k) \varrho_1^n \tilde{h}, \\
&\ge Z(n)\bar{M}(n,k) - (\varrho - \varrho_1)^{-1} \varrho^k \varrho_1^n \tilde{h},
\end{aligned}
$$

where $\bar{M}(n,k) := \bar{M}(n)\bar{M}(n+1)\cdots\bar{M}(n+k-1)$. Because $\bar{M}(n)$ increase coordinate-wise to $M$ as $n \to \infty$, so does $\bar{M}(n,k)$ to $M^k$. We now fix $\eta > 0$, and we may pick $n_0$ large enough to ensure
that for all $n \ge n_0$,

$$
\left\{
\begin{array}{ll}
(a) & \mathbb{P}(\varrho^{-n}Z(n)h|\chi_0^c \le \varepsilon) \ge 1 - \eta \qquad \text{(by Lemma 15)} \\
(b) & \forall x \in \mathcal{X}_0, \bar{M}_{xy}(n,k) \ge (1-2\varepsilon)\varrho^k h_x \tilde{h}_y \qquad (\mathcal{X}_0 \text{ finite and using (13)}); \\
(c) & \mathbb{P}(\forall n \ge n_0, Z(n+k) > Z(n)\bar{M}(n,k) - (\varrho - \varrho_1)^{-1}\varrho^k \varrho_1^n \tilde{h}) \ge 1 - \eta \qquad \text{(by Lemma 16)}; \\
(d) & (\varrho - \varrho_1)^{-1} \left(\frac{\varrho_1}{\varrho}\right)^n < \varepsilon \qquad \text{(since } \varrho_1 < \varrho\text{)}; \\
(e) & \mathbb{P}(\forall n \ge n_0, |\varrho^{-n}Z(n)h - W| \le \varepsilon) \ge 1 - \eta \qquad \text{(a.s. convergence of } W_n\text{).}
\end{array}
\right.
$$

Let $n \ge n_0$. By (d) we have

$$
\{Z(n+k) \ge Z(n)\bar{M}(n,k) - (\varrho - \varrho_1)^{-1} \varrho^k \varrho_1^n \tilde{h}\} \subset \left\{ \frac{Z(n+k)}{\varrho^{n+k}} \ge \varrho^{-(n+k)} Z(n)\bar{M}(n,k) - \varepsilon \tilde{h} \right\}.
$$

Thus using (c), the probability of the latter event is at least $1 - \eta$. Furthermore by (b), we have
$(Z(n)\bar{M}(n,k))_y \ge \varrho^k(1-2\varepsilon)(\sum_{x\in\mathcal{X}_0} Z_x(n)h_x)\tilde{h}_y = \varrho^k(1-2\varepsilon)Z(n)h|_{\mathcal{X}_0} \cdot \tilde{h}_y$. Whence we derive

$$
\mathbb{P} \left( \frac{Z_y(n+k)}{\varrho^{n+k}} \ge ((1-2\varepsilon)\varrho^{-n} Z(n)h|_{\mathcal{X}_0} - \varepsilon) \tilde{h}_y \right) \ge 1 - \eta.
$$

Now by (a) and (e), we have $\mathbb{P}(|\varrho^{-n}Z(n)h|_{\mathcal{X}_0} - W| \le 2\varepsilon) \ge 1 - 2\eta$. It follows that

$$
\mathbb{P} \left( \varrho^{-(n+k)} Z_y(n+k) \ge ((1-2\varepsilon)(W-2\varepsilon) - \varepsilon) \tilde{h}_y \right) \ge 1 - 3\eta,
$$

and since $\varepsilon$ and $\eta$ are arbitrary, standard arguments yield:

$$
\forall \varepsilon > 0, \quad \mathbb{P} \left( \varrho^{-n} Z_y(n) \ge W \tilde{h}_y - \varepsilon \right) \xrightarrow{n \to \infty} 1.
$$

Combined with the fact that $\mathbb{E}[\sum_y \varrho^{-n} Z_y(n) h_y] = \mathbb{E}[W_n] = \mathbb{E}[W] = \mathbb{E}[\sum_y W \tilde{h}_y h_y]$, applying Lemma 14 to $U_{n,x} = \varrho^{-n} Z_x(n) h_x$ and $V_x = W \tilde{h}_x h_x$ we get

$$
\mathbb{E} \left[ \sum_{x \in \mathcal{X}} \left| \frac{Z_x(n)}{\varrho^n} - W \tilde{h}_x \right| h_x \right] \to 0,
$$

which concludes the proof of (ii).

**Proof of (iii)**

(a): Offspring distributions with finite variance. Assume $\forall x, y \in \mathcal{X}$, $\text{Var}(L_y^{(x)}) < \infty$ and satisfy

$$
\sum_{x,y \in \mathcal{X}} \tilde{h}_x \text{Var}(L_y^{(x)}) \tilde{h}_y^{-2} < \infty.
$$

Then we can improve Lemma 16 using the mean matrix directly in place of the capped matrix $\bar{M}$.
Doing so and applying the same steps as in the proof of Lemma 16 with $\varrho_1 < \sqrt{\varrho}$, we obtain that
almost surely, for all $n$ large enough we have

$$
Z(n+k) \ge Z(n)M^k - (\varrho - \varrho_1)^{-1} \varrho^k \varrho_1^n \tilde{h}, \quad \forall k \in \mathbb{N}. \quad (14)
$$

This implies that for $n$ large enough, we have for any $y \in \mathcal{X}$,

$$
\frac{Z_y(n+k)}{\varrho^{n+k}} \ge \left( \frac{Z(n)}{\varrho^n} \frac{M^k}{\varrho^k} \right)_y - (\varrho - \varrho_1)^{-1} \left( \frac{\varrho_1}{\varrho} \right)^n \tilde{h}_y, \quad \forall k \in \mathbb{N}.
$$
Because $\varrho^{-k}M^k \to h\tilde{h}$, it is not hard to see that $\liminf_k \varrho^{-k}(vM^k)_y \ge vh \cdot \tilde{h}_y$ for any $v \in \mathbb{B}_h$. So letting $k \to \infty$, we get that for $n$ large enough, we have

$$ \liminf_{m \to \infty} \frac{Z_y(m)}{\varrho^m} \ge \frac{Z(n)h}{\varrho^n} \tilde{h}_y = W_n \tilde{h}_y, $$

and since the left-hand side does not depend on $n$, we have simply $\liminf_m \varrho^{-m}Z_y(m) \ge W\tilde{h}_y$, and the conclusion holds again by applying Scheffé's lemma for nonnegative functions, using the fact that $\sum_y W\tilde{h}_y h_y = W = \lim_n \sum_y \varrho^{-n}Z_y(n)h_y$.

(b): **Entropy condition.** The key will again be to show that

$$ \liminf_{m \to \infty} \varrho^{-m} Z(m) \ge W\tilde{h} \quad (15) $$

coordinate-wise, and conclude as above. Let us first justify that we can assume without loss of generality that the initial population $Z(0)$ is finite. In the general case, note that for any $\mathcal{X}_0 \subset \mathcal{X}$, we can decompose $Z(n) = Z^{\mathcal{X}_0}(n) + Z^{\mathcal{X}_0^c}(n)$, where $(Z^{\mathcal{X}_0}(n))_{n \ge 0}$ and $(Z^{\mathcal{X}_0^c}(n))_{n \ge 0}$ are versions of the branching process started respectively from $Z(0)|_{\mathcal{X}_0}$ and $Z(0)|_{\mathcal{X}_0^c}$. Note that we can similarly write $W = W^{\mathcal{X}_0} + W^{\mathcal{X}_0^c}$ with obvious definitions, and it is not hard to see that $W^{\mathcal{X}_0^c} \to 0$ as $\mathcal{X}_0 \uparrow \mathcal{X}$. Therefore, if (15) is shown to hold for any finite initial population, then in the general case we have $\liminf_m \varrho^{-m}Z(m) \ge \sup_{\mathcal{X}_0} W^{\mathcal{X}_0} \tilde{h} = W\tilde{h}$, where the supremum is taken on finite $\mathcal{X}_0 \subset \mathcal{X}$, and the proof will be complete. From now on, we therefore assume that $Z(0)$ is finite.

Recall the capped matrix $\bar{M}(n)$ from (12), defined for each $n \in \mathbb{N}$. From Lemma 16, we have $Z(n+1) \ge Z(n)\bar{M}(n) - \varrho_1^n \tilde{h}$ for all $n$ large enough almost surely, which we can rewrite (note that each of the following terms is nonnegative coordinate-wise)

$$ Z(n)M \le Z(n+1) + Z(n)(M - \bar{M}(n)) + \varrho_1^n \tilde{h}. $$

Iterating the inequality above by multiplying by $M$ on the right, we obtain that for $n$ large enough and for all $k \in \mathbb{N}$:

$$ \frac{Z(n)M^k}{\varrho^{n+k}} \le \frac{Z(n+k)}{\varrho^{n+k}} + \sum_{i=0}^{k-1} \frac{Z(n+i)}{\varrho^{n+i}} (M - \bar{M}(n+i)) \frac{M^{k-i-1}}{\varrho^{k-i}} + (\varrho - \varrho_1)^{-1} \left(\frac{\varrho_1}{\varrho}\right)^n \tilde{h}. \quad (16) $$

We will first take $k \to \infty$ and then $n \to \infty$ in this inequality to obtain $\liminf_m \varrho^{-m}Z(m) \ge W\tilde{h}$ coordinate-wise. As $\varrho_1 < \varrho$, the term $(\varrho - \varrho_1)^{-1}(\varrho_1/\varrho)^n \tilde{h}$ will disappear in the second limit, so the delicate term is the sum $R_{n,k} := \sum_{i=0}^{k-1} \frac{Z(n+i)}{\varrho^{n+i}} (M - \bar{M}(n+i)) \frac{M^{k-i-1}}{\varrho^{k-i}}$. First, note that

$$ \|R_{n,k}\|_h = R_{n,k}h = \frac{1}{\varrho} \sum_{i=0}^{k-1} \mathbb{E} \left[ \frac{Z(n+i)}{\varrho^{n+i}} (M - \bar{M}(n+i))h \right], $$

which is non-decreasing in $k$, and tends to

$$ \lim_{k \to \infty} \|R_{n,k}\|_h := \mathcal{R}_n = \frac{1}{\varrho} \sum_{i \ge n} \mathbb{E} \left[ \frac{Z(i)}{\varrho^i} (M - \bar{M}(i))h \right]. $$

Now $(\mathcal{R}_n)$ is non increasing, so to show that it converges almost surely to 0 it suffices to show that $\mathbb{E}[\mathcal{R}_n] \to 0$. Since we have assumed that $Z(0)$ has a finite number of positive coordinates, there exists $\alpha > 0$ such that $Z(0) \le \alpha\tilde{h}$ holds, which implies that $\mathbb{E}[Z(i)] = Z(0)M^i \le \alpha\varrho^i\tilde{h}$. Using this,
we compute

$$ 
\begin{aligned}
\mathbb{E}[\mathcal{R}_n] &= \frac{1}{\varrho} \sum_{i \ge n} \mathbb{E} \left[ \frac{Z(i)}{\varrho^i} (M - \bar{M}(n+i))h \right] \\
&\le \frac{\alpha}{\varrho} \sum_{i \ge n} \tilde{h}(M - \bar{M}(i))h
\end{aligned}
\qquad (17) $$

Let us show that this is the remainder of a convergent series, i.e. that the following converges:

$$ 
\begin{aligned}
\sum_{i \ge 1} \tilde{h}(M - \bar{M}(i))h &= \sum_{x,y \in \mathcal{X}} \tilde{h}_x \left( \sum_{i \ge 1} (M - \bar{M}(i))_{xy} \right) h_y. \\
&= \sum_{x,y \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ \sum_{i \ge 1} \left( L_y^{(x)} - \varrho_2^i h_y \tilde{h}_y^2 \right)_+ \right] h_y.
\end{aligned}
$$

Now notice that for any $a, b > 0$, we have

$$ \sum_{i \ge 1} (a - \varrho_2^i b)_+ \le a |\{i \in \mathbb{N} : \varrho_2^i b \le a\}| = a \left[ \frac{\log_+(a/b)}{\log \varrho_2} \right] \le a \frac{\log_+(a/b)}{\log \varrho_2}, $$

therefore we obtain

$$ \sum_{i \ge 1} \tilde{h}(M - \bar{M}(i))h \le \frac{1}{\log \varrho_2} \sum_{x,y \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ L_y^{(x)} \log_+(L_y^{(x)} h_y^{-1} \tilde{h}_y^{-2}) \right] h_y. $$

It results that Condition (6) stated in the remark following the statement of Theorem 5 implies the convergence of the series $\sum_{i \ge 1} \tilde{h}(M - \bar{M}(i))h$, hence the fact that $\mathcal{R}_n \to 0$ almost surely. This means $\lim_{n \to \infty} \lim_{k \to \infty} \|R_{n,k}\|_h = 0$ almost surely, and so we can take the limits in (16) to establish $\liminf_n \varrho^{-n} Z(n) \ge W \tilde{h}$, and conclude as in the proof of (iii).(a).

It remains to retrieve (6) from the condition stated in (iii).(b). Recalling that $-\log(\tilde{h}_y h_y) \ge 0$ for all $y \in \mathcal{X}$, we compute:

$$ 
\begin{aligned}
& \sum_{x,y \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ L_y^{(x)} \log_+(L_y^{(x)} h_y^{-1} \tilde{h}_y^{-2}) \right] h_y \\
\le & \sum_{x,y \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ L_y^{(x)} \log_+(L_y^{(x)} h_y) \right] h_y - 2 \sum_{x,y \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ L_y^{(x)} \log (\tilde{h}_y h_y) \right] h_y \\
\le & \sum_{x \in \mathcal{X}} \tilde{h}_x \mathbb{E} \left[ (L^{(x)} h) \log_+(L^{(x)} h) \right] - 2\varrho \sum_{y \in \mathcal{X}} \tilde{h}_y h_y \log (\tilde{h}_y h_y),
\end{aligned}
$$

where we have used the equality $\sum_x \tilde{h}_x \mathbb{E}[L_y^{(x)}] = (\tilde{h}M)_y = \varrho \tilde{h}_y$. This completes the proof of Theorem 5.

**Acknowledgements.** We thank Maxime Ligonnière for insightful discussions regarding the (local or not) extinction of the branching process, in particular concerning the positive recurrent example where $\{W = 0\} = \text{LocExt} \ne \text{Ext}$.

We acknowledge funding through the ANR project GARP (ANR-24-CE40-7154). Mathilde André acknowledges funding from the Center for Interdisciplinary Research in Biology (CIRB), Collège de France and the École Polytechnique.

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