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# Convergence in $L^p$ for a supercritical multi-type branching process in a random environment

Ion Grama, Quansheng Liu, Erwan Pin

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Ion Grama, Quansheng Liu, Erwan Pin. Convergence in $L^p$ for a supercritical multi-type branching process in a random environment. 2020. hal-02934079

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# Convergence in $L^p$ for a supercritical multi-type branching process in a random environment

Ion Grama, Quansheng Liu and Erwan Pin

Université de Bretagne-Sud

LMBA UMR CNRS 6205

Vannes, France.

e-mail: ion.grama@univ-ubs.fr quansheng.liu@univ-ubs.fr erwan.pin@univ-ubs.fr

**Abstract:** Consider a $d$-type supercritical branching process $Z_n^i = (Z_n^i(1), \dots, Z_n^i(d))$, $n \ge 0$, in an independent and identically distributed random environment $\xi = (\xi_0, \xi_1, \dots)$, starting with one initial particle of type $i$. In a previous paper we have established a Kesten-Stigum type theorem for $Z_n^i$, which implies that for any $1 \le i, j \le d$, $Z_n^i(j)/\mathbb{E}_\xi Z_n^i(j) \to W^i$ in probability as $n \to +\infty$, where $\mathbb{E}_\xi Z_n^i(j)$ is the conditional expectation of $Z_n^i(j)$ given the environment $\xi$, and $W^i$ is a non-negative and finite random variable. The goal of this paper is to obtain a necessary and sufficient condition for the convergence in $L^p$ of $Z_n^i(j)/\mathbb{E}_\xi Z_n^i(j)$, and to prove that the convergence rate is exponential. To this end, we first establish the corresponding results for the fundamental martingale ($W_n^i$) associated to the branching process ($Z_n^i$).

**MSC 2010 subject classifications:** Primary 60J80, 60K37; secondary 60J85.

**Keywords and phrases:** Multitype branching processes, random environment, $L^p$ convergence, products of random matrices, fundamental martingale.

## 1. Introduction

A significant advancement in the theory of branching processes was made with the introduction of a random environment such that the offspring distribution of generation $n$ depends on some random environment $\xi_n$ at time $n$, in contrast to a constant distribution assumed in the Galton-Watson process. This allows a more adequate modeling, and turned out to be very fruitful in theoretical as well as in practical senses. For the first fundamental results on branching processes in random environments, see Athreya and Karlin [1, 2]. The importance of the study of branching processes in random environments is mainly due to its wide application background, both in theory and in practical problems. For example, Kesten, Kozlov and Spitzer [20] used such a process to study limit properties of for random walks in random environments; biologists are currently paying special attention to the problems of genetic transformation, and such problems can be studied via a multi-type branching process in a random environment; see Bansaye [4] for application in cell contamination. Due to huge applications and important technique challenge, in recent years, there is a great progress in the study of branching processes in random environments, see e.g. the recent papers [26, 30, 31], the recent book by Kersting and Vatutin [23] and many references therein. In an earlier work [11], for a supercritical
multi-type branching process in an independent and identically distributed random environment, we have studied the convergence of the normalized population size and the non-degeneracy of its limit. In this paper, we will consider its convergence in $L^p$.

Let $Z_n = (Z_n(1), \dots, Z_n(d))$, $n \ge 0$, be a $d$-type branching process in an independent and identically distributed (i.i.d.) random environment $\xi = (\xi_0, \xi_1, \dots)$. For $n \ge 0$, denote by $M_n$ the matrix of the conditioned means of the offspring distribution of $n$-th generation given the environment: the $(i, j)$-th entry of $M_n$ is

$$M_n(i, j) = \mathbb{E}_{\xi}[Z_{n+1}(j) | Z_n = e_i],$$

where $\mathbb{E}_{\xi}$ denotes the conditional expectation given the environment $\xi$. Let $M_{0,n} = M_0 \cdots M_n$ be the product matrix. The process $(Z_n)_{n \ge 0}$ will be denoted $(Z_n^i)_{n \ge 0}$ when it starts with one initial particle of type $i$, that is when $Z_0 = e_i$, where $e_i$ is the unit vector whose $i$-th component is 1 and all the others are 0. In [11] we obtained an extension of the famous Kesten-Stigum result on the Galton-Watson process to the multi-type branching process in random environment (MBPRE). Assume that the MBPRE $(Z_n^i)_{n \ge 0}$ is in the supercritical regime, in the sense that

$$\gamma := \lim_{n \to +\infty} \frac{1}{n} \mathbb{E} \log \|M_{0,n-1}\| > 0,$$

where $\|M_{0,n-1}\|$ is the $L_1$-norm of the matrix $M_{0,n-1}$. Under the Furstengerg-Kesten condition H1 (see Section 2), we proved in [11, Theorem 2.11] that for all $1 \le i, j \le d$,

$$\frac{Z_n^i(j)}{\mathbb{E}_{\xi} Z_n^i(j)} = \frac{Z_{n-1}^i(j)}{M_{0,n-1}(i,j)} \to W^i \quad \text{in probability,} \quad (1.1)$$

where $W^i$ is a non-negative random variable independent of $j$; moreover, $W^i$ is non-degenerate for all $i$ if and only if

$$\mathbb{E}\left(\frac{Z_1^i(j)}{M_0(i, j)} \log^+ \frac{Z_1^i(j)}{M_0(i, j)}\right) < +\infty \quad \forall i, j = 1, \dots, d; \quad (1.2)$$

in addition $\mathbb{E}_{\xi} W^i = 1$ almost surely (a.s.) when (1.2) holds. By Sheffé's theorem, it follows that $\frac{Z_n^i(j)}{M_{0,n-1}(i,j)} \to W^i$ in $L^1$ if and only if (1.2) holds.

The main objective in this paper is to find a necessary and sufficient condition under which the normalized population size $Z_n^i(j)/M_{0,n-1}(i,j)$ converges to $W^i$ in $L^p$, $p > 1$, and to prove that the convergence rate is exponential, for all $1 \le i, j \le d$. In the single type case, it is known that such kind of results play an important role in the study of asymptotic properties of large deviations and Berry Esseen bounds in the central limit theorem on the process $(Z_n^i)$, see [16, 10]. The situation is the same in the multi-type case, as can be seen in the preprints [12, 13].

For a single type branching process in a random environment $(Z_n)_{n \ge 0}$, Guivarc'h and Liu [14, Theorem 1.3] established the (annealed) $L^p$ convergence criterion: they
showed that when $d = 1$, for each given $p > 1$, $(Z_n/m_{0,n-1})_{n \ge 0}$ converges in $L^p$ to a non negative random variable $W$ if and only if

$$ \mathbb{E}\left(\frac{Z_1}{m_0}\right)^p < +\infty \quad \text{and} \quad \mathbb{E}m_0^{1-p} < 1, \qquad (1.3) $$

where $m_{0,n-1} = m_0 \cdots m_{n-1}$, and $m_k$ denotes the conditioned mean of the offspring distribution at time $k$ given the environment. Huang and Liu [17, Theorem 1.5] proved that the $L^p$ convergence rate is exponential: if (1.3) holds, then

$$ \lim_{n \to +\infty} \delta^{-n} \left( \mathbb{E} \left| \frac{Z_n}{m_{0,n-1}} - W \right|^p \right)^{1/p} = 0 \quad \forall \delta > \delta_c(p), \qquad (1.4) $$

with

$$ \delta_c(p) = \begin{cases} (\mathbb{E}m_0^{1-p})^{1/p} & \text{if } p \in (1, 2), \\ \max \left\{ (\mathbb{E}m_0^{1-p})^{1/p}, (\mathbb{E}m_0^{-p/2})^{1/p} \right\} & \text{if } p \ge 2. \end{cases} \qquad (1.5) $$

For the MBPRE's case, the only result about the annealed $L^p$ convergence is a claim by Cohn [8] which concerns the $L^2$ convergence. Assume that the supercritical condition $\gamma > 0$ holds, that each entry of $M_0$ is bounded a.s. from below and above by two positive constants, and that all the conditional second moments of the offspring distributions given the environment are bounded a.s. by a constant. Assume also the integrability condition $\mathbb{E}|\log \sum_{i=1}^d (1 - \mathbb{P}(\|Z_1^i\| = 0))| < \infty$. Under these conditions Cohn [8] claimed that for each $j = 1, \cdots, d$,

$$ \frac{Z_n^i(j)}{\mathbb{E}_\xi Z_n^i(j)} \to W^i \quad \text{in } L^2 \qquad (1.6) $$

under the annealed law $\mathbb{P}$, where $W^i$ is a non degenerate random variable satisfying $\mathbb{E}W^i = 1$. However, the claim of Cohn [8] is false. To see this, it suffices to notice that when $d = 1$, (1.6) holds if and only if $\mathbb{E}(\frac{Z_1}{m_0})^2 < +\infty$ and $\mathbb{E}m_0^{-1} < 1$ by the criterion (1.3) of Guivarc'h and Liu [14, Theorem 1.3]. A quantitative condition (which ensures $\mathbb{E}m_0^{-1} < 1$ for $d = 1$) is missing in the claim of Cohn [8]. This shows that the annealed $L^p$ convergence is rather delicate even for $p = 2$. We mention that Jones [18], Biggins, Cohn and Nerman [5] have studied respectively the $L^2$ and $L^p$ convergence of multi-type branching processes in varying environment. Their results give sufficient conditions for quenched $L^p$ convergence for multi-type branching processes in random environments. In this paper, we deal with the annealed $L^p$ convergence, which is in general more delicate because there is an additional integral operation. Since we will always deal with the annealed $L^p$ convergence, for simplicity we will just say $L^p$ convergence in the following.

More precisely, we will find a necessary and sufficient condition for the $L^p$ convergence which extends the criterion (1.3) to the multi-type case, and establish the exponential convergence rate. Let $p > 1$ be such that $\mathbb{E}M_0(i,j)^{1-p} < +\infty$ for all $1 \le i, j \le d$, and define

$$ \kappa(1-p) = \lim_{n \to +\infty} (\mathbb{E}\|M_{0,n-1}\|^{1-p})^{1/n}. $$
It will be shown that the limit exists and is finite. Under the Furstenberg-Kesten condition **H1**, we will prove that $\frac{Z_n^i(j)}{M_{0,n-1}(i,j)} \to W^i$ in $L^p$ for any $1 \le i, j \le d$ if and only if

$$ \max_{1 \le i, j \le d} \mathbb{E} \left( \frac{Z_1^i(j)}{M_0(i, j)} \right)^p < +\infty \quad \text{and} \quad \kappa(1-p) < 1. \qquad (1.7) $$

(cf. Theorem 2.1); moreover, if (1.7) holds, then there exists $\delta \in (0, 1)$ such that

$$ \lim_{n \to +\infty} \delta^{-n} \left( \mathbb{E} \left| \frac{Z_n^i(j)}{M_{0,n-1}(i,j)} - W^i \right|^p \right)^{1/p} = 0 \qquad (1.8) $$

(cf. Theorem 2.2). For a single type branching process in random environment, we have $\kappa(1-p) = \mathbb{E}m_0^{1-p}$, so (1.7) coincides with (1.3), and (1.8) corresponds to (1.4) but with less information on the exact exponential rate.

The proof of (1.7) and (1.8) is based on the corresponding results for the associated fundamental martingale ($W_n^i$) introduced in [11]. Let us recall briefly its construction. For any $n, k \ge 0$, let $\rho_{n,n+k}$ be the spectral radius of $M_{n,n+k}$. Applying the famous Perron-Frobenius theorem (see e.g. [3]), $\rho_{n,n+k}$ is a positive eigenvalue of $M_{n,n+k}$, for which there exist positive right and left eigenvectors $U_{n,n+k}$ and $V_{n,n+k}$ with the normalizations $\|U_{n,n+k}\| = 1$ and $\langle V_{n,n+k}, U_{n,n+k} \rangle = 1$, where $\|x\|$ denotes the $L^1$ norm of the vector $x$, and $\langle x, y \rangle$ the scalar product of the vectors $x, y$. Then, under certain conditions, by the results of Hennion [15, Lemma 3.3 and Theorem 1] the limit

$$ U_{n,\infty} := \lim_{k \to \infty} U_{n,n+k} \qquad (1.9) $$

exists a.s., with $U_{n,\infty} > 0$ a.s. and $\|U_{n,\infty}\| = 1$; moreover, there exist random scalars $\lambda_n > 0$ a.s. called the pseudo-spectral radii of the random matrices ($M_n$), which satisfy a.s. the relation

$$ M_n U_{n+1,\infty} = \lambda_n U_{n,\infty}. \qquad (1.10) $$

The relation (1.10) can be iterated to obtain

$$ M_{n,n+k} U_{n+k+1,\infty} = \lambda_{n,n+k} U_{n,\infty}, \qquad (1.11) $$

where $\lambda_{n,n+k} = \prod_{r=n}^{n+k} \lambda_r$ for $n, k \ge 0$. Then, the non-negative martingale ($W_n^i$) is defined as follows [11]:

$$ W_0^i = 1, \quad W_n^i = \frac{\langle Z_n^i, U_{n,\infty} \rangle}{\lambda_{0,n-1} U_{0,\infty}(i)}, \quad n \ge 1. \qquad (1.12) $$

Assume for simplicity that the Furstenberg-Kesten condition **H1** is satisfied. Assume also that $p > 1$ is such that $\mathbb{E}M_0(i,j)^{1-p} < +\infty$ for all $1 \le i, j \le d$. Then we show that $W_n^i$ converges in $L^p$ to the random variable $W^i$ for any $1 \le i \le d$ if and only if (1.7) holds (cf. Theorem 2.3); moreover, if (1.7) is satisfied, then

$$ \lim_{n \to +\infty} \delta^{-n} \left( \mathbb{E} |W_n^i - W^i|^p \right)^{1/p} = 0 \quad \forall \delta > \delta_c(p), \qquad (1.13) $$
with

$$ \delta_c(p) = \begin{cases} \kappa(1-p)^{1/p} & \text{if } p \in (1, 2), \\ \max\{\kappa(1-p)^{1/p}, \kappa(-p/2)^{1/p}\} & \text{if } p \ge 2 \end{cases} \quad (1.14) $$

(cf. Theorem 2.4). In the case of the single type branching process, the martingale ($W_n$) coincides with the normalized population size ($Z_n/m_{0,n-1}$), so the relations (1.13) and (1.14) coincide exactly with (1.4) and (1.5). It is known that when $d=1$, the critical value $\delta_c(p)$ is the best possible for (1.13) to hold (see Huang and Liu [17]).

For the proof, we develop the approach in [17] where the case $d=1$ was considered. In addition to the complexity related to the products of random matrices, the main difficulty for the multi-dimensional case resides in the fact that $W_n^i$ depends on the whole environment sequence $\xi = (\xi_0, \xi_1, \dots)$, not just on the environment sequence until the present $(\xi_0, \dots, \xi_{n-1})$, contrary to the one-dimensional case. Let us give a short description of the approach. For $p \in (1, 2]$, we first control the quenched $L^p$ norm of the martingale difference $W_{n+1}^i - W_n^i$, using the branching property and the Marcinkiewicz-Zygmund inequality on the $L^p$ norm of sums of independent random variables. This permits us to obtain a bound of $E_\xi|W_{n+1}^i - W_n^i|^p$ in terms of $(\lambda_{0,n-1}U_{0,\infty}(i))^{1-p}$. To overcome the difficulty related to the dependence on the whole environment sequence, we condition on the future $T^n\xi = (\xi_n, \xi_{n+1}, \dots)$ to obtain $E_{T^n\xi}(\lambda_{0,n-1}U_{0,\infty}(i))^{1-p} \le C\kappa(1-p)^n$, which gives the correct convergence rate in $L^p$ for the martingale ($W_n^i$). For $p > 2$, we use an argument by induction. To get the convergence rate of the normalized population size $Z_n^i(j)/M_{0,n-1}(i,j)$, we prove that the difference $Z_n^i(j)/M_{0,n-1}(i,j) - W_n^i$ goes to 0 in $L^p$ exponentially fast, using the exponential convergence of the products of stochastic matrices due to Seneta [28]. For the necessity, we first establish some spectral properties of the important transfer operator $P_s$ for $s \le 0$ (see Section 3).

The main results will be presented in Section 2. In Section 3 we establish the spectral properties of the transfer operator $P_s$ that we will need. In Section 4 we prove the criterion for the convergence in $L^p$ of the martingales ($W_n^i$), as well as their exponential convergence rate. Similar results for the normalized population size $Z_n^i(j)/M_{0,n-1}(i,j)$ are proved in Section 5.

## 2. Notation and main results

Let $\mathbb{N} = \{0, 1, \dots\}$ be the set of non-negative integers. The indicator of an event $A$ is denoted by $1_A$. The symbol $\xrightarrow{\mathbb{P}}$ denotes the convergence in probability with respect to the annealed law $\mathbb{P}$. For an integer $d \ge 1$, let $\mathbb{R}^d$ be the $d$-dimensional space of vectors with real coordinates, equipped with the scalar product and the $L^1$-norm respectively defined by

$$ \langle x, y \rangle := \sum_{i=1}^{d} x(i)y(i) \quad \text{and} \quad \|x\| := \sum_{i=1}^{d} |x(i)|, \quad x, y \in \mathbb{R}^d. $$
Let $e_i$ be the $d$-dimensional vector with 1 in the $i$-th place and 0 elsewhere. Define also $\mathcal{M}_d(\mathbb{R})$ the set of $d \times d$ matrices with entries in $\mathbb{R}$, and the operator norm on $\mathcal{M}_d(\mathbb{R})$:

$$ \|M\| := \sup_{\|x\|=1} \|Mx\|, \quad M \in \mathcal{M}_d(\mathbb{R}). $$

For a matrix or a vector $X$, we write $X > 0$ to mean that each entry of $X$ is strictly positive.

Let us define precisely the multi-type branching process in random environment (MBPRE). Let $\xi = (\xi_n)_{n \ge 0}$ be the random environment, which is an independent and identically distributed (i.i.d.) sequence with values in an abstract space $\mathcal{X}$. To each realization of $\xi_n$, we associate $d$ probability generating functions : for $1 \le r \le d$,

$$ f_n^r(s) = \sum_{k_1, \dots, k_d=0}^{\infty} p_{k_1, \dots, k_d}^r(\xi_n) s_1^{k_1} \cdots s_d^{k_d}, \quad s = (s_1, \dots, s_d) \in [0, 1]^d. $$

A MBPRE ($Z_n$) in the random environment $\xi$ is a process with values in $\mathbb{N}^d$ such that for all $n \ge 0$,

$$ Z_{n+1} = \sum_{r=1}^{d} \sum_{l=1}^{Z_n(r)} N_{l,n}^{r}, \qquad (2.1) $$

where $Z_0 \in \mathbb{N}^d$ is fixed, $Z_n(j)$ represents the number of particles of type $j$ of some population in generation $n$, and $N_{l,n}^r(j)$ is the offspring of type $j$ at time $n+1$ of the $l$-th particle of type $r$ in generation $n$. The random vectors $N_{l,n}^r = (N_{l,n}^r(1), \dots, N_{l,n}^r(d))$, indexed by $l \ge 1$, $n \ge 0$, $r \in \{1, \dots, d\}$, are conditionally independent and have the same probability generating function $f_n^r$, given the environment $\xi$. Set $f_n := (f_n^1, \dots, f_n^d)$. When the process starts with one initial particle of type $i$, that is, when $Z_0 = e_i$, we write $Z_n^i$ instead of $Z_n$.

Denote by $\mathbb{P}_\xi$ the underline probability when the environment $\xi$ is given; it is called quenched law. Let $\tau$ be the law of the environment $\xi$. Then, the total probability $\mathbb{P}$, called annealed law, is defined by $\mathbb{P}(dx, d\xi) = \mathbb{P}_\xi(dx)\tau(d\xi)$. The expectation with respect to $\mathbb{P}_\xi$ and $\mathbb{P}$ are denoted respectively by $\mathbb{E}_\xi$ and $\mathbb{E}$. By our notation the quenched probability generating function of $N_{l,n}^r$ is

$$ f_n^r(s) = \mathbb{E}_\xi \left( \prod_{j=1}^{d} s_j^{N_{l,n}^r(j)} \right), \quad s = (s_1, \dots, s_d) \in [0, 1]^d. $$

We introduce the random mean matrices $M_n \in \mathcal{M}_d(\mathbb{R})$ whose entries are defined by

$$ M_n(i, j) := \frac{\partial f_n^i(\mathbf{1})}{\partial s_j} = \mathbb{E}_\xi [Z_{n+1}(j) | Z_n = e_i], \quad 1 \le i, j \le d, n \ge 0, $$

where $\frac{\partial f_n^i(\mathbf{1})}{\partial s_j}$ is the left derivative at $\mathbf{1} = (1, \dots, 1) \in \mathbb{R}^d$ of the $d$-dimensional probability generating function $f_n^i$ with respect to $s_j$. For each $1 \le i, j \le d$, $M_n(i, j)$
represents the conditioned mean of the number of children of type $j$ produced by a particle of type $i$ at time $n$. The sequence of random matrices $(M_n)$ is i.i.d. (because the sequence $(\xi_n)$ is i.i.d.). We define the products of these matrices by

$$M_{k,n} := M_k \cdots M_n, \quad 0 \le k \le n.$$

Notice that we have

$$\mathbb{E}_{\xi} Z_{n+1}^{i}(j) = M_{0,n}(i,j), \quad n \ge 0, \ 1 \le i, j \le d. \qquad (2.2)$$

For $n, k \ge 0$, denote by $\rho_{n,n+k}$ the spectral radius of $M_{n,n+k}$. By the Perron-Frobenius theorem (see e.g. [3]), $\rho_{n,n+k}$ is an eigenvalue of $M_{n,n+k}$. Let $U_{n,n+k}$ and $V_{n,n+k}$ be respectively the positive right and left eigenvectors associated to the eigenvalue $\rho_{n,n+k}$, with the normalizations $\|U_{n,n+k}\| = 1$ and $\langle V_{n,n+k}, U_{n,n+k} \rangle = 1$.

Let $\mathcal{G}_+^0$ be the subset of the matrices of $M_d(\mathbb{R})$ with strictly positive entries. According to the results of Hennion [15, Lemma 3.3 and Theorem 1], if $M_0$ is allowable in the sense that every row and column contains a strictly positive element, and if the positivity condition

$$\mathbb{P}\left(\bigcup_{n \ge 0} \{M_{0,n} \in \mathcal{G}_+^0\} > 0\right) > 0 \qquad (2.3)$$

holds, then the random vectors $U_{n,\infty}$ and the random scalars $\lambda_n$ are well defined by (1.9) and (1.10), and satisfy (1.11). Note that the sequences $(U_{n,\infty})$ and $(\lambda_n)$ are stationary ergodic. It is proved in [11, Theorem 1] that the sequence $(W_n^i)_{n \ge 0}$ defined by (1.12) is a non-negative martingale under $\mathbb{P}_\xi$ and $\mathbb{P}$, with respect to the filtration

$$\mathcal{F}_0 = \sigma(\xi), \quad \mathcal{F}_n = \sigma(\xi, N_{l,k}^r(j), 0 \le k \le n-1, 1 \le r, j \le d, l \ge 1) \quad \text{for } n \ge 1.$$

Thus $\mathbb{P}$-a.s. for all $1 \le i \le d$, the limit

$$W^i := \lim_{n \to +\infty} W_n^i \qquad (2.4)$$

exists and $\mathbb{E}_{\xi} W^i \le 1$ by Fatou's lemma.

Now we introduce a classification of MBPRE's. Under the following moment condition

$$\mathbb{E} \log^+ \|M_0\| < +\infty, \qquad (2.5)$$

by an argument of sub-additivity, the limite

$$\gamma := \lim_{n \to +\infty} \frac{1}{n} \mathbb{E} \log \|M_{0,n-1}\| = \inf_{n \ge 1} \frac{1}{n} \mathbb{E} \log \|M_{0,n-1}\|,$$

exists; it is called Lyapunov exponent of the sequence $(M_n)_{n \ge 0}$. Moreover, Furstenberg and Kesten established in [9] a strong law of large numbers for $\log \|M_{0,n-1}\|$:

$$\lim_{n \to +\infty} \frac{1}{n} \log \|M_{0,n-1}\| = \gamma \quad \mathbb{P}\text{-a.s.}$$
We say that a MBPRE is supercritical if $\gamma > 0$, critical if $\gamma = 0$, and subcritical if $\gamma < 0$. In this paper, the process $(Z_n)$ will always be supercritical, i.e. $\gamma > 0$, under the conditions that we will assume.

The non-degeneracy of the limit variables $W^i$ has been studied in [11]. In particular, when $\gamma > 0$, it has been proved in [11, Theorem 2.6] that the $X \log X$ condition

$$ \mathbb{E} \left( \frac{Z_1^i(j)}{M_0(i, j)} \log^+ \frac{Z_1^i(j)}{M_0(i, j)} \right) < +\infty \quad \forall 1 \le i, j \le d \quad (2.6) $$

is sufficient for the non-degeneracy of each $W^i$ in the sense that $\mathbb{P}(W^i > 0) > 0$, and that this condition is also necessary under the additional condition **H1** that we will introduce below. Moreover, when $W^i$ are non-degenerate, then

$$ \mathbb{P}_{\xi}(W^i > 0) > 0 \quad \text{and} \quad \mathbb{E}_{\xi}W^i = 1 \quad \text{a.s.,} \quad \text{and} \quad W_n^i \to W^i \text{ in } L^1. \quad (2.7) $$

In this paper, for a given $p > 1$, we study the convergence in $L^p$ of the fundamental martingale $(W_n^i)_{n \ge 0}$ and the normalized population size $Z_n^i(j)/\mathbb{E}_{\xi}Z_n^i(j)$, for all $i, j = 1, \dots, d$.

We first consider the martingale $(W_n^i)_{n \ge 0}$, $1 \le i \le d$. To formulate our results, we need to introduce some notation and condition. Set

$$ I = \{s \le 0 : \mathbb{E}M_0(i, j)^s < +\infty \quad \forall i, j = 1, \dots, d\}. $$

Obviously, by Hölder's inequality, $I$ is an interval, and if there exists $s \in I$ with $s < 0$, then $M_0 > 0$ P-a.s., so that condition (2.3) is satisfied. It will be seen in Proposition 3.1 that for $s \in I$ the limit

$$ \kappa(s) := \lim_{n \to +\infty} (\mathbb{E}\|M_{0,n-1}\|^s)^{1/n} \quad (2.8) $$

exists, with $\kappa(s) < +\infty$. Notice that $\kappa$ is a log-convex function on $I$. We will need the following condition of Furstenberg and Kesten [9]:

**H1.** There exists a constant $D > 1$ such that

$$ 1 \le \frac{\max_{1 \le i,j \le d} M_0(i, j)}{\min_{1 \le i,j \le d} M_0(i, j)} \le D. $$

Note that condition **H1** implies condition (2.3).

Our first theorem gives sufficient and necessary conditions for the $L^p$ convergence of the martingales $(W_n^i)$, $1 \le i \le d$.

**Theorem 2.1.** Let $p > 1$ be such that $1 - p \in I$. If

$$ \max_{1 \le i,j \le d} \mathbb{E} \left( \frac{Z_1^i(j)}{M_0(i, j)} \right)^p < +\infty \quad \text{and} \quad \kappa(1-p) < 1, \quad (2.9) $$

then $W_n^i \xrightarrow{n \to +\infty} W^i$ in $L^p$ for any $1 \le i \le d$. The converse is also valid when the Furstenberg-Kesten condition **H1** holds.
It is clear that condition (2.9) implies (2.6). Moreover, (2.9) also implies the supercritical condition $\gamma > 0$ when condition (2.5) holds, since by Jensen's inequality we have $\log \kappa(1-p) \ge (1-p)\gamma$.

Our second theorem shows that the $L^p$ convergence of $W_n^i$ has an exponential rate.

**Theorem 2.2.** Let $p > 1$ be such that $1-p \in I$. Assume (2.9).

$$1. \text{ If } 1 < p \le 2, \text{ then denoting } \delta_c(p) = \kappa(1-p)^{1/p} \text{ we have}$$
$$\limsup_{n \to +\infty} \delta_c(p)^{-n} (\mathbb{E}|W_n^i - W^i|^p)^{1/p} < +\infty. \quad (2.10)$$

$$2. \text{ If } p > 2, \text{ then } \delta_c(p) := \max \left\{ \kappa(1-p)^{1/p}, \kappa(-p/2)^{1/p} \right\} < 1, \text{ and}$$
$$\lim_{n \to +\infty} \delta^{-n} (\mathbb{E}|W_n^i - W^i|^p)^{1/p} = 0 \quad \forall \delta > \delta_c(p). \quad (2.11)$$

In the proof we shall see that in Part 1 the moment condition $\mathbb{E}(\frac{Z_1^i(j)}{M_0(i,j)})^p < +\infty$ for all $1 \le i, j \le d$ can be relaxed to $\mathbb{E}(W_1^i)^p < +\infty$ for all $1 \le i \le d$.

Note that for $p \ge 2$, by applying Hölder's inequality to $\mathbb{E}\|M_{0,n-1}\|^{-p/2}$ and then letting $n \to +\infty$, it is easy to see that $\kappa(-p/2)^{2/p} \le \kappa(1-p)^{1/(p-1)}$. Thus $\kappa(1-p) < 1$ implies $\kappa(-p/2) < 1$, so that $\delta_c(p) < 1$.

Now we investigate the convergence in $L^p$ of the normalized population size $\frac{Z_n^i(j)}{\mathbb{E}Z_n^i(j)} = \frac{Z_{0,n-1}^i(i,j)}{M_{0,n-1}(i,j)}$. Recall that under condition (2.5), **H1** and the supercriticality condition $\gamma > 0$, by the Kesten-Stigum type theorem for a supercritical MBPRE [11, Theorem 2.11], for all $1 \le i, j \le d$,

$$\frac{Z_n^i(j)}{M_{0,n-1}(i,j)} \xrightarrow{n \to +\infty} W^i, \quad (2.12)$$

and the convergence holds a.s. if additionally $\mathbb{E}(Z_1^i(j)/M_0(i,j))^p < +\infty$ and $\mathbb{E}\|M_0\|^{1-p} < +\infty$ for some $p > 1$ and all $1 \le i, j \le d$ (see [11, Theorem 2.13]). By [11, Theorem 2.11] and Sheffé's theorem, under the supercritical condition $\gamma > 0$ and the Furstengerg-Kesten condition **H1**, $\frac{Z_n^i(j)}{M_{0,n-1}(i,j)} \to W^i$ in $L^1$ if and only if (2.6) holds. From Theorem 2.1 and under condition **H1**, we obtain a criterion for the convergence in $L^p$ of $\frac{Z_n^i(j)}{M_{0,n-1}(i,j)}$.

**Theorem 2.3.** Assume condition **H1**. Let $p > 1$ be such that $1-p \in I$. Then $\frac{Z_n^i(j)}{M_{0,n-1}(i,j)} \xrightarrow{n \to +\infty} W^i$ in $L^p$ for any $1 \le i, j \le d$ if and only if (2.9) holds.

Finally, from Theorem 2.2, we deduce an exponential rate of the convergence in $L^p$ of $\frac{Z_n^i(j)}{M_{0,n-1}(i,j)}$.

**Theorem 2.4.** Assume condition **H1**. Let $p > 1$ be such that $1-p \in I$ and that (2.9) holds. Then there exists $\delta \in (0,1)$ such that

$$\lim_{n \to +\infty} \delta^{-n} \left( \mathbb{E} \left| \frac{Z_n^i(j)}{M_{0,n-1}(i,j)} - W^i \right|^p \right)^{1/p} = 0. \quad (2.13)$$
### 3. Spectral properties of the transfer operator $P_s$

We start this section by giving some notation. Denote by $\mathcal{S} = \{x \in \mathbb{R}^d : x \ge 0, \|x\| = 1\}$. For $x \in \mathcal{S}$ and $M \in \mathcal{G}_+^0$ (the set of matrices with strictly positive entries), define the projective action of $M$ on $\mathcal{S}$ by $M \cdot x := \frac{Mx}{\|Mx\|}$. Let $\mathcal{C}(\mathcal{S})$ be the space of continuous functions on $\mathcal{S}$ with real values. For any $\varphi \in \mathcal{C}(\mathcal{S})$, set

$$ \|\varphi\|_{\infty} = \sup_{x \in \mathcal{S}} \|\varphi x\|. $$

For $s \in I$, define the transfer operator $P_s$ as follows: for all $\varphi \in \mathcal{C}(\mathcal{S})$,

$$ P_s\varphi(x) := \mathbb{E}[\|M_0x\|^s\varphi(M_0 \cdot x)], \quad x \in \mathcal{S}. \quad (3.1) $$

Define also the conjugate operator $P_s^*$, such that for $s \in I$ and $\varphi \in \mathcal{C}(\mathcal{S})$,

$$ P_s^*\varphi(x) := \mathbb{E}[\|M_0^T x\|^s \varphi(M_0^T \cdot x)], \quad x \in \mathcal{S}. \quad (3.2) $$

In this section, we investigate the spectral properties of the transfer operator $P_s$ and its conjugate $P_s^*$ for $s \le 0$. These results extend some properties known in the case $s \ge 0$ (see [25, 6]) to the case $s < 0$. We also give some properties of $\kappa(s)$. The main result is given by the following proposition. We use the notation $\mu(\psi) = \int \psi d\mu$ to denote the integral of $\psi$ with respect the measure $\mu$.

**Proposition 3.1.** Assume that $s \in I$. Then $\kappa(s) < +\infty$, the spectral radius of $P_s$ is equal to $\kappa(s)$, and there exists a probability measure $\nu_s$ on $\mathcal{S}$ and a strictly positive function $r_s \in \mathcal{C}(\mathcal{S})$ such that

$$ \nu_s P_s = \kappa(s) \nu_s \quad \text{and} \quad P_s r_s = \kappa(s) r_s, $$

where $\nu_s P_s$ denotes the measure on $\mathcal{S}$ such that $(\nu_s P_s)(\psi) = \nu_s(P_s\psi)$ for all $\psi \in \mathcal{C}(\mathcal{S})$. Moreover, $\kappa(s)$ is also the spectral radius of $P_s^*$, and there exists a probability measure $\nu_s^*$ on $\mathcal{S}$ and a strictly positive function $r_s^* \in \mathcal{C}(\mathcal{S})$ such that

$$ \nu_s^* P_s^* = \kappa(s) \nu_s^* \quad \text{and} \quad P_s^* r_s^* = \kappa(s) r_s^*. $$

To prove the above proposition, we will use the following Lemma about the properties of $\kappa(s)$, $s \le 0$.

**Lemma 3.2.** Assume that $s \in I$. Then

$$ \kappa(s) = \lim_{n \to +\infty} (\mathbb{E}\|M_{0,n-1}\|^s)^{1/n} = \sup_{n \ge 1} (\mathbb{E}\|M_{0,n-1}\|^s)^{1/n} < +\infty, $$

and there exists $C_s > 0$ such that for all $x, y \in \mathcal{S}$ and $n \ge 1$,

$$ \mathbb{E}\|M_{0,n-1}\|^s \le \mathbb{E}\|M_{0,n-1}x\|^s \le \mathbb{E}\langle M_{0,n-1}x, y \rangle^s \le C_s \mathbb{E}\|M_{0,n-1}\|^s. $$

*Proof.* Notice that the sequence $(\mathbb{E}\|M_{0,n-1}\|^s)_{n \ge 1}$ is super-multiplicative for $s \in I$, so the limit $\kappa(s) = \lim_{n \to +\infty} (\mathbb{E}\|M_{0,n-1}\|^s)^{1/n}$ exists, and

$$ \kappa(s) = \sup_{n \ge 1} (\mathbb{E}\|M_{0,n-1}\|^s)^{1/n} \in \mathbb{R}_+ \cup \{+\infty\}. $$
Clearly, for all $x, y \in S$ and $n \ge 1$, we have the inequalities

$$ \mathbb{E}\|M_{0,n-1}\|^s \le \mathbb{E}\|M_{0,n-1}x\|^s \le \mathbb{E}\langle M_{0,n-1}x, y \rangle^s \le \mathbb{E}\left(\max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s\right). \quad (3.3) $$

Moreover, since the sequence of matrices ($M_n$) is i.i.d, for all $n, k \ge 1$ we have

$$ \begin{aligned} & \mathbb{E} \max_{1 \le i,j \le d} M_{0,n+k-1}(i,j)^s \\ & \le \mathbb{E} \max_{1 \le i,j,l \le d} (M_{0,n-1}(i,l)^s M_{n,n+k-1}(l,j)^s) \\ & \le \mathbb{E} \max_{1 \le i,l \le d} M_{0,n-1}(i,l)^s \mathbb{E} \max_{1 \le l,j \le d} M_{0,k-1}(l,j)^s. \end{aligned} $$

Hence $(\mathbb{E} \max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s)_{n \ge 1}$ is sub-multiplicative, so that

$$ \lim_{n \to +\infty} \left( \mathbb{E} \max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s \right)^{1/n} = \inf_{n \ge 1} \left( \mathbb{E} \max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s \right)^{1/n} $$

Combining this with (3.3), and letting $n \to +\infty$, we obtain

$$ \kappa(s) \le \lim_{n \to +\infty} \left( \mathbb{E} \max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s \right)^{1/n} \le \mathbb{E} \max_{1 \le i,j \le d} M_0(i,j)^s < +\infty. $$

Furthermore by simple calculations, for all $n \ge 3$ it holds that

$$ \begin{aligned} & \mathbb{E} \max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s \\ &= \mathbb{E} \max_{1 \le i,j \le d} \left( \sum_{1 \le l_1, l_2 \le d} M_0(i, l_1) M_{1,n-2}(l_1, l_2) M_{n-1}(l_2, j) \right)^s \\ &\le \mathbb{E} \left( \min_{1 \le i, l_1 \le d} M_0(i, l_1) \min_{1 \le l_2, j \le d} M_{n-1}(l_2, j) \sum_{1 \le l_1, l_2 \le d} M_{1,n-2}(l_1, l_2) \right)^s \\ &\le \left( \mathbb{E} \max_{1 \le i,j \le d} M_0(i,j)^s \right)^2 \mathbb{E} \|M_{0,n-3}\|^s. \end{aligned} $$

It follows that for all $n \ge 3$,

$$ \begin{aligned} & \mathbb{E} \max_{1 \le i,j \le d} M_{0,n-1}(i,j)^s \\ & \le \left( \mathbb{E} \max_{1 \le i,j \le d} M_0(i,j)^s \right)^2 \frac{\mathbb{E}(\|M_{0,n-3}\| \|M_{n-2}\| \|M_{n-1}\|^s)}{(\mathbb{E}\|M_0\|^s)^2} \\ & \le \left( \frac{\mathbb{E} \max_{1 \le i,j \le d} M_0(i,j)^s}{\mathbb{E}\|M_0\|^s} \right)^2 \mathbb{E}\|M_{0,n-1}\|^s. \end{aligned} $$

This, together with (3.3), proves the inequalities of Lemma 3.2 for $n \ge 3$ with

$$ C_s = \left( \frac{\mathbb{E} \max_{1 \le i,j \le d} M_0(i,j)^s}{\mathbb{E} \|M_0\|^s} \right)^2, $$

which is finite since $s \in I$. It is clear that the inequalities remain valid for $1 \le n \le 2$ by modifying slightly the value of $C_s$ (choosing it large enough). This concludes the proof of Lemma 3.2. $\square$
Proof of Proposition 3.1. We shall use an argument similar to that in the proof of [6, Proposition 4.4] where the case $s \ge 0$ is considered. Let $M^1(S)$ be the set of all probability measures on $S$, and $M_b^1(\mathbb{R})$ the set of all finite signed measures on $\mathbb{R}$ equipped with the total variation norm. Since $M^1(S)$ is a compact convex subset of the Banach space $M_b^1(\mathbb{R})$, by the Schauder-Tychonoff theorem applied to the continuous map $\nu \mapsto \nu P_s/\nu P_s(S)$, there exists an invariant probability measure $\nu_s \in M^1(S)$ of this map. Consequently, $\nu_s$ is an eigenmeasure of $P_s$:

$$ \nu_s P_s = [\nu_s P_s(S)] \nu_s. \qquad (3.4) $$

In the same way there exists a probability eigenmeasure $\nu_s^*$ of the operator $P_s^*$, associated to the eigenvalue $k(s) = \nu_s^* P_s^*(S)$:

$$ \nu_s^* P_s^* = k(s) \nu_s^*. \qquad (3.5) $$

Set

$$ r_s(x) := \frac{1}{k(s)} \int_S \mathbb{E}\langle M_0 x, y \rangle^s \nu_s^*(dy), \quad x \in S. $$

Since $s \in I$, it is clear that for all $x \in S$,

$$ 0 < \frac{1}{k(s)} \mathbb{E} \min_{1 \le i,j \le d} M_0(i,j)^s \le r_s(x) \le \frac{1}{k(s)} \mathbb{E} \max_{1 \le i,j \le d} M_0(i,j)^s < +\infty, $$

and that $r_s \in C(S)$. Moreover, for all $x \in S$ we have

$$ \begin{aligned} r_s(x) &= \frac{1}{k(s)} \int_S \mathbb{E}\langle x, M_0^T y \rangle^s \nu_s^*(dy) \\ &= \frac{1}{k(s)} \int_S \mathbb{E}[\|M_0^T y\|^s \langle x, M_0^T \cdot y \rangle^s] \nu_s^*(dy) \\ &= \frac{1}{k(s)} \int_S \langle x, y \rangle^s (\nu_s^* P_s^*)(dy) \\ &= \int_S \langle x, y \rangle^s \nu_s^*(dy). \end{aligned} \qquad (3.6) $$

Using Fubini's theorem, it follows from the definition of $r_s$ and (3.6) that for all $x \in S$,

$$ \begin{aligned} r_s(x) &= \frac{1}{k(s)} \mathbb{E}[\|M_0 x\|^s \int_S \langle M_0 \cdot x, y \rangle^s \nu_s^*(dy)] \\ &= \frac{1}{k(s)} \mathbb{E}[\|M_0 x\|^s r_s(M_0 \cdot x)] \\ &= \frac{1}{k(s)} P_s r_s(x). \end{aligned} \qquad (3.7) $$

So we have proved that $P_s r_s = k(s) r_s$. Now we show that

$$ k(s) = \rho(P_s) = \kappa(s), $$
where $\rho(P_s)$ is the spectral radius of $P_s$.

First we have $k(s) \le \rho(P_s)$, since $k(s)$ is a positive eigenvalue of $P_s$.

Next, we prove that $\rho(P_s) \le \kappa(s)$. By iteration of the operator $P_s$ (using the fact that $M_{0,n-1}$ has the same law as $M_{n-1} \cdots M_0$) and Lemma 3.2, it holds that for all $\varphi \in C(S)$ and $x \in S$,

$$P_s^n \varphi(x) = \mathbb{E}[\|M_{0,n-1}x\|^s \varphi(M_{0,n-1} \cdot x)] \le C_s \|\varphi\|_{\infty} \mathbb{E}\|M_{0,n-1}\|^s.$$

This implies that

$$\rho(P_s) = \lim_{n \to \infty} \sup \{ \| P_s^n \varphi \|_{\infty}^{1/n} : \| \varphi \|_{\infty} = 1 \} \le \lim_{n \to +\infty} (\mathbb{E} \| M_{0,n-1} \|^s)^{1/n} = \kappa(s).$$

We then prove that $\kappa(s) \le k(s)$. Iterating the relation $\nu_s^* P_s^* = k(s) \nu_s^*$, we obtain $\nu_s^* P_s^{*n} = k(s)^n \nu_s^*$, so that

$$k(s)^n = \nu_s^* P_s^{*n}(S) = \int_S \mathbb{E} \| M_{0,n-1}^T y \|^s \nu_s^*(dy) \ge \mathbb{E} \| M_{0,n-1}^T \|^s.$$

This implies $\kappa(s) \le k(s)$, since

$$\kappa(s) = \lim_{n \to +\infty} (\mathbb{E} \| M_{0,n-1} \|^s)^{1/n} = \lim_{n \to +\infty} (\mathbb{E} \| M_{0,n-1}^T \|^s)^{1/n}.$$

So we have proved the equalities $k(s) = \rho(P_s) = \kappa(s)$. This together with (3.5) and (3.7) implies

$$\nu_s^* P_s^* = \kappa(s) \nu_s^* \quad \text{and} \quad P_s r_s = \kappa(s) r_s.$$

Changing the roles of $P_s$ and $P_s^*$, by the same arguments we can prove that

$$\nu_s P_s = \kappa(s) \nu_s \quad \text{and} \quad P_s^* r_s^* = \kappa(s) r_s^*$$

for some strictly positive function $r_s^* \in C(S)$, and that $\kappa(s)$ is also the spectral radius of $P_s^*$. This concludes the proof of Proposition 3.1. $\square$

### 4. Convergence in $L^p$ of the martingale $W_n^i$

In this section, we prove Theorems 2.1 and 2.2 giving sufficient and necessary conditions for the $L^p$ convergence of $W_n^i$, $1 \le i \le d$, with an exponential speed. First we formulate the following result.

**Theorem 4.1.** Let $p > 1$ be such that $1 - p \in I$. Consider the assertions:

$$W_n^i \underset{n \to +\infty}{\longrightarrow} W^i \quad \text{in } L^p \quad \forall i = 1, \dots, d \qquad (4.1)$$

$$\mathbb{E}(W_1^i)^p < +\infty \quad \forall i = 1, \dots, d \quad \text{and} \quad \kappa(1-p) < 1. \qquad (4.2)$$

$$\mathbb{E}\left(\frac{Z_1^i(j)}{M_0(i, j)}\right)^p < +\infty \quad \forall i, j = 1, \dots, d \quad \text{and} \quad \kappa(1-p) < 1. \qquad (4.3)$$

If $1 < p \le 2$, then we have the implications: (4.3) $\Rightarrow$ (4.1) $\leftrightarrow$ (4.2). If $p > 2$, then we have: (4.3) $\Rightarrow$ (4.1) $\Rightarrow$ (4.2). When the Furstenberg-Kesten condition **H1** holds, then for each $p > 1$, (4.3) $\leftrightarrow$ (4.1) $\leftrightarrow$ (4.2).
It is clear that the assertions of Theorem 2.1 follow from Theorem 4.1. Theorem 4.1 is slightly stronger in the sense that for $1 < p \le 2$, it gives a sufficient and necessary condition without assuming the Furstenberg-Kesten condition **H1**.

The following Lemma will be useful to investigate the convergence in $L^p$, and it is a direct consequence of the Marcinkiewicz-Zygmund inequality in [7, Theorem 1.5], as stated in [27, Lemma 1.4].

**Lemma 4.2.** Let $(X_k)_{k \ge 1}$ be a sequence of i.i.d. random centered variables. Then for all $n \ge 1$ and $p > 1$:

$$ \mathbb{E} \left| \sum_{k=1}^{n} X_k \right|^p \le \begin{cases} (B_p)^p \mathbb{E} |X_k|^p n, & \text{if } 1 < p \le 2, \\ (B_p)^p \mathbb{E} |X_k|^{p \frac{p}{2}}, & \text{if } p > 2, \end{cases} $$

where $B_p = 2 \min \{k^{1/2} : k \in \mathbb{N}, k \ge \frac{p}{2}\}$.

In a last article, we proved the following result [11, Lemma 7.1]. It gives some properties on the products of random matrices $M_{n,n+k}$ under the Furstenberg-Kesten condition **H1**.

**Lemma 4.3.** Assume condition **H1**. Then:

1. for all $n, k \ge 0$ and $1 \le i, j, r \le d$, $\mathbb{P}$-a.s.,

$$ \frac{1}{D} \le \frac{M_{n,n+k}(i,j)}{M_{n,n+k}(i,r)} \le D \quad \text{and} \quad \frac{1}{D} \le \frac{M_{n,n+k}(i,j)}{M_{n,n+k}(r,j)} \le D; \quad (4.4) $$

2. for all $n, k \ge 0$ and $1 \le i, j \le d$, $\mathbb{P}$-a.s.,

$$ \frac{1}{dD^2} \le \frac{M_{n,n+k}(i,j)U_{n+k+1,\infty}(j)}{\lambda_{n,n+k}U_{n,\infty}(i)} \le 1. \quad (4.5) $$

**Proof of Theorems 4.1 and 2.2.** By iterating (2.1), it is easy to see that the process $(Z_n)_{n \ge 0}$ satisfies the relation

$$ Z_{n+k} = \sum_{j=1}^{d} \sum_{l=1}^{Z_n(j)} Z_{l,n,k}^{j}, \quad n \ge 0, k \ge 1, \quad (4.6) $$

where $Z_{l,n,k}^j(r)$ denotes the number of the offspring of type $r$ at time $n+k$ of the $l$-th particle of type $j$ in the generation $n$; conditional on the environment $\xi$, the random vectors $Z_{l,n,k}^j = (Z_{l,n,k}^j(1), \dots, Z_{l,n,k}^j(d))$ indexed by $l \in \mathbb{N}^*$ and $j \in \{1, \dots, d\}$ (for fixed $n$ and $k$) are independent, each has the probability generating function $f_n^j \circ f_{n+1}^j \circ \dots \circ f_{n+k-1}^j$. Combining (4.6), (1.12) and (1.11), we have, for all $n, k \ge 0$
and $1 \le i \le d$,

$$
\begin{align}
W_{n+k}^i - W_n^i &= \sum_{r=1}^d \frac{U_{n+k, \infty}(r)}{\lambda_{0,n+k-1} U_{0, \infty}(i)} \sum_{j=1}^d \sum_{l=1}^{Z_n^i(j)} Z_{l,n,k}^j(r) - W_n^i \nonumber \\
&= \sum_{j=1}^d \frac{U_{n, \infty}(j)}{\lambda_{0,n-1} U_{0, \infty}(i)} \sum_{l=1}^{Z_n^i(j)} \sum_{r=1}^d \frac{U_{n+k, \infty}(r) Z_{l,n,k}^j(r)}{\lambda_{n,n+k-1} U_{n, \infty}(j)} - W_n^i \nonumber \\
&= \sum_{j=1}^d \frac{U_{n, \infty}(j)}{\lambda_{0,n-1} U_{0, \infty}(i)} \sum_{l=1}^{Z_n^i(j)} (W_{l,n,k}^j - 1), \tag{4.7}
\end{align}
$$

where

$$
W_{l,n,k}^{j} := \frac{\langle Z_{l,n,k}^{j}, U_{n+k,\infty} \rangle}{\lambda_{n,n+k-1} U_{n,\infty}(j)}.
$$

Let $T$ be the shift operator of the environment sequence:

$$
T\xi = (\xi_1, \xi_2, \dots) \quad \text{if} \quad \xi = (\xi_0, \xi_1, \dots),
$$

and let $T^n$ be its $n$-fold iteration. It is clear that, given the environment $\xi$, the random variables $W_{l,n,k}^j$, $l \ge 1$, are i.i.d.; they are independent of $\xi_0, \dots, \xi_{n-1}$ and $Z_n^i$, and have the same distribution as $W_{n,k}^j$, where $(W_{n,k}^j)_{k \ge 0}$ is the martingale associated to a MBPRE starting with one individual of type $j$, in the shift random environment $T^n\xi$.

We divide the proof into 5 steps.

Step 1. We first prove that for $1 < p \le 2$, we have the implications (4.2) $\Rightarrow$ (4.1) of Theorem 4.1, and (4.2) $\Rightarrow$ (2.10) of Theorem 2.2. We assume that $1 < p \le 2$ and (4.2). Applying (4.7), the convexity of the function $x \mapsto x^p$ (together with the fact that $\sum_{j=1}^d U_{n,\infty}(j) = 1$), Lemma 4.2 and (1.11), for all $n \ge 0$, $k \ge 1$ and $1 \le i \le d$, $\mathbb{P}$-a.s., we have (for $1 < p \le 2$),

$$
\begin{align}
\mathbb{E}_{\xi} |W_{n+k}^i - W_n^i|^p &\le \mathbb{E}_{\xi} \left( \left| \sum_{j=1}^d \frac{U_{n,\infty}(j)}{\lambda_{0,n-1} U_{0,\infty}(i)} \right| \sum_{l=1}^{Z_n^i(j)} (W_{l,n,k}^j - 1) \right|^p \nonumber \\
&\le \sum_{j=1}^d \frac{U_{n,\infty}(j)}{(\lambda_{0,n-1} U_{0,\infty}(i))^p} \mathbb{E}_{\xi} \left( \left| \sum_{l=1}^{Z_n^i(j)} (W_{l,n,k}^j - 1) \right| \right)^p \nonumber \\
&\le B_p^p \sum_{j=1}^d \frac{U_{n,\infty}(j)}{(\lambda_{0,n-1} U_{0,\infty}(i))^p} \mathbb{E}_{\xi} Z_n^i(j) \mathbb{E}_{\xi} |W_{n,k}^j - 1|^p \nonumber \\
&\le B_p^p \sigma_{n,k}(p) \sum_{j=1}^d \frac{M_{0,n-1}(i,j) U_{n,\infty}(j)}{(\lambda_{0,n-1} U_{0,\infty}(i))^p} \nonumber \\
&= B_p^p \sigma_{n,k}(p) (\lambda_{0,n-1} U_{0,\infty}(i))^{1-p}, \tag{4.8}
\end{align}
$$
with

$$ \sigma_{n,k}(p) = \max_{1 \le j \le d} \mathbb{E}_{\xi} |W_{n,k}^j - 1|^p. \quad (4.9) $$

Using again (1.11) and Lemma 3.2 together with the fact that $M_{0,n-1}$ is independent of $T^n\xi$, we get that for all $s \in I$, $n \ge 1$ and $1 \le i \le d$, $\mathbb{P}$-a.s.

$$ \begin{cases} \mathbb{E}_{T^n\xi} \lambda_{0,n-1}^s = \mathbb{E}_{T^n\xi} \|M_{0,n-1}U_{n,\infty}\|^s \le C_s \kappa(s)^n; \\ \mathbb{E}_{T^n\xi} (\lambda_{0,n-1}U_{0,\infty}(i))^s = \mathbb{E}_{T^n\xi} \langle M_{0,n-1}U_{n,\infty}, e_i \rangle^s \le C_s \kappa(s)^n. \end{cases} \quad (4.10) $$

Taking expectation in (4.8), by (4.10) we get that for all $n \ge 0$, $k \ge 1$ and $1 \le i \le d$,

$$ \begin{aligned} \mathbb{E}|W_{n+k}^i - W_n^i|^p &\le B_p^p \mathbb{E}[\sigma_{n,k}(p)\mathbb{E}_{T^n\xi}(\lambda_{0,n-1}U_{0,\infty}(i))^{1-p}] \\ &\le B_p^p C_{1-p} \mathbb{E}\sigma_{0,k}(p)\kappa(1-p)^n. \end{aligned} \quad (4.11) $$

From condition (4.2) we have $\mathbb{E}\sigma_{0,1}(p) < +\infty$ and $\kappa(1-p) < 1$. So by the triangular inequality of $L^p$, it follows from (4.11) that for all $1 \le i \le d$, with $C = B_p[C_{1-p}\mathbb{E}\sigma_{0,1}(p)]^{1/p}$,

$$ \sup_{n \ge 0} (\mathbb{E}(W_n^i)^p)^{1/p} \le 1 + C \sum_{n=0}^{+\infty} \kappa(1-p)^{n/p} < +\infty. \quad (4.12) $$

Therefore for all $1 \le i \le d$, $(W_n^i)$ is a martingale bounded in $L^p$, so that it converges in $L^p$. This proves the implication (4.2) ⇒ (4.1) of Theorem 4.1. Furthermore (4.12) implies that $\sup_{k \ge 0} \mathbb{E}\sigma_{0,k}(p) < +\infty$. So, by letting $k \to +\infty$ in (4.11) we get (2.10) of Theorem 2.2.

**Step 2.** We next prove the implication (4.3) ⇒ (4.2) of Theorem 4.1 for any $p > 1$, which, in particular, will conclude the proof of Theorem 4.1 for $1 < p \le 2$. By (1.10) we have $0 \le \frac{M_{0}(i,j)U_{1,\infty}(j)}{\lambda_{0}U_{0,\infty}(i)} \le 1$ a.s. for all $1 \le i, j \le d$. So by the triangular inequality of $L^p$, it follows that for $p > 1$ and $1 \le i \le d$,

$$ (\mathbb{E}(W_1^i)^p)^{1/p} = \left( \mathbb{E} \left( \frac{\langle Z_1^i, U_{1,\infty} \rangle}{\lambda_0 U_{0,\infty}(i)} \right)^p \right)^{1/p} \quad (4.13) $$

$$ = \left( \mathbb{E} \left( \sum_{j=1}^{d} \frac{M_{0}(i,j) U_{1,\infty}(j)}{\lambda_{0} U_{0,\infty}(i)} \frac{Z_{1}^{i}(j)}{M_{0}(i,j)} \right)^{p} \right)^{1/p} \quad (4.14) $$

$$ \le \sum_{j=1}^{d} \left( \mathbb{E} \left( \frac{Z_1^i(j)}{M_0(i,j)} \right)^p \right)^{1/p} . \quad (4.15) $$

Therefore the implication (4.3) ⇒ (4.2) of Theorem 4.1 holds.

**Step 3.** We now prove that for $p > 2$, we have the implications (4.3) ⇒ (4.1) of Theorem 4.1, and (4.3) ⇒ (2.11) of Theorem 2.2. Assume $p > 2$ and (4.3). In the following $C > 0$ will be a constant which may depend on $p$ and which may differ from line to line. Applying (4.7), the inequality $(\sum_{j=1}^d x_j)^p \le d^{p-1} \sum_{j=1}^d x_j^p$, $x_j \ge 0$
for any $1 \le j \le d$, and Lemma 4.2, for all $n \ge 0$, $k \ge 1$ and $1 \le i \le d$, $\mathbb{P}$-a.s. we have

$$
\begin{align*}
& \mathbb{E}_{\xi} |W_{n+k}^i - W_n^i|^p \\
\le {}& d^{p-1} B_p^p \sum_{j=1}^d \left( \frac{U_{n,\infty}(j)}{\lambda_{0,n-1} U_{0,\infty}(i)} \right)^p \mathbb{E}_{\xi} (Z_n^i(j))^{p/2} \mathbb{E}_{\xi} |W_{n,k}^j - 1|^p \\
\le {}& C \sigma_{n,k}(p) \sum_{j=1}^d (U_{n,\infty}(j))^{p/2} \mathbb{E}_{\xi} \left( \frac{U_{n,\infty}(j) Z_n^i(j)}{\lambda_{0,n-1} U_{0,\infty}(i)} \right)^{p/2} (\lambda_{0,n-1} U_{0,\infty}(i))^{-p/2} \\
\le {}& C \sigma_{n,k}(p) \left( \sum_{j=1}^d U_{n,\infty}(j) \right) \mathbb{E}_{\xi} (W_n^i)^{p/2} (\lambda_{0,n-1} U_{0,\infty}(i))^{-p/2} \\
\le {}& C \sigma_{n,k}(p) \mathbb{E}_{\xi} (W_n^i)^{p/2} (\lambda_{0,n-1} U_{0,\infty}(i))^{-p/2}, \tag{4.16}
\end{align*}
$$

with $\sigma_{n,k}(p)$ defined as in (4.9) (for $p > 2$). Set $j_p \in \mathbb{N}$ the unique integer such that
$1 < \frac{p}{2j_p} \le 2$. For all $n \ge 0$, $1 \le i \le d$ and $1 \le j \le j_p$, define

$$
a_{n,j}^{i}(p) := (\lambda_{0,n-1} U_{0,\infty}(i))^{p/2^{j-p}} \mathbb{E}_{\xi} (W_{n}^{i})^{p/2^{j}}. \quad (4.17)
$$

Taking expectation in (4.16), we obtain that for all $n \ge 0$, $k \ge 1$ and $1 \le i \le d$,

$$
\mathbb{E} |W_{n+k}^i - W_n^i|^p \le C \mathbb{E} [\sigma_{n,k}(p) \mathbb{E}_{T^n \xi} a_{n,1}^i(p)]. \quad (4.18)
$$

To prove (4.1) of Theorem 4.1, and (2.11) of Theorem 2.2, it is enough to show that there exists a constant $C_1 > 0$ (which may depend on $p$) such that for all $n \ge 0$, $1 \le i \le d$, $1 \le j \le j_p$ and $\delta > \delta_c(p)$,

$$
\delta^{-n} (\mathbb{E}_{T^n \xi} a_{n,j}^i(p))^{1/p} \le C_1 \quad \mathbb{P}\text{-a.s.} \tag{4.19}
$$

In fact, combining (4.18) and (4.19) for $j=1$, it follows that for all $1 \le i \le d$ and
$\delta > \delta_c(p)$,

$$
\sup_{n \ge 0} (\mathbb{E}(W_n^i)^{1/p})^{1/p} \le 1 + C (\mathbb{E}\sigma_{0,1}(p))^{1/p} \sum_{n=0}^{+\infty} \delta^n. \quad (4.20)
$$

Condition (4.3) implies that $\mathbb{E}\sigma_{0,1}(p) < +\infty$ and $\delta_c(p) < 1$. Therefore, applying (4.20) with $\delta \in (\delta_c(p), 1)$, we deduce that $(W_n^i)$ is a martingale bounded in $L^p$, for all $1 \le i \le d$. Hence, $(W_n^i)$, $1 \le i \le d$, converge in $L^p$, and we have $\sup_{k \ge 0} \mathbb{E}\sigma_{0,k}(p) < +\infty$. This proves the implication (4.3) $\Rightarrow$ (4.1). Moreover, combining again (4.18) and (4.19), and letting $k \to +\infty$, we obtain that for all $n \ge 0$, $1 \le i \le d$ and $\delta \in (\delta_c(p), 1)$,

$$
(\mathbb{E}|W^i - W_n^i|^p)^{1/p} \le C\delta^n,
$$

which implies (2.11) and ends the proof of Theorem 2.2 in the case $p > 2$.
It remains to prove (4.19). We will prove it by iteration on $j$. First consider the case $j = j_p$. By definition of $j_p$ we have $1 < p/2^{j_p} \le 2$. So, by the triangular inequality in $L^{p/2^{j_p}}$ under $\mathbb{P}_\xi$ and (4.8), it follows that for all $1 \le i \le d$ and $n \ge 0$, $\mathbb{P}$-a.s.,

$$ 
\begin{aligned}
a_{n,j_p}^i(p)^{2^{j_p}/p} &\le (\lambda_{0,n-1} U_{0,\infty}(i))^{1-2^{j_p}} \\
&\quad + (\lambda_{0,n-1} U_{0,\infty}(i))^{1-2^{j_p}} \sum_{l=0}^{n-1} (\mathbb{E}_\xi |W_{l+1}^i - W_l^i|^{p/2^{j_p}})^{2^{j_p}/p} \\
&\le (\lambda_{0,n-1} U_{0,\infty}(i))^{1-2^{j_p}} \\
&\quad + C \sum_{l=0}^{n-1} \left[ \sigma_{l,1} \left( \frac{p}{2^{j_p}} \right) \right]^{2^{j_p}/p} (\lambda_{0,l-1} U_{0,\infty}(i))^{2^{j_p}/p-2^{j_p}} \lambda_{l,n-1}^{1-2^{j_p}}. \quad (4.21)
\end{aligned}
$$

Taking the $L^{p/2^{j_p}}$-norm under $\mathbb{P}_{T^{n}\xi}$ on both sides, and using the triangular inequality in $L^{p/2^{j_p}}$ and inequalities (4.10), we obtain that for all $1 \le i \le d$ and $n \ge 0$, $\mathbb{P}$-a.s.,

$$ 
\begin{aligned}
& (\mathbb{E}_{T^{n}\xi} a_{n,j_p}^i(p))^{2^{j_p}/p} \\
&\le (\mathbb{E}_{T^{n}\xi} (\lambda_{0,n-1} U_{0,\infty}(i))^{p/2^{j_p}-p})^{2^{j_p}/p} \\
&\quad + C \sum_{l=0}^{n-1} \left\{ \mathbb{E}_{T^{n}\xi} \left[ \mathbb{E}_{T^l\xi} \left[ (\lambda_{0,l-1} U_{0,\infty}(i))^{1-p} \right] \sigma_{l,1} \left( \frac{p}{2^{j_p}} \right) \lambda_{l,n-1}^{p/2^{j_p}-p} \right] \right\}^{2^{j_p}/p} \\
&\le C \kappa \left( \frac{p}{2^{j_p}} - p \right)^{\frac{n2^{j_p}}{p}} + C \sum_{l=0}^{n-1} \left\{ \kappa (1-p)^l \mathbb{E}_{T^{n}\xi} \left[ \sigma_{l,1} \left( \frac{p}{2^{j_p}} \right) \lambda_{l,n-1}^{p/2^{j_p}-p} \right] \right\}^{2^{j_p}/p}.
\end{aligned}
\quad (4.22) $$

Notice that if $1 \le j \le j_p$, then we have $1 - p < \frac{p}{2^j} - p < -\frac{p}{2}$. Since $\kappa$ is log-convex on $I$, we obtain that

$$ \max_{1 \le j \le j_p} \left\{ \kappa \left( \frac{p}{2^j} - p \right) \right\} \le \max \left\{ \kappa(1-p), \kappa \left(-\frac{p}{2}\right) \right\} = \delta_c(p)^p. \quad (4.23) $$

We now deal with the second term in (4.22), by calculating first the conditional expectation $\mathbb{E}_{T^{l+1}\xi}$. By the triangular inequalities of $L^{p/2^j}$ under $\mathbb{P}_\xi$ and $\mathbb{P}_{T^{l+1}\xi}$, and inequalities (4.10), it holds that for all $l \ge 0$ and $1 \le j \le j_p$, $\mathbb{P}$-a.s.,

$$ 
\begin{aligned}
& \left\{ \mathbb{E}_{T^{l+1}\xi} \left( \sigma_{l,1} \left( \frac{p}{2^j} \right) \lambda_l^{p/2^j - p} \right) \right\}^{2^j/p} \\
&= \left\{ \mathbb{E}_{T^{l+1}\xi} \left( \max_{1 \le r \le d} \mathbb{E}_\xi |W_{l,k}^r - 1|^{p/2^j} \lambda_l^{p/2^j - p} \right) \right\}^{2^j/p} \\
&\le \left\{ \mathbb{E}_{T^{l+1}\xi} \left( \max_{1 \le r \le d} \mathbb{E}_\xi (W_{l,1}^r)^{p/2^j} \lambda_l^{p/2^j - p} \right) \right\}^{2^j/p} + \left( \mathbb{E}_{T^{l+1}\xi} \lambda_l^{p/2^j - p} \right)^{2^j/p} \\
&\le \left\{ \sum_{r=1}^d \mathbb{E}_{T^{l+1}\xi} \left[ \mathbb{E}_\xi (W_{l,1}^r)^{p/2^j} \lambda_l^{p/2^j - p} \right] \right\}^{2^j/p} + C \kappa \left( \frac{p}{2^j} - p \right)^{2^j/p}.
\end{aligned}
$$
Therefore, using inequality (4.23), condition (4.3) and again the triangular inequal-
ities of $L^{p/2^j}$ under $\mathbb{P}_\xi$ and $\mathbb{P}_{T^{l+1}\xi}$, we see that for all $l \ge 0$ and $1 \le j \le j_p$,
$\mathbb{P}$-a.s.,

$$
\begin{align*}
& \left( \mathbb{E}_{T^{l+1}\xi} \left( \sigma_{l,1} \left( \frac{p}{2^j} \right) \lambda_l^{p/2^j - p} \right) \right)^{2^j/p} \\
\le & d^{2^j/p} \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} \left[ \lambda_l^{-p} \mathbb{E}_\xi \left( \lambda_l W_{l,1}^r \right)^{p/2^j} \right] \right)^{2^j/p} + C \\
\le & C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} \left[ \lambda_l^{-p} \mathbb{E}_\xi \left( \lambda_l W_{l,1}^r \mathbb{1}_{\{\lambda_l W_{l,1}^r \le 1\}} \right)^{p/2^j} \right] \right)^{2^j/p} \\
& + C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} \left[ \lambda_l^{-p} \mathbb{E}_\xi \left( \lambda_l W_{l,1}^r \mathbb{1}_{\{\lambda_l W_{l,1}^r > 1\}} \right)^{p/2^j} \right] \right)^{2^j/p} + C \\
\le & C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} \left[ \lambda_l^{1-p} \mathbb{E}_\xi W_{l,1}^r \right] \right)^{2^j/p} + C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} \left( W_{l,1}^r \right)^p \right)^{2^j/p} + C.
\end{align*}
$$

We know that $(W_{l,k}^r)_{k \ge 0}$ is the martingale associated to a MBPRE starting with
one individual of type $r$, in the shift random environment $T^l \xi$. In particular we
have $\mathbb{E}_\xi W_{l,1}^r = 1$ a.s. Therefore, applying again (4.10), (4.23) and condition (4.3),
it follows that for all $l \ge 0$ and $1 \le j \le j_p$, $\mathbb{P}$-a.s.,

$$
\begin{align*}
& \left( \mathbb{E}_{T^{l+1}\xi} \left( \sigma_{l,1} \left( \frac{p}{2^j} \right) \lambda_l^{p/2^j - p} \right) \right)^{2^j/p} \\
\le & C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} \lambda_l^{1-p} \right)^{2^j/p} + C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} (W_{l,1}^r)^p \right)^{2^j/p} + C \\
\le & C \kappa (1-p)^{2^j/p} + C \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} (W_{l,1}^r)^p \right)^{2^j/p} + C \\
\le & C \left( 1 + \max_{1 \le r \le d} \left( \mathbb{E}_{T^{l+1}\xi} (W_{l,1}^r)^p \right)^{2^j/p} \right). \tag{4.24}
\end{align*}
$$

Then, by a similar calculation as in (4.13), for all $1 \le r \le d$ and $l \ge 0$, $\mathbb{P}$-a.s., we
have

$$
\begin{align*}
(\mathbb{E}_{T^{l+1}\xi} (W_{l,1}^r)^p)^{1/p} &= \left( \mathbb{E}_{T^{l+1}\xi} \left( \frac{\langle N_l^r, U_{l+1,\infty} \rangle}{\lambda_l U_{l,\infty}(r)} \right)^p \right)^{1/p} \\
&= \left( \mathbb{E}_{T^{l+1}\xi} \left( \sum_{j=1}^d \frac{M_l(r,j) U_{l+1,\infty}(j)}{\lambda_l U_{l,\infty}(r)} \frac{N_l^r}{M_l(r,j)} \right)^p \right)^{1/p} \\
&\le \sum_{j=1}^d \left( \mathbb{E}_{T^{l+1}\xi} \left( \frac{N_l^r(j)}{M_l(r,j)} \right)^p \right)^{1/p} \\
&= \sum_{j=1}^d \left( \mathbb{E} \left( \frac{Z_1^r(j)}{M_0(r,j)} \right)^p \right)^{1/p} < +\infty. \tag{4.25}
\end{align*}
$$
Putting together (4.24) and (4.25), we get that for all $l \ge 0$ and $1 \le j \le j_p$, $\mathbb{P}$-a.s.,

$$ \mathbb{E}_{T^{l+1}\xi} \left( \sigma_{l,1} \left( \frac{p}{2^j} \right) \lambda_l^{p/2^j - p} \right) \le C. \quad (4.26) $$

Therefore, for all $n \ge 0$ and $0 \le l \le n-1$, $\mathbb{P}$-a.s., (the value of the constant $C$ may change from line to line),

$$ \begin{aligned} \mathbb{E}_{T^n \xi} \left[ \sigma_{l,1} \left( \frac{p}{2^{j_p}} \right) \lambda_{l,n-1}^{p/2^{j_p}-p} \right] &= \mathbb{E}_{T^n \xi} \left[ \mathbb{E}_{T^{l+1} \xi} \left( \sigma_{l,1} \left( \frac{p}{2^{j_p}} \right) \lambda_l^{p/2^{j_p}-p} \right) \lambda_{l+1,n-1}^{p/2^{j_p}-p} \right] \\ &\le C \mathbb{E}_{T^n \xi} \lambda_{l+1,n-1}^{p/2^{j_p}-p} \\ &\le C \left[ \kappa \left( \frac{p}{2^{j_p}} - p \right) \right]^{n-1-l} \\ &\le C \delta_c(p)^{(n-1-l)p}, \end{aligned} $$

where the last two inequalities hold by (4.10) and (4.23). Combining this with (4.22) and (4.23), we obtain that for all $1 \le i \le d$ and $n \ge 0$, $\mathbb{P}$-a.s.,

$$ \begin{aligned} \left( \mathbb{E}_{T^n \xi} a_{n,j_p}^i (p) \right)^{2^{j_p}/p} &\le C \delta_c(p)^{n 2^{j_p}} \\ &\le C \delta_c(p)^{n 2^{j_p}} + C \sum_{l=0}^{n-1} \left( \delta_c(p)^{lp} \delta_c(p)^{(n-1-l)p} \right)^{2^{j_p}/p} \\ &\le C \left( 1 + \delta_c(p)^{-2^{j_p}} n \right) \delta_c(p)^{n 2^{j_p}}. \end{aligned} $$

So (4.19) holds for $j = j_p$.

Now suppose that (4.19) holds for $j+1 \le j_p$ with $j \ge 1$. We will prove that it still holds for $j$. By recurrence this will prove that (4.19) holds for all $j = 1, \dots, j_p$. Since $j+1$ satisfies (4.19), for all $n \ge 0$, $1 \le i \le d$, and $\delta > \delta_c(p)$,

$$ \delta^{-n} \left( \mathbb{E}_{T^n \xi} a_{n,j+1}^i (p) \right)^{1/p} \le C \quad \mathbb{P}\text{-a.s.} \quad (4.27) $$

By definition of $j_p$ we have $p/2^j > 2$. Corresponding to (4.22), with the same argument as in its proof but applying (4.16) instead of (4.8), we obtain that for all $1 \le i \le d$ and $n \ge 0$, $\mathbb{P}$-a.s.,

$$ \begin{aligned} & \left( \mathbb{E}_{T^n \xi} a_{n,j}^i (p) \right)^{2^j/p} \\ &\le C \kappa \left( \frac{p}{2^j} - p \right)^{n 2^j/p} + C \times \\ & \sum_{l=0}^{n-1} \left\{ \mathbb{E}_{T^n \xi} \left[ \sigma_{l,1} \left( \frac{p}{2^j} \right) \mathbb{E}_{\xi} \left( W_l^i \right)^{\frac{p}{2^{j+1}} - \lambda_{0,l-1} U_{0,\infty}(i) \right) \right. \right. \\ & \left. \left. \lambda_{l,n-1}^{p/2^j - p} \right] \right\}^{2^j/p} \\ &= C \kappa \left( \frac{p}{2^j} - p \right)^{n 2^j/p} + C \sum_{l=0}^{n-1} \left( \mathbb{E}_{T^n \xi} \left[ \sigma_{l,1} \left( \frac{p}{2^j} \right) \lambda_{l,n-1}^{p/2^j - p} \mathbb{E}_{T^l \xi} a_{l,j+1}^i (p) \right] \right)^{2^j/p}. \end{aligned} \quad (4.28) $$
This enables us to obtain the desired bound of $\mathbb{E}_{T^n\xi}a_{n,j}^i(p)$ from that of $\mathbb{E}_{T^n\xi}a_{n,j+1}^i(p)$. In fact, combining this with the recurrence hypothesis (4.27), together with (4.23), (4.26) and (4.10), we obtain that for all $n \ge 0$, $1 \le i \le d$, and $\delta > \delta_c(p)$,

$$ (\mathbb{E}_{T^n\xi} a_{n,j}^i(p))^{2j/p} \le C \delta_c(p)^{n2^j} + C \times \\ \sum_{l=0}^{n-1} \delta^{l2^j} \left\{ \mathbb{E}_{T^n\xi} \left[ \mathbb{E}_{T^{l+1}\xi} \left[ \sigma_{l,1} \left( \frac{p}{2^j} \right) \lambda_l^{p/2^j-p} \right] \lambda_{l+1,n-1}^{p/2^j-p} \right] \right\}^{2^j/p} \\ \le C \delta_c(p)^{n2^j} + C \sum_{l=0}^{n-1} \delta^{l2^j} \kappa \left( \frac{p}{2^j} - p \right)^{(n-1-l)2^j/p} \\ \le C(\delta_c(p)^{n2^j} + n\delta^{(n-1)2^j}) \\ \le C(1 + \delta^{-2^j}n)\delta^{n2^j}. \qquad (4.29) $$

So (4.19) also holds for $j$. Therefore, by recurrence, we have proved that (4.19) holds for all $j = 1, \dots, j_p$.

**Step 4.** For any $p > 1$, we prove the implication (4.1) ⇒ (4.2) of Theorem 4.1. Assume $p > 1$ and condition (4.1), that is, the martingale $(W_n^i)_{n \ge 0}$ converges in $L^p$, for all $1 \le i \le d$. In particular we have $\mathbb{E}(W_1^i)^p < +\infty$, $\mathbb{E}(W^i)^p < +\infty$, and $\mathbb{E}(W^i) = 1$ for all $1 \le i \le d$. It was observed in [11, Theorem 2.6] that $\mathbb{E}_\xi(W^i) = 1$ a.s. when $W^i$ are non-degenerate. In fact $\mathbb{E}_\xi(W^i) = 1$ a.s. whenever $\mathbb{E}(W^i) = 1$, because $\mathbb{E}_\xi(W^i) \le 1$ by Fatou's lemma.

Notice that for all $1 \le i \le d$, the limit variables $W^i$ satisfy

$$ W^i = \sum_{j=1}^{d} \frac{U_{1,\infty}(j)}{\lambda_0 U_{0,\infty}(i)} \sum_{l=1}^{Z_1^j} W^j(l, T\xi) \quad \mathbb{P}\text{-a.s.,} $$

where for all $1 \le j \le d$, under the conditional law $\mathbb{P}_\xi$, $(W^j(l, T\xi))_{l \ge 1}$ is a sequence of i.i.d. random variables, also independent of $Z_1^j$, with common distribution $\mathbb{P}_\xi(W^j(l, T\xi) \in \cdot) = \mathbb{P}_{T\xi}(W^j \in \cdot)$. So, by the strict sub-additivity of the function $x \mapsto x^p$ on $\mathbb{R}_+$, we get that for all $1 \le i \le d$,

$$ \mathbb{E}_{\xi}(W^i)^p > \sum_{j=1}^{d} \frac{M_0(i, j) U_{1, \infty}(j)^p}{\lambda_0^p U_{0, \infty}(i)^p} \mathbb{E}_{T\xi}(W^j)^p \quad \mathbb{P}\text{-a.s.,} $$

using the fact that $\mathbb{E}_\xi(W^i) = 1$ a.s. This, together with (1.10), implies that for all $1 \le i \le d$, $\mathbb{P}$-a.s.,

$$ \begin{aligned} \mathbb{E}_{\xi}(W^i)^p U_{0,\infty}(i)^{p-1} &> \lambda_0^{1-p} \sum_{j=1}^{d} \frac{M_0(i,j) U_{1,\infty}(j)}{\lambda_0 U_{0,\infty}(i)} \mathbb{E}_{T\xi}(W^j)^p U_{1,\infty}(j)^{p-1} \\ &> \lambda_0^{1-p} \min_{1 \le j \le d} (\mathbb{E}_{T\xi}(W^j)^p U_{1,\infty}(j)^{p-1}). \end{aligned} $$
Therefore we obtain

$$ \min_{1 \le i \le d} (\mathbb{E}_{\xi}(W^i)^p U_{0,\infty}(i)^{p-1}) > \lambda_0^{1-p} \min_{1 \le i \le d} (\mathbb{E}_{T\xi}(W^i)^p U_{1,\infty}(i)^{p-1}) \quad \text{P-a.s.} \quad (4.30) $$

On the other hand, by (3.1) and (1.10), the transfer operator $P_{1-p}$ satisfies the following property: for all $\varphi \in \mathcal{C}(S)$, P-a.s.,

$$ P_{1-p}\varphi(U_{1,\infty}) = \mathbb{E}_{T\xi}[\|M_0 U_{1,\infty}\|^{1-p}\varphi(M_0 \cdot U_{1,\infty})] = \mathbb{E}_{T\xi}[\lambda_0^{1-p}\varphi(U_{0,\infty})]. \quad (4.31) $$

Using (4.31) with $\varphi = r_{1-p}$, and combining this with Proposition 3.1, we get

$$ \mathbb{E}_{T\xi}[\lambda_0^{1-p} r_{1-p}(U_{0,\infty})] = \kappa(1-p)r_{1-p}(U_{1,\infty}) \quad \text{P-a.s.} \quad (4.32) $$

Moreover, by Proposition 3.1 we know that $r_{1-p}$ is a strictly positive continuous function on $S$. This implies that

$$ 0 < \mathbb{E} \left[ \min_{1 \le i \le d} \left( \mathbb{E}_{\xi}(W^i)^p U_{0,\infty}(i)^{p-1}) r_{1-p}(U_{0,\infty}) \right] \right. \\ \left. \le \|r_{1-p}\|_{\infty} \min_{1 \le i \le d} \mathbb{E}(W^i)^p < +\infty. \right] \quad (4.33) $$

Therefore, putting together (4.30), (4.32) and (4.33), we obtain

$$ \begin{aligned} & \mathbb{E} \left[ \min_{1 \le i \le d} \left( \mathbb{E}_{\xi}(W^i)^p U_{0,\infty}(i)^{p-1}) r_{1-p}(U_{0,\infty}) \right] \right] \\ & > \mathbb{E} \left[ \mathbb{E}_{T\xi} \left[ \lambda_0^{1-p} r_{1-p}(U_{0,\infty}) \right] \min_{1 \le i \le d} \left( \mathbb{E}_{T\xi}(W^i)^p U_{1,\infty}(i)^{p-1}) \right] \right] \\ & = \kappa(1-p) \mathbb{E} \left[ \min_{1 \le i \le d} \left( \mathbb{E}_{T\xi}(W^i)^p U_{1,\infty}(i)^{p-1}) r_{1-p}(U_{1,\infty}) \right] \right] \\ & = \kappa(1-p) \mathbb{E} \left[ \min_{1 \le i \le d} \left( \mathbb{E}_{\xi}(W^i)^p U_{0,\infty}(i)^{p-1}) r_{1-p}(U_{0,\infty}) \right] \right], \end{aligned} $$

so $\kappa(1-p) < 1$. This ends the proof of (4.1) ⇒ (4.2).

Step 5. To conclude the proof, it remains to show that under the Furstenberg-Kesten condition **H1** we have (4.2) ⇒ (4.3) for all $p > 1$. By (4.5) in Lemma 4.3, we know that, under **H1**, we have for all $n \ge 0$ and $1 \le i, j \le d$

$$ \frac{1}{dD^2} \le \frac{M_{0,n-1}(i,j)U_{n,\infty}(j)}{\lambda_{0,n-1}U_{0,\infty}(i)} \le 1 \quad \text{P-a.s.} $$

Therefore we obtain that for all $n \ge 0$ and $1 \le i, j \le d$,

$$ \begin{aligned} \frac{Z_n^i(j)}{M_{0,n-1}(i,j)} &\le dD^2 \frac{M_{0,n-1}(i,j)U_{n,\infty}(j)}{\lambda_{0,n-1}U_{0,\infty}(i)} \frac{Z_n^i(j)}{M_{0,n-1}(i,j)} \\ &= dD^2 \frac{Z_n^i(j)U_{n,\infty}(j)}{\lambda_{0,n-1}U_{0,\infty}(i)} \le dD^2 W_n^i. \end{aligned} \quad (4.34) $$

The implication (4.2) ⇒ (4.3) follows from (4.34) with $n = 1$.

This ends the proof of Theorems 4.1 and 2.2. □
**5. Convergence in $L^p$ of the normalized population size $\frac{Z_n^i(j)}{\mathbb{E}_\xi Z_n^i(j)}$**

In this section we give proof of Theorems 2.3 and 2.4 about the convergence in $L^p$ of $\frac{Z_n^i(j)}{\mathbb{E}_\xi Z_n^i(j)}$ and its exponential convergence rate, under the Furstenberg-Kesten condition **H1**.

**5.1. Auxiliary results**

We need some preliminary results concerning the products of random matrices $M_{n,n+k}$. The following proposition was established by Hennion in [15, Theorem 1], which provides an analog of the Perron-Frobenius theorem for products of random matrices.

**Proposition 5.1.** Assume that $M_0 > 0$ $\mathbb{P}$-a.s. Then for all $n \ge 0$ and $1 \le i, j \le d$, as $k \to +\infty$, $\mathbb{P}$-a.s.,

$$M_{n,n+k}(i,j) \underset{k \to +\infty}{\sim} \rho_{n,n+k} U_{n,n+k}(i) V_{n,n+k}(j).$$

For $1 \le i \le d$, let $(\Pi_n^i)_{n \ge 0}$ be the sequence of random matrices in $M_d(\mathbb{R})$ such that for all $1 \le j, r \le d$,

$$\Pi_0^i(j,r) := \delta_{i,r}, \quad \Pi_n^i(j,r) := \frac{M_{0,n-1}(i,r)M_n(r,j)}{M_{0,n}(i,j)}, \quad n \ge 1.$$

By definition all the entries of the $i$-th column of $\Pi_0^i$ are equal to 1, the others are 0; each $\Pi_n^i$ is a stochastic matrix. For $n, k \ge 0$ let

$$\Pi_{n+k,n}^i := \Pi_{n+k}^i \cdots \Pi_n^i$$

be the products of the matrices $\Pi_n^i$. Clearly each $\Pi_{n+k,n}^i$ is a non-negative stochastic random matrix. The following lemma concerns the convergence of the products $\Pi_{n+k,n}^i$ of random matrices and its exponential rate as $k \to +\infty$, which will be very useful for the study of the $L^p$ convergence of the normalized population size $\frac{Z_n^i(j)}{\mathbb{E}_\xi Z_n^i(j)}$.

**Lemma 5.2.** Assume the Furstenberg-Kesten condition **H1**. Then for all $n \ge 0$ and $1 \le i \le d$, as $k \to +\infty$, the sequence $(\Pi_{n+k,n}^i)_{k \ge 0}$ converges $\mathbb{P}$-a.s. to the random matrix $\Pi_{\infty,n}^i$ such that:

(1) For all $1 \le j, r \le d$,

$$\Pi_{\infty,0}^i(j,r) := \Pi_0^i(j,r), \quad \Pi_{\infty,n}^i(j,r) := \frac{M_{0,n-1}(i,r)U_{n,\infty}(r)}{\lambda_{0,n-1}U_{0,\infty}(i)}, \quad n \ge 1.$$

(2) There exist $C > 0$ and $\delta \in (0, 1)$ such that for all $k \ge 0$ and $1 \le i \le d$,

$$\sup_{n \ge 0} \| \Pi_{n+k,n}^i - \Pi_{\infty,n}^i \| \le C \delta^k \quad \mathbb{P}\text{-a.s.} \qquad (5.1)$$
For the proof of Lemma 5.2, we will use the following result established by Seneta [28, Theorem 4.19], which gives the convergence of products of stochastic matrices with an exponential rate.

**Lemma 5.3 ([28]).** Assume that $(P_n)_{n \ge 0}$ is a sequence of stochastic matrices such that for some $\varepsilon \in (0, 1)$, all $n \ge 0$ and all $1 \le i, j \le d$,

$$P_n(i, j) \ge \varepsilon.$$

Then, for all $n \ge 0$ the product $P_{n+k,n} := P_{n+k} \cdots P_n$ converges as $k \to +\infty$ to some matrix $P_{\infty,n}$, and there exist two constants $C > 0$ and $\delta \in (0, 1)$ depending only on $\varepsilon$ such that

$$\sup_{n \ge 0} \| P_{n+k,n} - P_{\infty,n} \| \le C \delta^k.$$

**Proof of Lemma 5.2.** (1) First we show by induction on $k$ that for all $n \ge 1$, $k \ge 0$ and $1 \le i, j, r \le d$,

$$\Pi_{n+k,n}^i(j, r) = \frac{M_{0,n-1}(i, r)M_{n,n+k}(r, j)}{M_{0,n+k}(i, j)}. \qquad (5.2)$$

Obviously, by definition of $\Pi_n^i$, (5.2) holds for $k = 0$. Assume that (5.2) holds for some $k \ge 0$. Then, for all $n \ge 1$ and $1 \le i, j, r \le d$ we have

$$\begin{aligned} \Pi_{n+k+1,n}^i(j,r) &= \sum_{l=1}^d \Pi_{n+k+1}^i(j,l) \Pi_{n+k,n}^i(l,r) \\ &= \sum_{l=1}^d \frac{M_{0,n+k}(i,l) M_{n+k+1}(l,j) M_{0,n-1}(i,r) M_{n,n+k}(r,l)}{M_{0,n+k+1}(i,j) M_{0,n+k}(i,l)} \\ &= \frac{M_{0,n-1}(i,r)}{M_{0,n+k+1}(i,j)} \sum_{l=1}^d M_{n,n+k}(r,l) M_{n+k+1}(l,j) \\ &= \frac{M_{0,n-1}(i,r) M_{n,n+k+1}(r,j)}{M_{0,n+k+1}(i,j)}. \end{aligned}$$

So (5.2) holds for $k + 1$. Therefore by reduction (5.2) holds for all $k \ge 0$.

Combining (5.2) with Proposition 5.1, we deduce that for all $n \ge 1$ and $1 \le i, j, r \le d$, P-a.s. as $k \to +\infty$,

$$\begin{aligned} \Pi_{n+k,n}^i(j,r) &\sim \frac{M_{0,n-1}(i,r)\rho_{n,n+k}U_{n,n+k}(r)V_{n,n+k}(j)}{\rho_{0,n+k}U_{0,n+k}(i)V_{0,n+k}(j)} \\ &\sim \frac{\rho_{n,n+k}V_{n,n+k}(j) M_{0,n-1}(i,r)U_{n,\infty}(r)}{\rho_{0,n+k}V_{0,n+k}(j) U_{0,\infty}(i)} \\ &= \left( \prod_{l=0}^{n-1} \frac{\rho_{l+1,n+k}V_{l+1,n+k}(j)}{\rho_{l,n+k}V_{l,n+k}(j)} \right) \frac{M_{0,n-1}(i,r)U_{n,\infty}(r)}{U_{0,\infty}(i)}. \end{aligned} \qquad (5.3)$$
By [11, Proposition 2.2] we know that for all $l, n \ge 0$ and $1 \le j \le d$,

$$ \lambda_l = \lim_{k \to +\infty} \frac{\rho_{l,n+k} V_{l,n+k}(j)}{\rho_{l+1,n+k} V_{l+1,n+k}(j)} \quad \mathbb{P}\text{-a.s.} $$

This, together with (5.3), implies that for $n \ge 1$ and $1 \le j, r \le d$, as $k \to +\infty$,

$$ \Pi_{n+k,n}^{i}(j,r) \to \Pi_{\infty,n}^{i}(j,r) = \frac{M_{0,n-1}(i,r)U_{n,\infty}(r)}{\lambda_{0,n-1}U_{0,\infty}(i)} \quad \mathbb{P}\text{-a.s.} $$

Hence, as $k \to \infty$, $\mathbb{P}$-a.s., $\Pi_{k,0}^i = \Pi_{k,1}^i \Pi_0^i \to \Pi_{\infty,1}^i \Pi_0^i = \Pi_{\infty,0}^i$, where

$$ \Pi_{\infty,0}^{i}(j,r) = \sum_{l=1}^{d} \Pi_{\infty,1}^{i}(j,l) \Pi_{0}^{i}(l,r) = \sum_{l=1}^{d} \Pi_{\infty,1}^{i}(j,l) \delta_{i,r} = \delta_{i,r}, \quad 1 \le j, r \le d. $$

(2) By (4.4) in Lemma 4.3, we have, for all $n \ge 1$ and $1 \le i, j, r \le d$,

$$ \frac{1}{\Pi_n^i(j,r)} = \sum_{l=1}^{d} \frac{M_{0,n-1}(i,l)M_n(l,j)}{M_{0,n-1}(i,r)M_n(r,j)} \le dD^2 \quad \mathbb{P}\text{-a.s.,} $$

or equivalently

$$ \Pi_n^i(j,r) \ge \frac{1}{dD^2} \quad \mathbb{P}\text{-a.s.} \qquad (5.4) $$

Since $(\Pi_n^i)_{n \ge 0}$ is a sequence of positive stochastic matrices satisfying (5.4), by Lemma 5.3, there exist two constants $C > 0$ and $\delta \in (0, 1)$ such that for all $k \ge 0$ and $1 \le i \le d$,

$$ \sup_{n \ge 0} \| \Pi_{n+k,n}^i - \Pi_{\infty,n}^i \| \le C \delta^k, \quad \mathbb{P}\text{-a.s.} $$

This concludes the proof of Lemma 5.2. $\square$

## 5.2. Proof of Theorems 2.3 and 2.4

For all $n \ge 0$ and $1 \le i, j \le d$, set

$$ \bar{Z}_n^i(j) := \frac{Z_n^i(j)}{\mathbb{E}_\xi Z_n^i(j)} = \frac{Z_n^i(j)}{M_{0,n-1}(i,j)}. $$

First we show that (2.9) is a necessary condition for the convergence in $L^p$ of the normalized population size $\bar{Z}_n^i(j)$, $1 \le i, j \le d$. Assume that $(\bar{Z}_n^i(j))_{n \ge 0}$ converges in $L^p$ for all $1 \le i, j \le d$. For $n \ge 0$ and $1 \le i \le d$ we have, by the definition of $W_n^i$ and (1.11),

$$ W_n^i = \sum_{j=1}^{d} \frac{M_{0,n-1}(i,j)U_{n,\infty}(j)}{\lambda_{0,n-1}U_{0,\infty}(i)} \bar{Z}_n^i(j) \le \max_{1 \le j \le d} \bar{Z}_n^i(j). $$
This implies that the martingale $(W_n^i)_{n \ge 0}$, $1 \le i \le d$, is bounded in $L^p$, hence convergences in $L^p$. So by Theorem 2.1, condition (2.9) holds.

Now we prove that (2.9) is sufficient for the convergence in $L^p$ of $\overline{Z}_n^i(j)$, $1 \le i, j \le d$, and establish meanwhile Theorem 2.4 about its convergence rate. Assume (2.9). By the definition of the branching process $(Z_n^i)$ (cf. (4.6)), we have the following decomposition: for all $1 \le i, j \le d$ and $n, k \ge 1$,

$$ 
\begin{align} 
\overline{Z}_{n+k}^i(j) &= \sum_{r=1}^d \frac{M_{n,n+k-1}(r,j)}{M_{0,n+k-1}(i,j)} \sum_{l=1}^{Z_n^i(r)} \frac{Z_{l,n,k}^r(j)}{M_{n,n+k-1}(r,j)} \nonumber \\
&= \sum_{r=1}^d \frac{M_{n,n+k-1}(r,j)}{M_{0,n+k-1}(i,j)} Z_n^i(r) \nonumber \\
&\quad + \sum_{r=1}^d \frac{M_{n,n+k-1}(r,j)}{M_{0,n+k-1}(i,j)} \sum_{l=1}^{Z_n^i(r)} \left( \frac{Z_{l,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right). \tag{5.5}
\end{align} 
$$

Combining (5.5) and (5.2), we get that for all $1 \le i, j \le d$ and $n, k \ge 1$,

$$ 
\begin{align} 
\overline{Z}_{n+k}^i(j) &= \sum_{r=1}^d \Pi_{n+k-1,n}^i(j,r) \overline{Z}_n^i(r) \nonumber \\
&\quad + \sum_{r=1}^d \frac{\Pi_{n+k-1,n}^i(j,r)}{M_{0,n-1}(i,r)} \sum_{l=1}^{Z_n^i(r)} \left( \frac{Z_{l,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right) \nonumber \\
&= \langle \Pi_{n+k-1,n}^i \overline{Z}_n^i, e_j \rangle + R_{n,k}^i(j), \tag{5.6}
\end{align} 
$$

with

$$ R_{n,k}^i(j) := \sum_{r=1}^d \frac{\Pi_{n+k-1,n}^i(j,r)}{M_{0,n-1}(i,r)} \sum_{l=1}^{Z_n^i(r)} \left( \frac{Z_{l,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right). $$

Notice that by the definition of $W_n^i$ and that of $\Pi_{\infty,n}^i$ (cf. Lemma 5.2 (1)),

$$ W_n^i = \sum_{r=1}^d \Pi_{\infty,n}^i(j,r) \overline{Z}_n^i(r) = \langle \Pi_{\infty,n}^i \overline{Z}_n^i, e_j \rangle \tag{5.7} $$

for any $1 \le i, j \le d$. Using (5.6) and (5.7), together with the triangular inequality
in $L^p$ under $\mathbb{P}$, we obtain that for all $1 \le i, j \le d$ and $n, k \ge 1$,

$$ 
\begin{aligned}
& \left( \mathbb{E} |\bar{Z}_{n+k}^i(j) - W^i|^p \right)^{1/p} \\
&= \left( \mathbb{E} \left| \Pi_{n+k-1,n}^i \bar{Z}_n^i, e_j \right| \right) - W^i + R_{n,k}^i(j) |^p \right)^{1/p} \\
&= \left( \mathbb{E} |W_n^i - W^i| + \left( \Pi_{n+k-1,n}^i - \Pi_{\infty,n}^i \right) \bar{Z}_n^i, e_j \right) + R_{n,k}^i(j) |^p \right)^{1/p} \\
&\le \left( \mathbb{E} |W_n^i - W^i|^p \right)^{1/p} + \left( \mathbb{E} \left\| \left( \Pi_{n+k-1,n}^i - \Pi_{\infty,n}^i \right) \bar{Z}_n^i \right\|^p \right)^{1/p} \\
&\quad + \max_{1 \le j \le d} \left( \mathbb{E} |R_{n,k}^i(j)|^p \right)^{1/p} \\
&= J_1^i(n) + J_2^i(n, k) + J_3^i(n, k). \tag{5.8}
\end{aligned}
$$

In the following $C > 0$ will be a constant which may depend on $p$ and which may differ from line to line.

*Control of $J_1^i(n)$.* By condition (2.9) and Theorem 2.2 we get that there exists $\delta_1 \in (\delta_c(p), 1)$ such that for all $n \ge 1$ and $1 \le i \le d$,

$$ J_1^i(n) = \left( \mathbb{E} |W_n^i - W^i|^p \right)^{1/p} \le C \delta_1^n. \tag{5.9} $$

*Control of $J_2^i(n, k)$.* Applying the relation (5.1) of Lemma 5.2, we get that there exists $\delta_2 \in (0, 1)$ such that for all $n, k \ge 1$ and $1 \le i \le d$,

$$ J_2^i(n, k) = \left( \mathbb{E} \left\| \left( \Pi_{n+k-1,n}^i - \Pi_{\infty,n}^i \right) \bar{Z}_n^i \right\|^p \right)^{1/p} \le C \left( \mathbb{E} \left\| \bar{Z}_n^i \right\|^p \right)^{1/p} \delta_2^k. \tag{5.10} $$

Combining (4.34) and Theorem 2.1, and using condition (2.9), we obtain that, for all $1 \le i \le d$,

$$ \sup_{n \ge 0} \left( \mathbb{E} \| \bar{Z}_n^i \|^p \right)^{1/p} \le d^2 D^2 \sup_{n \ge 0} \left( \mathbb{E} (W_n^i)^p \right)^{1/p} < +\infty. \tag{5.11} $$

This, together with (5.10), implies that for all $n, k \ge 1$ and $1 \le i \le d$,

$$ J_2^i(n, k) \le C \delta_2^k. \tag{5.12} $$

*Control of $J_3^i(n, k)$ for $1 < p \le 2$.* Assume that $1 < p \le 2$. Using the convexity of the function $x \mapsto x^p$ (together with $\sum_{r=1}^d \Pi_{n+k-1,n}^i(j, r) = 1$) and Lemma 4.2,
for all $n, k \ge 1$ and $1 \le i, j \le d$, $\mathbb{P}$-a.s., we have

$$
\begin{align*}
\mathbb{E}_{\xi} |R_{n,k}^i(j)|^p &\le \mathbb{E}_{\xi} \left( \sum_{r=1}^d \frac{\Pi_{n+k-1,n}^i(j,r)}{M_{0,n-1}(i,r)} \left| \sum_{l=1}^{Z_n^i(r)} \left( \frac{Z_{l,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right) \right| \right)^p \\
&\le \mathbb{E}_{\xi} \left( \sum_{r=1}^d \frac{\Pi_{n+k-1,n}^i(j,r)}{M_{0,n-1}(i,r)^p} \left| \sum_{l=1}^{Z_n^i(r)} \left( \frac{Z_{l,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right) \right| \right)^p \\
&\le B_p^p \sum_{r=1}^d \frac{\Pi_{n+k-1,n}^i(j,r)}{M_{0,n-1}(i,r)^p} \mathbb{E}_{\xi} Z_n^i(r) \mathbb{E}_{\xi} \left| \frac{Z_{1,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right|^p \\
&= B_p^p \sum_{r=1}^d \Pi_{n+k-1,n}^i(j,r) M_{0,n-1}(i,r)^{1-p} \mathbb{E}_{\xi} \left| \frac{Z_{1,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right|^p \\
&\le B_p^p \bar{\sigma}_{n,k}(p) \max_{1 \le r \le d} M_{0,n-1}(i,r)^{1-p},
\end{align*}
$$

where

$$
\bar{\sigma}_{n,k}(p) = \max_{1 \le r, j \le d} \mathbb{E}_{\xi} \left| \frac{Z_{1,n,k}^r(j)}{M_{n,n+k-1}(r,j)} - 1 \right|^p.
$$

So, by taking expectation and using the independence between $\bar{\sigma}_{n,k}(p)$ and $M_{0,n-1}$,
we get that for all $n, k \ge 1$ and $1 \le i \le d$,

$$
J_3^i(n, k)^p \le B_p^p \mathbb{E} \bar{\sigma}_{0,k}(p) \sum_{r=1}^d \mathbb{E} [M_{0,n-1}(i,r)^{1-p}]. \quad (5.13)
$$

By (5.11) we have

$$
\sup_{k \ge 0} \mathbb{E} \bar{\sigma}_{0,k}(p) \le d^2 \max_{1 \le r, j \le d} \sup_{k \ge 0} \mathbb{E} |\bar{Z}_k^r(j) - 1|^p < +\infty. \quad (5.14)
$$

Therefore, putting together the relations (5.13) and (5.14) with Lemma 3.2, we get
that for $n, k \ge 1$ and $1 \le i \le d$,

$$
\begin{align*}
J_3^i(n, k)^p &\le C \sup_{k \ge 0} \mathbb{E} \bar{\sigma}_{0,k}(p) \sum_{r=1}^d \mathbb{E} [\langle M_{0,n-1} e_r, e_i \rangle^{1-p}] \\
&\le C \kappa (1-p)^n \\
&\le C \delta_c (p)^{np} \tag{5.15}
\end{align*}
$$

(recall that the value of C may change from line to line by our convention).

Control of $J_3^i(n, k)$ for $p \ge 2$. Assume that $p \ge 2$. Similar to the preceding case, by the convexity of $x \mapsto x^p$ (together with $\sum_{r=1}^d \Pi_{n+k-1,n}^i(j,r) = 1$) and Lemma
4.2, for all $n, k \ge 1$ and $1 \le i, j \le d$, $\mathbb{P}$-a.s.,

$$
\begin{align*}
\mathbb{E}_{\xi}|R_{n,k}^{i}(j)|^{p} &\le \mathbb{E}_{\xi}\left(\sum_{r=1}^{d} \frac{\Pi_{n+k-1,n}^{i}(j,r)}{M_{0,n-1}(i,r)}\left|\sum_{l=1}^{Z_{n}^{i}(r)} \left(\frac{Z_{l,n,k}^{r}(j)}{M_{n,n+k-1}(r,j)} - 1\right)\right|\right)^{p} \\
&\le B_{p}^{p} \sum_{r=1}^{d} \frac{\Pi_{n+k-1,n}^{i}(j,r)}{M_{0,n-1}(i,r)^{p}} \mathbb{E}_{\xi}(Z_{n}^{i}(r))^{p/2} \mathbb{E}_{\xi}\left|\frac{Z_{1,n,k}^{r}(j)}{M_{n,n+k-1}(r,j)} - 1\right|^{p} \\
&\le B_{p}^{p} \bar{\sigma}_{n,k}(p) \sum_{r=1}^{d} \Pi_{n+k-1,n}^{i}(j,r) \mathbb{E}_{\xi}(\bar{Z}_{n}^{i}(r))^{p/2} M_{0,n-1}(i,r)^{-p/2} \\
&\le B_{p}^{p} \bar{\sigma}_{n,k}(p) \max_{1 \le r \le d}\{\mathbb{E}_{\xi}(\bar{Z}_{n}^{i}(r))^{p/2} M_{0,n-1}(i,r)^{-p/2}\}. \tag{5.16}
\end{align*}
$$

Notice that (5.14) still holds when $p \ge 2$. Therefore, taking expectation in (5.16) and using (5.14), we obtain that for all $n, k \ge 1$ and $1 \le i \le d$,

$$
\begin{align}
J_3^i(n, k)^p &\le B_p^p \mathbb{E} \bar{\sigma}_{0,k}(p) \sum_{1 \le r \le d} \mathbb{E} \left[ \mathbb{E}_{\xi} \left( \bar{Z}_n^i(r) \right)^{p/2} M_{0,n-1}(i,r)^{-p/2} \right] \nonumber \\
&\le C \sum_{r=1}^d \mathbb{E} \left[ \mathbb{E}_{\xi} \left( \bar{Z}_n^i(r) \right)^{p/2} M_{0,n-1}(i,r)^{-p/2} \right]. \tag{5.17}
\end{align}
$$

Using (4.4) in Lemma 4.3 and (1.11), for all $n \ge 1$ and $1 \le i, r \le d$, $\mathbb{P}$-a.s., we have

$$
\begin{align*}
M_{0,n-1}(i,r) &\ge \frac{1}{dD} \|M_{0,n-1}(i,\cdot)\| \\
&\ge \frac{1}{dD} \langle M_{0,n-1}(i,\cdot), U_{n,\infty} \rangle \\
&= \frac{1}{dD} \lambda_{0,n-1} U_{0,\infty}(i).
\end{align*}
$$

Combining this with (5.17) and (4.34), we get that for $p > 2$, $n, k \ge 1$ and $1 \le i \le d$,

$$
J_3^i(n, k)^p \le C \mathbb{E} \left[ \mathbb{E}_{\xi} (W_n^i)^{p/2} (\lambda_{0,n-1} U_{0,\infty}(i))^{-p/2} \right] = C \mathbb{E} a_{n,1}^i(p),
$$

where $a_{n,1}^i(p)$ is defined in (4.17) with $j=1$. This, together with (4.19) (which
holds under condition (2.9)), implies that there exists $\delta_3 \in (\delta_c(p), 1)$ such that for
all $n, k \ge 1$ and $1 \le i \le d$,

$$
J_3^i(n, k)^p \le C \delta_3^{np}. \tag{5.18}
$$

Combining (5.8), (5.9), (5.12), (5.15) and (5.18), we obtain that for all $n, k \ge 1$
and $1 \le i, j \le d$,

$$
(\mathbb{E} |\bar{Z}_{n+k}^i(j) - W^i|^p)^{1/p} \le C (\delta_1^n + \delta_2^k + \delta_3^n).
$$
Applying this inequality with $n$ replaced by $\lfloor n/2 \rfloor$ (the integral part of $n/2$) and taking $k = n - \lfloor n/2 \rfloor$, we see that for all $n \ge 1$ and $1 \le i, j \le d$,

$$ \left( \mathbb{E} |\bar{Z}_n^i(j) - W^i|^p \right)^{1/p} \le C (\delta_1^{n/2} + \delta_2^{n/2} + \delta_3^{n/2}) \le C \delta^n, $$

with $\delta = \max\{\delta_1^{1/2}, \delta_2^{1/2}, \delta_3^{1/2}\} < 1$. Therefore, for any $1 \le i, j \le d$ the normalized population size $\bar{Z}_n^i(j)$ convergences in $L^p$ to $W^i$ with an exponential speed, which gives (2.11). This concludes the proof of Theorems 2.3 and 2.4.

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