Coralie Fritsch
email: coralie.fritsch@inria.fr

Denis Villemonais
email: denis.villemonais@univ-lorraine.fr

Nicolás Zalduendo
email: zalduend1@univ-lorraine.fr.

THE MULTI-TYPE BISEXUAL GALTON-WATSON BRANCHING PROCESS

In this work we study the bisexual Galton-Watson process with a finite number of types, where females and males mate according to a "mating function" and form couples of different types. We assume that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. In the absence of a linear reproduction operator which is the key to understand the behaviour of the model in the asexual case, we build a concave reproduction operator and use a concave Perron-Frobenius theory to ensure the existence of eigenelements. Using this tool, we find a necessary and sufficient condition for almost sure extinction as well as a law of large numbers. Finally, we study the almost sure long-time convergence of the rescaled process through the identification of a supermartingale, and we give sufficient conditions to ensure a convergence in L 1 to a non-degenerate limit.

Introduction

The single-type bisexual Galton-Watson branching process is a modification of the standard Galton-Watson process. It assumes that there exist two disjoint classes (sexes), males and females who together form mating units (or couples) which can accomplish reproduction. This process was first introduced by Daley in [START_REF] Daley | Extinction conditions for certain bisexual galton-watson branching processes[END_REF]: it consists of a population where, in every generation n = 1, 2, 3, . . . , F n females and M n males form Z n = ξ(F n , M n ) mating units, for ξ a suitable and deterministic mating function. Each mating unit reproduces independently of the others and with identical distribution giving birth to the new generation of males and females.

Example 1. Some examples of mating functions are

(1) ξ(x, y) = x min{1, y} called the promiscuous mating model.

(2) ξ(x, y) = min{x, y} called the perfect fidelity mating model.

(3) ξ(x, y) = x, which corresponds to the classical Galton-Watson model. Daley studied in [START_REF] Daley | Extinction conditions for certain bisexual galton-watson branching processes[END_REF] the properties for the first two mating functions of the previous example. Since Daley's work, extinction conditions have been studied for models with a more general family of superadditive mating functions (see for instance [START_REF] David | A necessary condition for extinction in those bisexual galton-watson branching processes governed by superadditive mating functions[END_REF], [START_REF] Bruss | A note on extinction criteria for bisexual galton-watson processes[END_REF], [START_REF] Daley | Bisexual galton-watson branching processes with superadditive mating functions[END_REF]) and, in the last decades, results on the limit behaviour of this kind of processes were obtained (see for example [START_REF] Alsmeyer | The bisexual galton-watson process with promiscuous mating: extinction probabilities in the supercritical case[END_REF], [START_REF] Alsmeyer | Asexual versus promiscuous bisexual galton-watson processes: the extinction probability ratio[END_REF], [START_REF] González | On the limit behaviour of a superadditive bisexual galton-watson branching process[END_REF], [START_REF] González | On the l 2-convergence of a superadditive bisexual galton-watson branching process[END_REF]). From these works, new models of two-sex populations have been developed, such as processes in random or varying environment ( [START_REF] Ma | Bisexual galton-watson branching processes in random environments[END_REF], [START_REF] Ma | Two-sex branching processes with offspring and mating in a random environment[END_REF], [START_REF] Molina | Bisexual galton-watson branching process in varying environments[END_REF]), processes with immigration ([GMM00], [START_REF] González | On the limit behaviour of a superadditive bisexual galton-watson branching process[END_REF], [START_REF] Ma | The asymptotic properties of supercritical bisexual galton-watson branching processes with immigration of mating units[END_REF]), processes with mating function depending on the number of couples ( [START_REF] Molina | Bisexual galton-watson branching process with population-size-dependent mating[END_REF], [START_REF] Molina | On L α -convergence (1 ≤ α ≤ 2) for a bisexual branching process with population-size dependent mating[END_REF], [START_REF] Xing | On the extinction of a class of population-size-dependent bisexual branching processes[END_REF]) and more recently, processes with random mating ([JMM17], [START_REF] Molina | Population-dependent two-sex branching processes with random mating: rates of growth[END_REF]). The interested reader can also consult the surveys of Hull [START_REF] David | A survey of the literature associated with the bisexual galton-watson branching process[END_REF], Alsmeyer [START_REF] Alsmeyer | Bisexual galton-watson processes: A survey[END_REF] and Molina [START_REF] Molina | Two-sex branching process literature[END_REF] for a wide description of the work accomplished on this family of processes.

The asexual multi-type Galton-Watson branching process with a finite number of types (see [START_REF] Edward | The theory of branching processes[END_REF]) is a discrete time Markov Chain on N p , for p a fixed integer, and can be thought as a system of particles where each particle is characterized by a type among p options. Each particle reproduces independently and with an offspring distribution that depends only on its type, and if we define X i,j ∈ L 1 the number of type j particles produced by a type i progenitor, then, under some assumptions on the process, the greatest eigenvalue of the matrix H i,j = E(X i,j ) determines if the process will be eventually extinct with probability 1.

Our focus of study is the multi-type bisexual process with a finite number of types. Although specific models were studied in the two-sex population literature (see [START_REF] Karlin | Criteria for extinction of certain population growth processes with interacting types[END_REF], [START_REF] David | A reconsideration of galton's problem (using a two-sex population)[END_REF]), no general mathematical description for a superadditive multi-type model has yet been established. The aim of this paper is to fill this gap.

We will consider a general model, which includes the natural extension of Daley's bisexual model to multi-type processes, where the vector of couples in the n-th generation is defined by

Z n = (Z n,1 , . . . , Z n,p ) = ξ((F n,1 , . . . , F n,n f ), (M n,1 , . . . , M n,nm )), (1)
where ξ : N n f × N nm → N p is a positive function such that ξ(0, 0) = 0 and (F n,1 , . . . , F n,n f ) and (M n,1 , . . . , M n,nm ) are the vectors of females and males respectively with p, n f , n m ∈ N = {0, 1, 2, . . . } the number of types of couples, females and males respectively. Each couple reproduces independently from the others and produces females and males according to a distribution that depends only on its type, such that

F n+1,j = p i=1 Z n,i k=1 X (k,n+1) i,j , for 1 ≤ j ≤ n f , M n+1,j = p i=1 Z n,i k=1 Y (k,n+1) i,j , for 1 ≤ j ≤ n m ,
where, (X (k,n) , Y (k,n) ) k,n∈N is a family of i.i.d. copies of ((X i,j ) 1≤i≤p,1≤j≤n f , (Y i,j ) 1≤i≤p,1≤j≤nm ) where X i,j represents the number of female offspring of type j produced by one couple of type i, and similarly for the males and Y i,j . We recover Daley's process by setting p = n f = n m = 1.

Since types may also encode gender, and to simplify our notation, we present in Section 2 our definitions and main results in a more general setup, since they hold true for asexual and multi-sexual Galton-Watson processes (as they appear for instance for several plants species [START_REF] Ross | Understanding apomixis: recent advances and remaining conundrums[END_REF] and [BLVD + 11, BT12]). We present a law of large numbers (Theorem 1), necessary and sufficient conditions for extinction (Theorem 3) as well as asymptotic behaviour of the process (Theorems 4 and 6). We compare our results with existing works at the end of Section 2. Section 3 is devoted to the proof of Theorem 1. In Section 4, we turn our attention to some extra properties, such as the eigenvalue problem for concave functions, which play a fundamental role in the study of the extinction conditions, and prove Theorem 3. Section 5 is devoted to the proof of Theorem 4. Finally, we identify and establish properties on a supermartingale and prove Theorem 6 in Section 6.

Notation: Unless otherwise stated, for z ∈ (R + ) p , we denote by z i the i-th component of z. We set S := {z ∈ (R + ) p , |z| = 1} to be the unitary ball for the 1-norm on (R + ) p , with |z| = z 1 + • • • + z p , and define S * as the elements on S with strictly positive components. All random variables and random vectors are defined on the same probability space (Ω, F, P), we denote E for the expectation associated to P and we let L 1 = L 1 (Ω, F, P) be the set of integrable random variables defined on this space.

Model description and main results

2.1. Model description. For p, q ∈ N * = {1, 2, . . . }, we consider the process (Z n ) n∈N with values in N p and the process (W n ) n∈N with values in N q , where Z n and W n represent the mating units (of p different types) and the individuals population (where individuals are of q different types) respectively at the n-th generation. We assume that at each generation n ≥ 1, Z n is entirely determined by W n , through a mating function ξ : N q → N p satisfying ξ(0) = 0, that is Z n = (Z n,1 , . . . , Z n,p ) = ξ(W n,1 , . . . , W n,q ), where the individuals population W n at the n-th generation is produced by the Z n-1 mating units of the previous generation. Moreover, we assume that each mating unit reproduces independently from the others and is such that

W n,j = p i=1 Z n-1,i k=1 V (k,n) i,j
, for 1 ≤ j ≤ q, with (V (k,n) ) k,n∈N a family of i.i.d. copies of V = (V i,j ) 1≤i≤p;1≤j≤q , where V 1,• , . . . , V p,• are p mutually independent random vectors with values in N q . The random variable V (k,n) i,j represents the number of offspring of type j produced by the k-th mating unit of type i of the (n -1)-th generation. Note that the offspring V (k,n) i,j 1 and V (k,n) i,j 2 of type j 1 and j 2 produced by the same mating unit are not necessary independent.

Considering the empty sum as zero, (Z n ) n∈N forms a discrete time Markov Chain on N p with absorbing state 0. We define the probability of extinction with initial condition z ∈ N p as

q z = P(∃n ∈ N, Z n = 0|Z 0 = z),
and declare that the process will be almost surely extinct if q z = 1 for all z ∈ N p .

Although our model is sufficiently general to describe multi-type multi-sexual Galton-Watson branching process (see Example 2), keeping in mind our motivation for the definition of this process, we call (Z n ) n∈N the multi-type bisexual Galton-Watson branching process (from now on, multi-type bGWbp).

Throughout the paper, we make the assumption that the mating function ξ of the process is superadditive, that is,

ξ(x 1 + x 2 ) ≥ ξ(x 1 ) + ξ(x 2 ), ∀x 1 , x 2 ∈ N q . ( 2
)
The intuition for this type of processes is that two populations form a bigger number of mating units together rather than separate. The idea of a superadditive bisexual Galton-Watson process for the single-type case was first introduced by Hull in [START_REF] David | A necessary condition for extinction in those bisexual galton-watson branching processes governed by superadditive mating functions[END_REF] and necessary and sufficient conditions for certain extinction were given by Daley et al. in [START_REF] Daley | Bisexual galton-watson branching processes with superadditive mating functions[END_REF].

Remark 1. It will be useful in the proof to observe that, if we consider two superadditive functions ξ 1 , ξ 2 such that ξ 1 (x) ≤ ξ 2 (x), ∀x ∈ N q , then a multi-type bGWbp with mating function ξ 1 is stochastically dominated from above by a process with the same offspring distribution but with mating function ξ 2 .

Example 2 (Multi-type bisexual Galton-Watson process). We recover the multi-type bGWbp setup presented in the introduction if we set q = n f + n m and (W n,1 , . . . , W n,n f ) as the vector of females and (W n,n f +1 , . . . , W n,n f +nm ) as the vector of males of the n -th generation. We can also extend this definition to a multi-sexual process by separating the set of possible types into a larger number of sexes.

We assume that all the random variables (V i,j ) 1≤i≤p,1≤j≤q are integrable and define the matrix

V ∈ R p,q by V i,j = E(V i,j ), ∀ 1 ≤ i ≤ p, 1 ≤ j ≤ q.
We assume that, for all j ∈ {1, . . . , q}, p i=1 V i,j > 0. We now define M : R p

+ -→ (R + ∪ {+∞}) p by M(z) = lim r→+∞ ξ(rzV) r = sup r≥1 ξ(rzV) r , ( 3
)
where ξ is any superadditive extension of ξ to R q + (see Remark 2 below). The limit is well defined and equal to the supremum according to Fekete's Lemma.

As we will see, the convergence is in fact uniform on any compact subset of S where M is continuous (see Proposition 11). In addition M is concave on R p + (see Proposition 10). Remark 2. A superadditive function ξ on N q can always be extended to a superadditive function on R q + (for instance setting x ∈ R q + → ξ( x )) and it will appear that M does not depend on the choice of this extension (see Proposition 11).

Remark 3. The functional M plays a similar role as the reproduction matrix in the classical multi-type Galton-Watson case (see Example (3) below). However in our case, it is not necessarily linear, but only concave. In order to analyse the function M and its iterates, we make use of concave Perron-Frobenius theory and, more precisely, of the results developed by Krause [START_REF] Krause | Relative stability for ascending and positively homogeneous operators on banach spaces[END_REF] (see Section 4.1).

Main results.

Our first main result is a law of large numbers which relates M with the behaviour of Z in a large initial population setting and which is proved in Section 3.2. We set

M n = M • • • • • M composed n times
with the convention that M 0 is the identity function, and, for all i ∈ {1, . . . , p}, M i is the i th component of M.

Theorem 1 (Law of large numbers). Let (z m ) m≥1 be a random sequence in N p and z ∞ ∈ R p + \ {0} a deterministic value such that z m ∼ m→+∞ mz ∞ almost surely. For all m ≥ 1, denote by (Z m n ) n≥0 a multi-type bGWbp with common mating function and offspring distribution, but with initial configuration Z m 0 = z m . Define M as in (3) and assume it is finite over S. Then, for all n ≥ 0,

Z m n ∼ m→+∞ m M n (z ∞ ) almost surely. If in addition (z m /m) m≥1 is independent of the random variables V (k,n) i,j
and uniformly integrable, then Z m n /m converges to M n (z ∞ ) in L 1 . As a consequence, considering for z ∈ R p + the sequence z m = mz for all m ∈ N, we have the following corollary (the second equality is a classical consequence for superadditive sequences).

Corollary 2. We have for all

z ∈ R p + , M(z) = lim m→+∞ E(Z 1 | Z 0 = mz ) m = sup m≥1 E(Z 1 | Z 0 = mz ) m .
The function M extends to the multi-type case the mean growth rate introduced in the single-type case by Bruss in [START_REF] Bruss | A note on extinction criteria for bisexual galton-watson processes[END_REF] and used by Daley et al. in [DHT86] to study the extinction conditions for the process. Note also that, in the situation where z m = mz , one can adapt the proof of Klebaner [START_REF] Fima | Population-dependent branching processes with a threshold[END_REF] to obtain convergence in law in Theorem 1, as detailed in [START_REF] Adam | Persistance et vitesse d'extinction pour des modèles de populations stochastiques multitypes en temps discret[END_REF]. However, almost sure and L 1 convergence obtained in Theorem 1 are needed in the proofs of the following results.

For the rest of the results of this section, we add the following transitivity and primitivity assumptions for the process.

Assumption 1. We assume that:

(1) The process is transitive, which means that

P lim n→∞ |Z n | ∈ {0, +∞} | Z 0 = z = 1, ∀z ∈ N p .
(2) The process is primitive, that is, for all i ∈ {1, . . . , p}, there exist n i , k i ∈ N big enough such that for all m ≥ n i ,

E(Z m |Z 0 = k i e i ) > 0
where e i is the i-th canonical vector in R p .

Assumption 1.1 is a classical transitivity condition. The following cases are some examples where it holds.

(1) For all i ∈ {1, . . . , p}, P(W 1 = 0|Z 0 = e i ) > 0.

(2) In the context of the bisexual setting, ∀i ∈ {1, . . . , p}, P(M 1 = 0|Z 0 = e i ) > 0 and ξ(x, 0) = 0, ∀x ∈ R n f . (3) ∃ ∈ {1, . . . , p}, P(|Z 1 | = 2|Z 0 = e ) > 0 and the process is strongly primitive, that is Assumption 1.2 is satisfied with k i = 1 for all i ∈ {1, . . . , p} (see Appendix A for details). Note that the (not strong) primitivity of the process is not sufficient. In fact, choosing q = p = 2 and

ξ(x, y) = y 2 , x , V 1 ∼ 1 2 δ (2,0) + 1 2 δ (0,2) , V 2 ∼ δ (0,1) .
leads to a (not strongly) primitive branching process, satisfying

P(|Z 1 | = 2|Z 0 = (1, 0)) = 1/2 > 0 and such that {(1, 0), (0, 2)} is a recurrent class.
The following result provides a necessary and sufficient condition for almost sure extinction. This result is proved in Section 4.2.

Theorem 3 (Extinction criterion). Assume that Assumption 1 holds and that M is finite over S. Then there exist a unique λ * > 0 and a unique z * ∈ S * such that M(z * ) = λ * z * , and we have

q z = 1, ∀z ∈ N p ⇐⇒ λ * ≤ 1.
If λ * > 1 or if there exists z ∈ N p such that one of the components of M(z ) is not finite, then there exists r > 0 such that, if |z| > r, then q z < 1.

Before turning to the next result, we point out that the last theorem encompasses the well known extinction criteria for the classical multi-type Galton-Watson process and single-type bisexual Galton-Watson process. Further examples are provided in Section 2.3.

Example 3 (Multi-type Galton-Watson process). The Galton-Watson case corresponds to the case where p = q and the mating function is given by the identity function, ξ(x) = x, so that the process forms a classical asexual multi-type Galton-Watson process. In this case it is easy to see that M(z) = zV and hence M is a linear function and λ * is its greatest eigenvalue. We thus recover the well known fact that λ * ≤ 1 is a necessary and sufficient condition for certain extinction.

Example 4 (Single-type bisexual Galton-Watson process). The single-type bisexual Galton-Watson case corresponds to the case where p = 1 and q = 2. In this case it is easy to see that M(z) = rz for some r ≥ 0, that z * = 1 and that λ * = r. We recover the fact that λ * ≤ 1 is a necessary and sufficient condition for certain extinction (see [START_REF] Daley | Bisexual galton-watson branching processes with superadditive mating functions[END_REF]).

In the following theorem, we deal with the long-time behaviour of the process. In particular we prove that, on the non-extinction event {Z n = 0, ∀n ≥ 0}, the process will almost surely follow the direction of the eigenvector z * . The proof of this theorem is in Section 5. We emphasize that, for this result, we do not assume any L log L type condition. As far as we know, this result is new even in the p = 1 and q = 2 (single type) case.

In what follows, given x, y ∈ R p , we denote

|x, y| := {z ∈ R p , x ≤ z ≤ y}.
Theorem 4 (Long time behaviour). Assume that Assumption 1 holds and that M is finite over S. Let λ * and z * given by Theorem 3 and assume that λ * > 1. Then there exists n 0 ≥ 1 such that, for all ε ∈ (0, 1) and all η ∈ (0, 1), there exists r > 0 such that

P (Z n 0 = 0 and ∀n ≥ n 0 , Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) ≥ 1 -η, ∀|z| ≥ r.
In addition, on the non-extinction event {Z n = 0, ∀n ≥ 0}, and up to a

P(• | Z 0 = z)-negligible event, for all k ≥ 0, lim n→+∞ Z n+k |Z n | = (λ * ) k z * .
On the event of extinction, M(Z n ) vanishes for n large enough almost surely, which entails that

Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )|
for all n large enough. For λ * > 1 (so that extinction is not almost sure), this also holds true with probability one, as proved alongside Theorem 4. We thus obtain Corollary 5. Assume that Assumption 1 holds and that M is finite over S. Then, for all ε ∈ (0, 1) and all z ∈ N p ,

P (∃N ≥ 0 such that, ∀n ≥ N, Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) = 1. ( 4
)
Remark 4. As it will be clear from the proof, the value of n 0 in Theorem 4 is in fact chosen deterministically as the minimal n ∈ N such that M n (z) > 0 for all z ∈ N p \ {0}, which exists thanks to Proposition 14 below.

We now state a theorem related to the rescaled processes of mating units and children and we prove that they both have a non-negative limit with the same direction as the vector z * given by Theorem 3. We also show that the event of extinction coincides (almost surely) with the event where this limit is equal to zero. This last part is well known in the classical branching case. As far as we know, it is new even in the p = 1 and q = 2 (single type) bisexual branching case. This theorem is proved in Section 6.

Theorem 6 (Asymptotic profile). Assume that Assumption 1 holds, that M is finite over S. Then, for all z ∈ N p , there exists a real non-negative random variable C such that

Z n (λ * ) n P(•|Z 0 =z) a.s. ---------→ n→+∞ Cz * and W n (λ * ) n P(•|Z 0 =z) a.s. ---------→ n→+∞ 1 λ * Cz * V, (5)
with λ * and z * given by Theorem 3. Assume in addition that C is non-degenerate at 0, which means that P(C > 0 | Z 0 = z) > 0 for all z ∈ N p such that q z < 1. Then, for all z ∈ N p and up to a P(• | Z 0 = z) negligible event,

{C = 0} = {∃n ∈ N, Z n = 0}.
Remark 5. We observe that the transitivity assumption implies that the condition P(C > 0 | Z 0 = z) > 0 for all z ∈ N p such that q z < 1 is actually true if and only if

P(C > 0 | Z 0 = z) > 0 for some z ∈ N p .
A natural question that arises is to find conditions so that the previous convergence holds also in L 1 and the limit is non-degenerate at 0. The following proposition, proved in Section 6.3, deals with this question, for which we consider the function P : R p + -→ R p + given by

∀z ∈ R p + , P(z) = lim n→+∞ |M n (z)| (λ * ) n , ( 6
E(|P(Z 1 ) -P(M( z ))| | Z 0 = z ) ≤ U (y), (7)
with y → U (y) /y non-increasing and +∞ 1 U (y) y 2 dy < +∞, then the convergence in Theorem 6 is in L 1 and the random variable C is non-degenerate at 0.

The existence of the function U in the previous theorem may be difficult to check. In the following proposition we state sufficient conditions to ensure its existence, under a V log V condition and extra assumptions on the functions P, ξ and M. The proof of this proposition is in Section 6.4. Proposition 8. Assume that M is finite over S. In addition assume that both functions P and ξ are Lipschitz, that E(V i,j log V i,j ) < +∞ for all i ∈ {1, . . . , p}, j ∈ {1, . . . , q}, and that there exists

α > 0 such that ξ(zV) |z| - M(z) |z| = O(|z| -α ), ∀z ∈ R p + \ {0}.
Then, the condition (7) is satisfied.

Remark 6. Since originally ξ is only defined over N q , the statement "ξ is a Lipschitz function" must be interpreted as "there exists an extension of ξ from N q to R q + , that is Lipschitz".

The previous conditions are not necessary conditions to ensure the existence of the function U in (7). In fact, in Proposition 9, we state that in the model of Example 6 below, for which there is no Lipschitz extension for ξ over all R q + , the V log V condition is sufficient to ensure the L 1 convergence to a non-degenerate random variable in (5).

Examples in the context of bGWbp.

The following examples are in the context of the multi-type bisexual Galton-Watson process. We recall that in this case q = n f + n m , where n f , n m ∈ N * are respectively the number of types for females and males. In order to be consistent with our notation, we write ξ((x 1 , . . . , x n f ), (y 1 , . . . , y nm )) = ξ(x 1 . . . , x n f , y 1 , . . . y nm ). In this context, for i ∈ {1, . . . p} and j ∈ {1, . . . n f } we set X i,j = V i,j , and similarly, for j ∈ {1, . . . , n m }, Y i,j = V i,n f +j . Finally, we define the matrices X ∈ R p,n f and Y ∈ R p,nm given by

X i,j = E(X i,j ), Y i,j = E(Y i,j ).
Example 5 (Perfect fidelity mating function). Consider the case where n f = n m = p and the mating function ξ(x, y) = min{x, y} := (min{x i , y i }) i≤p , which is a natural extension of the perfect fidelity case presented by Daley ([Dal68]) to the multi-dimensional case. In this case, we have min{kzX, kzY} k = min{zX, zY}, ∀k ≥ 1.

Hence the function M takes the form M(z) = min{zX, zY}. Let us discuss different particular instances of this model.

• If X ≤ Y are aperiodic irreducible non-negative matrices, then M(z) = zX and so M is a linear function and λ * is its greatest eigenvalue. In the super-critical case (i.e. λ * > 1), the asymptotic profile of the types of the process in the non-extinction event is given by its positive left eigenvector. We thus observe that, despite the interaction between males and females, the extinction and growth characterization of the process is similar to the classical Galton-Watson case.

• The case X proportional with Y can be handled similarly: Let (X i,j ) 1≤i≤p,1≤j≤p and (Y i,j ) 1≤i≤p,1≤j≤p defined by

X i,j = U i,j =1 ε i,j, and Y i,j = U i,j i= (1 -ε i,j, ),
where (U i,j ) 1≤i≤p,1≤j≤p is a random integrable array with mean U and (ε i,j, ) 1≤i≤p,1≤j≤p, ∈N is an array of i.i.d. {0, 1} valued random variables independent from U . The variable U i,j describes the number of children of type j from a mating unit of type i and ε i,j, determines if the -th child is a female or a male. Note that in this example X i,j and Y i,j are not independent. Then, setting α = P(ε i,j, = 1), we have

X = αU and Y = (1 -α)U.
As a consequence,

M(z) = min{α, 1 -α} zU
and M is a linear function. Assume now that U is an aperiodic irreducible non-negative matrix with greatest eigenvalue λ U and positive left eigenvector z U . Then λ * = min{α, 1 -α}λ U and, in the super-critical case, the asymptotic profile of the types of the process (Z n ) n∈N on the non-extinction event is given by z * = z U . • Let us now consider a non-linear case. Assume that X = αI p + β1 p and Y = α

I p + β 1 p ,
where α, α ≥ 0 and β, β > 0 are constants, I p is the identity matrix of size p × p and 1 p is the matrix of size p × p filled with ones. Then, for all i ∈ {1, . . . , p},

M i (z) = min αz i + β|z|, α z i + β |z| .
Note that, for any permutation σ of {1, . . . , p}, we have

M(z σ(1) , . . . , z σ(p) ) = (M σ(1) (z), . . . , M σ(p) (z)).
Hence z * is stable by permutation of its components, so that z * = ( 1 /p, . . . , 1 /p). We deduce that

λ * = |M(z * )| = p min{α/p + β, α /p + β } = min{α + βp, α + β p}.
Remark 7. The previous example also covers a polygamous mating by one of the sexes, if we fix d ∈ N and let ξ(x, y) = min{x, dy} = (min{x i , dy i }) i≤p , as Daley did [START_REF] Daley | Extinction conditions for certain bisexual galton-watson branching processes[END_REF] in the single-type case. In this situation, we recover the same criterion.

Example 6 (Completely promiscuous mating function). The case studied by Karlin and Kaplan [START_REF] Karlin | Criteria for extinction of certain population growth processes with interacting types[END_REF] corresponds to the case where the number of couples is equal to the number of females present in every generation (in particular this implies that n f = p) given the condition that there is at least one male of each type present in every generation. In other words, they consider the mating function

ξ((x 1 , . . . , x p ), (y 1 , . . . , y nm )) = (x 1 , . . . , x p ) nm i=1
1 {y i >0} .

The function M in this case corresponds to

M(z) = zX1 {zY>0} .
We assume that X is aperiodic irreducible and that ∀j ≤ n m , ∀i ≤ p : Y i,j > 0, (this last condition ensures that Assumption 1 holds) then zY > 0 for all z ∈ R p + \ {0}. In particular M(z) = zX for all z ∈ R p + , which implies that the unique unitary positive eigenvector of M and its corresponding eigenvalue are the ones of X given by the Perron-Frobenious Theorem. This result already appeared in [START_REF] Karlin | Criteria for extinction of certain population growth processes with interacting types[END_REF].

In addition, Proposition 7 entails the following original convergence property, proved in Section 6.4. In particular, this model satisfies the conditions of Theorem 6. As stated before, Example 6 does not satisfy the assumptions of Proposition 8 since ξ : N q → N p is not Lipschitz. Proposition 9. Consider the model in Example 6 above. Assume in addition that E(X i,j log X i,j ) is finite for all i, j ∈ {1, . . . p}. Then, the rescaled process Zn /(λ * ) n converges almost surely and in L 1 to a non-degenerate random vector, with the same direction as z * .

Characterization of M and proof of Theorem 1

Let (Z n ) n∈N be a multi-type bGWbp with superadditive mating function ξ and consider the function M associated to this process given by (3). We start by stating and proving some properties related to this function in Section 3.1 and prove Theorem 1 in Section 3.2.

Characterization of M.

In this section we give some fundamental properties of the operator M defined in Section 2, and relate it to the behaviour of the number of mating units in the population. We start by proving that M is concave and positively homogeneous, then we prove that it does not depend on the chosen extension for ξ to R q and state first properties of this function.

Definition 1. A function F : R p + -→ R p + is said to be (1) Concave if F (αx + (1 -α)y) ≥ αF (x) + (1 -α)F (y), for all α ∈ [0, 1] and all x, y ∈ R p + . (2) Positively homogeneous if for all α > 0, F (αx) = αF (x) for all x ∈ R p + . (3) Primitive if there exists n 0 ≥ 1 such that F m (x) > 0 for all m ≥ n 0 and x ∈ R p + \ {0}.
Proposition 10. The function M is positively homogeneous and concave.

Proof. Let α > 0, then

M(αz) = lim k→+∞ ξ(αkzV) k = α lim k→+∞ ξ(αkzV) αk = αM(z),
and so M is positively homogeneous. Using this and the fact that ξ (and hence M) is a superadditive function, we deduce that M is a concave mapping.

Proposition 11. For all z ∈ R p + , we have

M(z) = lim r→+∞ ξ( rzV ) r = sup r≥1 ξ( rzV ) r . ( 8
)
In addition, for any compact set K ⊂ S such that M is continuous on K, M is either bounded on K or infinite on K. In the former case,

sup z∈K M(z) - ξ(rzV) r ----→ r→+∞ 0, ( 9
)
and, in the latter case,

inf z∈K ξ(rzV) r ----→ r→+∞ +∞. ( 10
)
Proof. For the first assertion consider z ∈ R p + , u ∈ {0, 1} p given by u i = 1 {(zV) i =0} and let n ∈ N * be such that zV ≥ 1 n u. Then, for all r > 0,

ξ( (r + n)zV ) r + n ≥ ξ( rzV + u ) r + n ≥ ξ(rzV) r r r + n .
Taking the limit when r → +∞, we conclude that

M(z) ≤ lim r→+∞ ξ( rzV ) r .
The reverse inequality is direct using the fact that ξ is non-decreasing in all its components, which concludes the proof of the first equality in (8). The second equality is a consequence of Fekete's Lemma and the fact that r → ξ( rzV ) is superadditive.

For the second part, let K be a compact subset of S such that M is continuous on K. Note that, M being continuous, it is either bounded or equal to +∞ on K.

We first consider the case where M is bounded on K. Since M(z) ≥ ξ(rzV) r for all z ∈ S and r > 0, we only have to prove that lim sup

r→+∞ sup z∈K M(z) - ξ(rzV) r ≤ 0.
Assume the contrary. Then there exist ε > 0 and two sequences (

z n ) n∈N ∈ K N and (r n ) n∈N ∈ (0, +∞) N such that r n +∞ and ξ(r n z n V) r n ≤ M(z n ) -ε.
Since K is compact, there exists, up to a subsequence, z ∞ ∈ K such that z n → z ∞ . In particular, for all δ ∈ (0, 1), there exists n δ,ε such that, for all n ≥ n δ,ε ,

z n ≥ (1 -δ)z ∞ and M(z n ) ≤ M(z ∞ ) + ε/2 and hence ξ ((1 -δ)r n z ∞ V) r n ≤ M(z ∞ ) -ε/2.
By definition of M and Proposition 10, the left hand side converges to

M((1 -δ)z ∞ ) = (1 -δ)M(z ∞ ) when n → +∞, and hence (1 -δ)M(z ∞ ) ≤ M(z ∞ ) -ε/2.
Since this is true for all δ > 0, ε > 0 and since M(z ∞ ) < +∞ by assumption, this is a contradiction. We thus proved (9).

The proof of (10) is similar. Assume that it does not hold true. Then there exist A > 0 and two sequences (z n ) n∈N ∈ K N and (r n ) n∈N ∈ (0, +∞) N such that r n +∞ and

ξ(r n z n V) r n ≤ A.
This implies that, up to a subsequence, z n converges to z ∞ ∈ K and that, for any δ ∈ (0, 1),

(1 -δ)M(z ∞ ) ≤ A.
But M is equal to +∞ on K and hence we obtained +∞ ≤ A, which is a contradiction. This concludes the proof of Proposition 11.

As a consequence of Proposition 11, we have the following result in the case where M is continuous.

Corollary 12. Assume M is finite and continuous over S. Then, we have

M(z) |z| - ξ(zV) |z| |z|→+∞ -----→ 0.
The following lemma, used in the proof of the forthcoming Proposition 14 and Proposition 15, relates the iterations of M with the expectation of the process. It is proved in the Section 3.2, after the proof of Theorem 1 and Corollary 2. Note that the proof of this lemma is based on Corollary 2, however neither Theorem 1 nor Corollary 2 makes use of Lemma 13, Proposition 14 or Proposition 15.

Lemma 13. For all n ≥ 0 and all z ∈ N p , we have

M n (z) ≥ E(Z n |Z 0 = z), ∀n ∈ N.
The following result states that the primitivity of the bGWbp entails the primitivity of M. Proposition 14. Assume that Assumption 1.2 holds, that is, (Z n ) n∈N is primitive, then M is a primitive function.

Proof. Since (Z n ) n∈N is primitive, we can find N, k ∈ N big enough so that for all i ∈ {1, . . . , p} and m ≥ N we have that E(Z m |Z 0 = ke i ) > 0. Hence, for m ≥ N and z ∈ N p \ {0}, using Lemma 13 and the superadditivity of ξ and then of z

→ E(Z m |Z 0 = z), kM m (z) = M m (kz) ≥ E(Z m |Z 0 = kz) ≥ p i=1 z i E(Z m |Z 0 = ke i ) > 0,
and so M m (z) > 0, which concludes the proof.

We finish this subsection by stating one last property on M.

Proposition 15. Assume that Assumption 1.2 holds. We have inf z∈S |M(z)| > 0 and, for all compact subset K ⊂ S * and for all i ∈ {1, . . . , p}, inf z∈K M i (z) > 0.

Proof. We start by proving the first assertion. Since M is primitive by Proposition 14, there exists n 0 ≥ 1 such that M n 0 (e i ) > 0 for all i ∈ {1, . . . , p}. In particular, M(e i ) = 0 for all i ∈ {1, . . . , p}. Using the concavity of M, we deduce that

inf z∈S |M(z)| ≥ inf z∈S p i=1 z i M(e i ) ≥ min i∈{1,...,p} |M(e i )| > 0.
Let us now prove the second assertion. For z ∈ K, if M i (z) = +∞, by Proposition 10, M i (z ) = +∞ for all z ∈ K and the result follows directly. If M i (z) < +∞, since M i is concave, it is locally Lipschitz on S * and hence z → M i (z) is continuous on the compact set K. It is thus sufficient to prove the result for any fixed z ∈ S * . For this, we simply observe that, for any two z, z ∈ S * , we have, using the fact that z → M i (z) is positively homogeneous and increasing,

M i (z) ≥ min j∈{1,...,p} z j max j∈{1,...,p} z j M i (z ).
Hence z → M i (z) is either null or positive on S * . Since M is primitive by Proposition 14, z → M i (z) is not null (take for instance z = M n 0 (e i )), which concludes the proof.

3.2. Proof of Theorem 1. We start with the following lemma, where we do not assume that M is finite over S.

Lemma 16. Let (z k ) k≥0 be a random sequence in N p such that z k ∼ k→+∞ kz ∞ ∈ R p + almost surely. We have 1 k ξ p i=1 z k,i m=1 V (m) i,• -→ M(z ∞ ),
almost surely when k → +∞.

The proof of this lemma is inspired by [START_REF] Daley | Bisexual galton-watson branching processes with superadditive mating functions[END_REF].

Proof. Since all the variables V i,j are integrable, then thanks to the strong law of large numbers, we have that for all i ≤ p, j ≤ q,

1 n n m=1 V (m) i,j a.s. -----→ n→+∞ V i,j .
Assume first that z ∞,i > 0. In this case, since z k,i → +∞ almost surely we deduce that

1 z k,i z k,i m=1 V (m) i,j a.s. ----→ k→+∞ V i,j
and hence 1 kz ∞,i

z k,i m=1 V (m) i,j a.s. ----→ k→+∞ V i,j . Fix 0 < ε < min i,j/V i,j =0
V i,j . Hence, with probability one there exists k 0 (random

) such that if k ≥ k 0 , then kz ∞,i (V i,j -ε) ≤ z k,i m=1 V (m) i,j ≤ kz ∞,i (V i,j + ε).
Assume now that z ∞,i = 0. Then, almost surely, there exists k 0 such that for all k ≥ k 0 , z k,i = 0, so that the last inequality also holds true.

We consider again the general case z ∞ ≥ 0. Summing on i we obtain that, almost surely, there exists k 0 such that, for all k ≥ k 0 and all j ≤ q,

p i=1 V i,j =0 kz ∞,i (V i,j -ε) ≤ p i=1 z k,i m=1 V (m) i,j ≤ p i=1 V i,j =0 kz ∞,i (V i,j + ε),
where we used the fact that

V (m) i,j = 0 almost surely if V i,j = 0. Define the matrices V ε + := (V i,j + ε)1 V i,j =0 1≤i≤p,1≤j≤q and V ε -:= (V i,j -ε)1 V i,j =0 1≤i≤p,1≤j≤q .
Since the function ξ is superadditive, in particular it is non decreasing. Hence, we get

ξ(kz ∞ V ε -) k ≤ 1 k ξ   p i=1 z k,i m=1 V (m) i,j 1≤j≤q   ≤ ξ(kz ∞ V ε + ) k . ( 11
)
Let define

δ = ε min i,j/V i,j =0 V i,j
, and note that δ < 1 thanks to our choice of ε.

Assume first that M(z ∞ ) < +∞ and note that lim sup

k→+∞ ξ(kz ∞ V ε + ) k = lim sup k→+∞ 1 k ξ          p i=1 V i,j =0 kz ∞,i 1 + ε V i,j V i,j     1≤j≤q      ≤ lim k→+∞ 1 k ξ   p i=1 kz ∞,i (1 + δ)V i,j 1≤j≤q   = (1 + δ)M(z ∞ ),
and similarly, lim inf

k→+∞ ξ(kz ∞ V ε -) k ≥ (1 -δ)M(z ∞ ).
Hence, taking k → +∞ in (11), we obtain

(1 -δ)M(z ∞ ) ≤ lim inf k→+∞ 1 k ξ   p i=1 z k,i m=1 V (m) i,j q j=1   ≤ (1 + δ)M(z ∞ )
Finally, taking ε → 0, then δ goes to 0 and we conclude the desired result when M(z ∞ ) < +∞.

If M (z ∞ ) = +∞ for some ∈ {1, . . . , p}, the inequality lim inf

k→+∞ ξ (kz ∞ V ε -) k ≥ (1 -δ)M (z ∞ ) still
holds and so the result follows in this case.

We now proceed with the proof of Theorem 1.

Proof of Theorem 1. We first prove the almost sure convergence in Step 1, and then the L 1 convergence in Step 2.

Step 1. Almost sure convergence. The result is trivial for n = 0. By Lemma 16, we have

Z m 1 m = 1 m ξ p i=1 z m,i k=1 V (1,k) i,• a.s. ----→ m→∞ M(z ∞ ), (12)
If M (z ∞ ) > 0 for some ∈ {1, . . . , p}, then this proves that

Z m 1, ∼ m→+∞ m M (z ∞ ) almost surely. If M (z ∞ ) = 0, then ξ (kz ∞ V)
vanishes for all k ≥ 1 and hence Z 1, 1 Z 0 ≤Cz∞ = 0 almost surely for all C > 0. Since z m ∼ m→+∞ mz ∞ , we deduce that there exists a (random) m 0 ≥ 1 such that, for all m ≥ m 0 , Z m 1, = 0. Thus we proved that Z m 1 ∼ m→+∞ m M(z ∞ ) almost surely, which proves the result when n = 1.

Assume now that Z m n ∼ m→+∞ mM n (z ∞ ) a.s. for some n ≥ 1. Then the previous step with

z m = Z m n entails that Z m n+1 ∼ m→∞ m M(M n (z ∞ )) = m M n+1 (z ∞ ) a.s.
This concludes the proof of the first assertion in Theorem 1.

Step 2. Convergence in L 1 . We prove now the L 1 -convergence. Denote 1 q ∈ N q , 1 q = (1, . . . , 1), and fix z 0 ∈ N p such that z 0 V ≥ 1 q . Consider the bGWbp with initial position Z m 0 = z m , m ≥ 1, and denote by W m 1 the number of children in the first generation. We have W m 1 ≤ |W m 1 |1 q ≤ |W m 1 |z 0 V, and so using the second equality in (3),

Z m 1 = ξ(W m 1 ) ≤ ξ (|W m 1 |z 0 V) ≤ M (|W m 1 |z 0 ) .
Using Proposition 10, we deduce that

Z m 1 ≤ |W m 1 |M(z 0 ). ( 13
)
By assumption, the random vector

U (m) := z m /m + 1
is uniformly integrable, and we have z m ≤ mU (m) almost surely, so that

0 ≤ |W m 1 | ≤ p i=1 mU (m) i k=1 V (k,1) i,• . ( 14
)
Since U (m) is independent from the other terms, we have

E    1 m q j=1 p i=1 mU (m) i k=1 V (k,1) i,j -V i,j    ≤ u∈(N\{0}) p q j=1 p i=1 1 mu i E mu i k=1 V (k,1) i,j -V i,j |u| P U (m) = u . ( 15
)
Using the law of large numbers, we deduce that, for each i ∈ {1, • • • , p} and j ∈ {1, • • • , q},

1 m E m k=1 V (k,1) i,j -V i,j -----→ m→+∞ 0.
In particular, this ensures that the family

f m := max u∈(N\{0}) p q j=1 p i=1 1 mu i E mu i k=1 V (k,1) i,j -V i,j
converges to 0 when m → +∞. In addition, for all A > 0,

u∈(N\{0}) p q j=1 p i=1 1 mu i E mu i k=1 V (k,1) i,j -V i,j |u| P U (m) = u ≤ u∈(N\{0}) p |u|≤A f m |u| P U (m) = u + u∈(N\{0}) p |u|>A max n∈N f n |u| P U (m) = u ≤ f m A + E U (m) 1 |U (m) |>A max n∈N f n .
Since the family (U (m) ) m≥0 is uniformly integrable and choosing A large enough, E U (m) 1 |U (m) |>A can be made arbitrarily small uniformly in m, and, for any fixed A, choosing m large enough, the term f m A can be chosen arbitrarily small. Using (15), this implies that

E    1 m q j=1 p i=1 mU (m) i k=1 V (k,1) i,j -V i,j    -----→ m→+∞ 0.
In particular, this shows that

1 m q j=1 p i=1 mU (m) i k=1 V (k,1) i,j
converges in L 1 and is thus uniformly integrable. By inequalities (14) and (13), this entails that (Z m 1 /m) m≥1 is uniformly integrable too. Now, since we also proved that Z m 1 /m converges almost surely to M(z ∞ ), this implies that Z m 1 /m converges in L 1 to M(z ∞ ).

As above, the result for general n ≥ 1 derives by iteration, which concludes the proof of Theorem 1.

We now turn to the proof of Corollary 2.

Proof of Corollary 2. Theorem 1 yields Corollary 2 when M takes finite values. In the situation where M is not finite valued, we consider the vector 1 p = (1, . . . , 1) ∈ N p and introduce the superadditive function ξ(x) = |x|1 p .

Then, for all α ∈ N, we define the superadditive mating function

ξ (α) (x) = min{ξ(x), α ξ(x)} := min{ξ(x) i , α ξ(x) i } 1≤i≤p ,
and we denote by M (α) the function associated if we consider ξ (α) as mating function with the same offspring distribution as the original process. We can check that M (α) (z) = min{M(z), α ξ(zV)} and so we obtain that M (α) (z) M(z) as α → +∞, for all z ∈ R p + . Since clearly ξ (α) ξ, using the Monotone Convergence Theorem, for all z ∈ N p and i ∈ {1, . . . , p},

E (α) (Z 1 |Z 0 = z) -----→ α→+∞ E(Z 1 |Z 0 = z), ( 16
)
where E (α) is the probability law associated to the process with mating function ξ (α) . In particular, using Corollary 2 for the finite valued M (α) , for all z ∈ R p + ,

M(z) = sup α>0 M (α) (z) = sup α>0 sup m≥1 E (α) (Z 1 |Z 0 = mz ) m = sup m≥1 sup α>0 E (α) (Z 1 |Z 0 = mz ) m = sup m≥1 E(Z 1 |Z 0 = mz ) m .
Since m → E(Z 1 |Z 0 = mz ) defines a superadditive sequence, we deduce that

M(z) = lim m≥1 E(Z 1 |Z 0 = mz ) m ,
which concludes the proof of Corollary 2.

Let us now prove Lemma 13, hence also concluding the proof of Propositions 14 and 15.

Proof of Lemma 13. Let z ∈ N p . For n = 1, we use Corollary 2 and obtain that

M(z) = lim k→∞ E(Z 1 |Z 0 = kz) k = sup k∈N E(Z 1 |Z 0 = kz) k ≥ E(Z 1 |Z 0 = z).
Assume now that the inequality is true for some n ∈ N. Using the fact that M is increasing (since it is superadditive), we obtain

M n+1 (z) ≥ M(E(Z n |Z 0 = z)) ≥ E(M(Z n )|Z 0 = z) ≥ E E(Z 1 |Z 0 = z ) |z =Zn |Z 0 = z = E(Z n+1 |Z 0 = z),
where in the second step we have used Jensen's inequality, since M is concave by Proposition 10, and the last inequality is due to the Markov property. The proof is then complete. 4. Existence of the eigenelements and proof of Theorem 3 4.1. The concave eigenvalue problem. Consider A a real strictly positive N × N matrix. A well-known result that goes back to Perron [START_REF] Perron | Zur theorie der matrices[END_REF] states that

lim n→∞ A n x λ n = c(x)v, ∀x ∈ R p + ,
where λ is the greatest eigenvalue of A with v its corresponding eigenvector and c is a suitable function. This result and its consequences are among the main tools used to study the asymptotic behaviour of the classical multi-type Galton-Watson process, applied to the expectation matrix associated with the process. In this section we give similar results: a theorem that goes back to Ulrich Krause [START_REF] Krause | Relative stability for ascending and positively homogeneous operators on banach spaces[END_REF] that provides us with the necessary tools to study the extinction conditions for the multi-type bGWbp.

Theorem 17 (See [START_REF] Krause | Relative stability for ascending and positively homogeneous operators on banach spaces[END_REF] Section 4). Consider M : R p + -→ R p + a concave, primitive and positively homogeneous mapping. Then, (1) The eigenvalue problem M (z) = λz has a unique solution

(λ * , z * ) ∈ R × S * , with λ * > 0. If (λ, x) ∈ R × R p + \ {0}
is another solution of the problem, then it must hold that x = rz * for some r > 0 and λ = λ * .

(2) The function

L : (R + ) p -→ R + z * given by L(x) = lim k→∞ M k (x) (λ * ) k exists on R p
+ and holds L(x) = P(x)z * where P : R p + -→ R + is a concave and positively homogeneous mapping with P(x) > 0 for all x ∈ R p + \ {0}.

(3)

lim k→∞ M k (x) |M k (x)| = z * for all x ∈ R p + \ {0}. (4) lim k→∞ |M k+1 (x)| |M k (x)| = λ * = lim k→∞ |M k (x)| 1 k for all x ∈ R p + \ {0}. (5) The convergence toward L(x) = P(x)z * is uniform on x ∈ S.
4.2. Proof of Theorem 3. If M is finite, the existence of λ * and z * are guaranteed by Theorem 17, Propositions 10 and Proposition 14. Note that, in the following of the proof, we make use of Theorem 4 whose proof is developed in the next section and, except from the existence and uniqueness of λ * and z * which are yet established, does not use Theorem 3.

If M takes finite values and λ * ≤ 1, then, by assertion (2) in Theorem 17, for all z ∈ N p , (M n (z)) n∈N is a bounded sequence. From Lemma 13,

M n (z) ≥ E(Z n |Z 0 = z), hence E(Z n |Z 0 = z)
is bounded for all n ∈ N and so Z n does not converge to +∞ with positive probability. The conclusion is then given by Assumption 1.1 since then lim n→∞ |Z n | can only be almost surely 0, which finishes the proof of the theorem in the case λ * ≤ 1.

If M(z) < +∞ for all z ∈ S and λ * > 1, Theorem 4 entails that, for all ε ∈ (0, 1), there exists n 0 ∈ N and r > 0 such that, if Z 0 = z ∈ N p with |z| > r, we have that with positive probability

Z n 0 = 0 and Z n ≥ (1 -ε) n-n 0 M n-n 0 (Z n 0 ) for all n ≥ n 0 , with (1 -ε) n-n 0 M n-n 0 (Z n 0 ) = 0 (since M is
primitive), and so we obtain q z < 1.

Assume now that there exist z 0 ∈ S and i 0 ∈ {1, . . . , p} such that (M(z 0 )) i 0 = +∞. Consider, in the same way as for the proof of Corollary 2, the vector 1 p = (1, . . . , 1) ∈ N p and the function ξ(x) = |x|1 p .

For α ∈ N we define the function

ξ (α) (x) = min{ξ(x), α ξ(x)},
which is superadditive, and we define M (α) the function associated if we consider ξ (α) as mating function with the same offspring distribution as the original process. We can check that M (α) (z) = min{M(z), α ξ(zV)} and so we obtain that M (α) (z) M(z) as α → +∞, for all z ∈ R p + . Note that in particular

M (α) (z 0 ) i 0 α→+∞ -----→ +∞. ( 17
)
Since clearly ξ (α) ξ, using the Monotone Convergence Theorem, for all m ≥ 1 and i ∈ {1, . . . , p},

E (α) (Z m |Z 0 = ke i ) -----→ α→+∞ E(Z m |Z 0 = ke i ). ( 18
)
where E (α) is the probability law associated to the process with mating function ξ (α) . By Assumption 1.2, there exists c 0 > 0, m ≥ 1 and k ≥ 1 such that for all i ∈ {1, . . . , p},

E(Z m |Z 0 = ke i ) ≥ c 0 1 p .
By (18), there exists α 0 > 0 (which depends on m) such that for all α > α 0 and all i ∈ {1, . . . , p},

E (α) (Z m |Z 0 = ke i ) ≥ c 0 1 p 2 ≥ c 0 2 max j≤p z 0,j z 0 .
This implies, by Lemma 13, that for all i ∈ {1, . . . , p},

M m (α) (e i ) ≥ c 0 2k max j≤p z 0,j z 0 .
Hence, by (17),

M m+1 (α) (e i ) i 0 ≥ c 0 2k max j≤p z 0,j (M (α) (z 0 )) i 0 α→+∞ -----→ +∞. This implies that inf z∈S M m+1 (α) (z) i 0 α→+∞ -----→ +∞.
We remark that, since M (α) is bounded over S, concave, positively homogeneous and primitive, there exists λ α > 0 and x α ∈ S such that M (α) (x α ) = λ α x α . Using this we have that

λ m+1 α = |M m+1 (α) (x α )| α→+∞ -----→ +∞.
We conclude that there exists α 0 big enough such that λ α 0 > 1 and thanks to the previous computations the process with mating function ξ α 0 will not be almost surely extinct. Since ξ (α 0 ) ≤ ξ, this process is stochastically dominated by the original process, and so we can find r α 0 > 0 such that for all z ∈ N p with |z| > r α 0 , given {Z 0 = z}, the original process has a positive probability of survival.

Proof of Theorem 4

In order to prove Theorem 4, we first prove that, if λ * > 1 and under the assumption that M is bounded over S, we have that for all ε ∈ (0, 1), δ ∈ (0, 1 /p],

lim |z|→+∞ z∈U δ P (∀n ∈ N, Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )|) = 1, ( 19
)
where U δ is the set given by

U δ = {z ∈ N p : z ≥ |z|δ1 p }, (20)
where we recall that 1 p = (1, . . . , 1) ∈ N p and for all a, b ∈ R p + , |a, b| := {z ∈ R p + , a ≤ z ≤ b}. We remark that, for δ > 0, U δ is non-empty if and only if δ ∈ (0, 1 /p].

Then, we prove that for any initial values, either the process goes to extinction or it reaches a set U δ in finite time. Both results then lead to the proof of Theorem 4.

The second result is stated in Lemma 22, the first one is stated in Lemma 21 and is based on Lemmas 18, 19 and 20 for which we introduction the following additional notation.

For any ε ∈ (0, 1) and n ≥ 1, we consider the sequence of events

A ε n := {∀i ∈ {1, . . . , n}, Z i ∈ |(1 -ε)M(Z i-1 ), (1 + ε)M(Z i-1
)|}, or simply A n when there is no risk of ambiguity. We also set A ε 0 = Ω.

Lemma 18. Assume that M is bounded on S. For any δ ∈ (0, 1 /p] and ε ∈ (0, 1), there exists c 0 > 0 such that for all z ∈ U δ and all n ∈ N,

P (A ε n | Z 0 = z) ≥ 1 - n i=1 c 0 E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z .
where

f (x) = x p i=1 q j=1 P(V i,j > x) + p i=1 q j=1 E(V 2 i,j 1 V i,j ≤x ) x .
Proof. We prove this lemma in two steps.

Step 1. We first consider the case n = 1. That is, we prove that for δ ∈ (0, 1 /p] and ε ∈ (0, 1), there exists c 1 such that, for z ∈ U δ ,

P (Z 1 ∈ |(1 -ε)M(z), (1 + ε)M(z)| | Z 0 = z) ≥ 1 -c 1 |z| p i=1 q j=1 P(V i,j > |z|) -c 1 p i=1 q j=1 E(V 2 i,j 1 V i,j ≤|z| ) |z| . ( 21
)
For Z 0 = z, we have

Z 1 = ξ(W 1,1 , . . . , W 1,q ) with W 1,j = p i=1 z i k=1 V (k,1) i,j
for 1 ≤ j ≤ q. Fix δ 1 ∈ (0, 1) and r 1 > 0 (depending on δ 1 ) such that, for all z ∈ N p with |z| ≥ r 1 ,

(1 -δ 1 )V i,j ≤ V ≤|z| i,j := E V i,j 1 V i,j ≤|z| ≤ (1 + δ 1 )V i,j .
For all |z| ≥ r 1 with z ∈ U δ , we have

P((W 1,1 , . . . , W 1,q ) ≥ (1 -δ 1 ) 2 zV | Z 0 = z) ≥ P((W 1,1 , . . . , W 1,q ) ≥ (1 -δ 1 )zV ≤|z| | Z 0 = z) ≥ 1 - q j=1 P(W 1,j < (1 -δ 1 ) p i=1 z i V ≤|z| i,j | Z 0 = z)
but for j ∈ {1, . . . , q},

P(W 1,j < (1 -δ 1 ) p i=1 z i V ≤|z| i,j | Z 0 = z) ≤ P(∃i ∈ {1, . . . , p}, k ∈ {1, . . . , z i } s.t V (k,1) i,j > |z| | Z 0 = z) + P p i=1 z i k=1 V (k,1) i,j 1 V (k,1) i,j ≤|z| -V ≤|z| i,j < -δ 1 p i=1 z i V ≤|z| i,j | Z 0 = z ≤ p i=1 z i P(V i,j > |z|) + Var p i=1 z i k=1 V (k,1) i,j 1 V (k,1) i,j ≤|z| | Z 0 = z δ 2 1 p i=1 z i V ≤|z| i,j 2 ≤ |z| p i=1 P(V i,j > |z|) + p i=1 z i k=1 Var V (k,1) i,j 1 V (k,1) i,j ≤|z| | Z 0 = z δ 2 1 δ 2 (1 -δ 1 ) 2 p i=1 V i,j 2 |z| 2 ≤ |z| p i=1 P(V i,j > |z|) + p i=1 E V 2 i,j 1 V i,j ≤|z| δ 2 1 δ 2 (1 -δ 1 ) 2 p i=1 V i,j 2 |z| ,
where we used the independence of the random variables V (k,1) i,j , the fact that z i ≥ δ|z| for all i ∈ {1, . . . , p}, and V ≤|z| i,j ≥ (1 -δ 1 )V i,j . Proceeding similarly for the event {(W 1,1 , . . . , W 1,q ) ≤ (1 + δ 1 ) 2 zV}, we deduce that there exists a constant c > 0 such that

P(W 1 ∈ |(1 -δ 1 ) 2 zV, (1 + δ 1 ) 2 zV| | Z 0 = z) ≥ 1 -c|z| p i=1 q j=1 P(V i,j > |z|) -c p i=1 q j=1 E(V 2 i,j 1 V i,j ≤|z| ) |z| , (22)
where W 1 = (W 1,1 , . . . , W 1,q ). If we now apply Proposition 11 with the compact set U δ ∩ S, for all ε > 0, there exists

r 2 > 0 such that if > r 2 (1 -δ 1 ) 2 , ξ( uV) -M(u) ≤ ε , ∀u ∈ U δ ∩ S,
and we deduce that for all z ∈ U δ such that |z| ≥ r 2 ,

ξ((1 -δ 1 ) 2 zV) (1 -δ 1 ) 2 |z| - M(z) |z| ≤ ε .
Hence, for all z ∈ U δ with |z| ≥ r 2 ,

ξ((1 -δ 1 ) 2 zV) (1 -δ 1 ) 2 |z| ≥ M(z/|z|) -ε .
In addition, since M is concave on S, it is locally Lipschitz on S * and in particular each of its components are uniformly bounded away from 0 on U δ ∩ S ⊂ S * by a constant m 1 > 0 (which depends on δ ∈ (0, 1 /p]). Hence, for all z ∈ U δ with |z| ≥ r 2 ,

ξ((1 -δ 1 ) 2 zV) (1 -δ 1 ) 2 |z| ≥ M(z/|z|)(1 -ε /m 1 ).
Similarly, there exist r 3 > 0 and m 2 > 0 such that, for all z ∈ U δ with |z| ≥ r 3 ,

ξ((1 + δ 1 ) 2 zV) (1 + δ 1 ) 2 |z| ≤ M(z/|z|)(1 + ε /m 2 ). Hence, for z ∈ U δ with |z| ≥ r 1 ∨ r 2 ∨ r 3 , {(W 1,1 , . . . , W 1,q ) ∈ |(1 -δ 1 ) 2 zV, (1 + δ 1 ) 2 zV|} ⊂ {Z 1 ∈ |M(z)(1 -ε /m 1 )(1 -δ 1 ) 2 , M(z)(1 + ε /m 2 )(1 + δ 1 ) 2 |}. ( 23
)
Choosing ε and δ 1 small enough, we deduce that there exists r 4 > 0 such that, for all z ∈ U δ with |z| ≥ r 4 , (21) holds true for some constant c 1 . Up to a change in the constant c 1 , we deduce that this is true for all z ∈ U δ .

Step 2. We iterate now the result obtained in the previous step. We have, for all z ∈ U δ .

P (Z 1 ∈ |(1 -ε)M(z), (1 + ε)M(z)| | Z 0 = z) ≥ 1 -c 1 f (|z|)
for some constant c 1 > 0. Then, observing that

Z 1 ∈ |(1 -ε)M(z), (1 + ε)M(z)| implies that Z 1 ∈ ||z|(1 -ε)M(z/|z|), (1 + ε)|z|M(z/|z|)| and hence that Z 1 /|Z 1 | ≥ (1 -ε)M(z/|z|) (1 + ε)|M(z/|z|)| ≥ δ 1 1 p , with δ 1 := (1 -ε) min u∈U δ ∩S, i∈{1,...,p} M(u), e i (1 + ε) sup u∈S |M(u)| .
where min u∈U δ ∩S, i∈{1,...,p} M(u), e i > 0 by Proposition 15. Now, applying the same reasoning as in Step 1 but with δ 1 instead of δ, we deduce that there exists a constant c 2 > 0 such that, on the event {Z 0 = z},

P (Z 2 ∈ |(1 -ε)M(Z 1 ), (1 + ε)M(Z 1 )| | Z 1 ) ≥ 1 Z 1 ∈|(1-ε)M(z),(1+ε)M(z)| (1 -c 2 f (|Z 1 |)) = 1 A ε 1 (1 -c 2 f (|Z 1 |))
. And hence, using Markov's property at time 1,

P (A ε 2 | Z 0 = z) ≥ E P(Z 2 ∈ |(1 -ε)M(Z 1 ), (1 + ε)M(Z 1 )| | Z 1 )1 A ε 1 | Z 0 = z ≥ P(A ε 1 | Z 0 = z) -E 1 A ε 1 c 2 f (|Z 1 |) | Z 0 = z ≥ 1 -c 1 f (|z|) -c 2 E 1 A ε 1 f (|Z 1 |) | Z 0 = z
Iterating this procedure, we deduce that there exists a positive sequence (c n ) n≥1 such that, for all n ≥ 1,

P (A ε n | Z 0 = z) ≥ 1 - n i=1 c i E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z .
According to Theorem 17 (5), there exists n 0 such that

sup u∈S (λ * ) -n 0 M n 0 (u) -P(u)z * ≤ 1 2 inf S P min i∈{1,...,p} z * i , so that M n 0 (u) ∈ |(λ * ) n 0 P(u)z * /2, 3(λ * ) n 0 P(u)z * /2|, ∀u ∈ S.
Then, under A ε n 0 , we have

Z n 0 ∈ |(1 -ε) n 0 M n 0 (z), (1 + ε) n 0 M n 0 (z)| ⊂ |(1 -ε) n 0 (λ * ) n 0 P(z)z * /2, 3(1 + ε) n 0 (λ * ) n 0 P(z)z * /2| , so that (recall that |z * | = 1) Z n 0 /|Z n 0 | ≥ (1 -ε) n 0 (λ * ) n 0 P(z)z * /2 3(1 + ε) n 0 (λ * ) n 0 P(z)/2 ≥ min z * (1 -ε) n 0 3(1 + ε) n 0 1 p .
Since δ := min z * (1-ε) n 0 3(1+ε) n 0 does not depend on δ, we deduce from Step 1 that there exists c 1 which does not depend on δ such that

P(A ε n 0 +1 | Z 0 , . . . , Z n 0 ) ≥ 1 A ε n 0 1 -c 1 f (|Z n 0 |) .
Iterating the procedure of the beginning of the proof, we deduce that there exist c 2 , . . . , c n 0 which do not depend on δ > 0 such that, for all n ∈ {n 0 + 1, . . . , 2n 0 },

P (A ε n | Z 0 = z) ≥ 1 - n 0 i=1 c i E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z - n i=n 0 +1 c i-n 0 E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z Under A ε 2n 0 , we have Z 2n 0 ∈ |(1 -ε) n 0 M n 0 (Z n 0 ), (1 + ε) n 0 M n 0 (Z n 0 )| ⊂ |(1 -ε) n 0 (λ * ) n 0 P(Z n 0 )z * /2, 3(1 + ε) n 0 (λ * ) n 0 P(Z n 0 )z * /2| ,
hence, using the same computations as above, we have

P(A ε 2n 0 +1 | Z 0 , . . . , Z 2n 0 ) ≥ 1 A ε 2n 0 1 -c 1 f (|Z 2n 0 |) ,
with the same constant c 1 . Iterating the procedure of the beginning of the proof, we deduce that, for all n ∈ {2n 0 + 1, . . . , 3n 0 },

P (A ε n | Z 0 = z) ≥ 1 - n 0 i=1 c i E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z - 2n 0 i=n 0 +1 c i-n 0 E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z - n i=2n 0 +1 c i-2n 0 E 1 A ε i-1 f (|Z i-1 |) | Z 0 = z
Proceeding by induction and taking c 0 = max i∈{1,...,n 0 } c i ∨ c i , this concludes the proof of the lemma.

We prove now a useful auxiliary lemma.

Lemma 19. Let (x n ) n≥0 be a positive sequence such that, for some α > 1 and c > 0,

x n ≥ cα n-k x k , ∀n ≥ k ≥ 0. ( 24
)
Then n≥0 f (x n ) ≤ 2α c(α -1) p i=1 q j=1 E V 2 i,j (cx 0 ) ∨ V i,j
with f defined in Lemma 18.

Proof. We first consider the first part and then the second part of f (x n ), for each i ∈ {1, . . . , p} and j ∈ {1, . . . , q}.

We have, using Fubini's Theorem, n≥0

x n P(V i,j > x n ) = E   n≥0 x n 1 V i,j >xn   ≤ E   1 V i,j >min n∈N xn N ij n=0 x n   ,
where N ij := max{n ≥ 0, x n < V i,j }, with the convention that max ∅ = -1 (note that, since α > 1, x n → +∞ so that N ij < +∞). Inequality (24) entails that min n∈N x n ≥ cx 0 and that, for all

n ≤ N ij , x n ≤ α n-N ij x N ij /c ≤ α n-N ij V i,j /c. We deduce that n≥0 x n P(V i,j > x n ) ≤ E   1 V i,j >cx 0 V i,j N ij n=0 α n-N ij /c   ≤ E 1 V i,j >cx 0 V i,j α c(α -1) ≤ α c(α -1) E V 2 i,j (cx 0 ) ∨ V i,j ,
where we used the fact that

1 V i,j >cx 0 ≤ V i,j
(cx 0 )∨V ij almost surely. Using again Fubini's Theorem, we have

n≥0 E V 2 i,j 1 V i,j ≤xn x n = E   V 2 i,j n≥0 1 V i,j ≤xn x n   ≤ E   V 2 i,j +∞ n=N ij 1 x n    ,
where

N ij = min{n ≥ 0, x n ≥ V i,j } = N ij + 1. Inequality (24) entails that, for all n ≥ N ij , 1/x n ≤ α N ij -n /(cx N ij ). Hence, using the fact that x N ij ≥ V i,j by definition of N ij , we obtain n≥0 E V 2 i,j 1 V i,j ≤xn x n ≤ E   V 2 i,j 1 x N ij ∨ V i,j +∞ n=N ij α N ij -n c    ≤ E V 2 i,j (cx 0 ) ∨ V i,j α c(α -1)
,

where we used the fact that x N ij ≥ cx 0 by (24). Summing over i ∈ {1, . . . , p} and j ∈ {1, . . . , q}, this concludes the proof of Lemma 19. Now we state a second auxiliary lemma, where we prove that for all z ∈ N p , the sequence (M n (z)) n∈N holds the property (24).

Lemma 20. Assume that M is bounded on S and λ * > 1. There exists a constant c 0 ∈ (0, 1] and λ ∈ (1, λ * ) such that, for all z ∈ N p and all n ≥ 1 and k ∈ {0, . . . , n},

|M n (z)| ≥ c 0 λ n-k |M k (z)|. Proof. Let δ ∈ (0, 1) such that λ := 1-δ 1+δ λ * > 1. Fix z ∈ N p \ {0}. If there exists x > 0 such that z ∈ |(1 -δ)xz * , (1 + δ)xz * |, then |z| ≤ |(1 + δ)xz * | = (1 + δ)x,
and, since M is increasing and positively homogeneous,

|M(z)| ≥ |M((1 -δ)xz * )| = |(1 -δ)xM(z * )| = |(1 -δ)xλ * z * | = (1 -δ)xλ * ≥ λ|z|. ( 25
)
Moreover, Theorem 17 (5) entails that there exists n 0 ≥ 1 such that, for any z ∈ S and n ≥ n 0 ,

M n (z) ∈ |(1 -δ)(λ * ) n C(z)z * , (1 + δ)(λ * ) n C(z)z * |, so that, according to (25), for n ≥ k ≥ n 0 |M n+1 (z)| ≥ λ|M n (z)| ≥ λ n+1-k |M k (z)| ≥ λ n+1-n 0 |M n 0 (z)|,
By homogeneity of M, this extends to all z ∈ N p . For all n ∈ {0, . . . , n 0 } and all z ∈ (N \ {0}) p , we have, for all k ∈ {0, . . . , n},

|M n+1 (z)| ≥ M M n (z) |M n (z)| |M n (z)| ≥ inf S |M| |M n (z)| ≥ (inf S |M|) n+1-k |M k (z)| ≥ c n+1-k 1 λ n+1-k |M k (z)|,
where c 1 = λ -1 inf S |M|. Setting c 0 = 1 ∧ c n 0 1 concludes the proof of Lemma 20.

We are now in position to compute the limit (19).

Lemma 21. Assume that M is bounded on S and λ * > 1. For any δ ∈ (0, 1 /p] and ε ∈ (0, 1), we have

P   n≥1 A ε n | Z 0 = z   ---------→ |z|→+∞, z∈U δ 1.
Proof. Take c 0 > 0 and λ ∈ (1, λ * ) from Lemma 20. We assume without loss of generality that α := (1 -ε)λ > 1. For all i ≥ k ≥ 1, on the event A i , we have then

|Z i | ≥ |(1 -ε) k M k (Z i-k )| ≥ c 0 λ k (1 -ε) k |Z i-k | = c 0 α k |Z i-k |.
Hence, according to Lemma 19, almost surely, on the event {Z 0 = z},

n≥1 1 A n-1 f (|Z n-1 |) ≤ 2α c 0 (α -1) p i=1 q j=1 E V 2 i,j (c 0 |z|) ∨ V i,j
.

We deduce that

n≥1 E 1 A n-1 f (|Z n-1 |) | Z 0 = z ≤ 2α c 0 (α -1) p i=1 q j=1 E V 2 i,j (c 0 |z|) ∨ V i,j
.

Letting n → +∞ in Lemma 18, we obtain that there exists c 0 > 0 such that, for all z ≥ δ|z|1 p (recall that A n-1 ⊂ A n for all n ≥ 1),

P   n≥1 A n | Z 0 = z   = lim n→+∞ P n k=1 A n | Z 0 = z ≥ 1 -c 0 n≥1 E 1 A n-1 f (|Z n-1 |) | Z 0 = z ≥ 1 - 2c 0 α c 0 (α -1) i,j E V 2 i,j (c 0 |z|) ∨ V i,j .
But V i,j is integrable for all i, j and hence, by dominated convergence theorem,

E V 2 i,j (c 0 |z|) ∨ V i,j -----→ |z|→+∞ 0.
This concludes the proof of Lemma 21.

Lemma 21 is useful for large starting values z such that z ≥ δ|z|1 p . In order to use it for all initial positions, we show that such values are eventually reached by the process in finite time.

Lemma 22. Assume that M is bounded on S. There exist δ 0 ∈ (0, 1 /p] and n 0 ≥ 1 such that, for all r > 0, for all η ∈ (0, 1), there exists ρ > 0 such that

inf z∈N p , |z|≥ρ P (τ δ 0 ,r ≤ n 0 | Z 0 = z) ≥ 1 -η, where τ δ 0 ,r = inf{n ≥ 0, Z n ≥ |Z n |δ 0 1 p and |Z n | ≥ r}.
In addition,

inf z∈N p P (τ δ 0 ,r ∧ T 0 < +∞ | Z 0 = z) = 1, with T 0 := inf{n ≥ 0, |Z n | = 0} the extinction time of the process.
Proof. Let n 0 ≥ 1 and δ 1 > 0 such that, for all i ∈ {1, . . . , p}, M n 0 (e i ) > δ 1 1 p . Such n 0 and δ 1 exist as M is primitive by Proposition 14. In order to prove the first inequality of the lemma, we show that there exists α > 0 such that, for all η ∈ (0, 1), there exists k 1 > 0 such that

P (Z n 0 ≥ |z|δ 2 1 p | Z 0 = z) ≥ 1 -η/2 , ∀|z| ≥ k 1 (26)
with δ 2 := δ 1/2p, and there exists k 2 > 0 such that

P (α |z| > |Z n 0 | | Z 0 = z) ≥ 1 -η/2 ∀|z| ≥ k 2 . ( 27
)
Once this is proved, we set δ 0 = δ 2 /α (which does not depend on η). Then for r > 0, setting ρ = k 1 ∨ k 2 ∨ r /δ 2 p, we obtain from the general property P(A ∩ B) ≥ P(A) + P(B) -1 that, for all z such that |z| ≥ ρ,

P (τ δ 0 ,r ≤ n 0 | Z 0 = z) ≥ P (Z n 0 ≥ |Z n 0 |δ 0 1 p and |Z n 0 | ≥ r and |Z n 0 | < α |z| | Z 0 = z) ≥ P (Z n 0 ≥ α|z| δ 0 1 p and |Z n 0 | < α |z| | Z 0 = z) ≥ P (Z n 0 ≥ |z|δ 2 1 p | Z 0 = z) + P (|Z n 0 | < α |z| | Z 0 = z) -1 ≥ 1 -η,
where we used for the second inequality that |α|z|δ 0 1 p | ≥ r for all |z| ≥ ρ.

Let us prove (26) and ( 27). By definition of n 0 ≥ 1 and δ 1 > 0 and by Theorem 1, we deduce that there exists k 0 ≥ 1 such that, for all k ≥ k 0 and all i ∈ {1, . . . , p},

P(Z n 0 ≥ kδ 1 1 p /2 | Z 0 = ke i ) ≥ (1 -η/2) 1 /p .
Using the auxiliary lemma in Appendix A, we obtain

P Z n 0 ≥ p i=1 1 z i ≥k 0 z i δ 1 1 p /2 | Z 0 = z ≥ p i=1 P (Z n 0 ≥ 1 z i ≥k 0 z i δ 1 1 p /2 | Z 0 = z i e i ) ≥ p i=1 (1 -η/2) 1 /p = 1 -η/2. If in addition |z| ≥ k 1 := pk 0 , then δ 1 2 p i=1 1 z i ≥k 0 z i ≥ δ 1 2 (|z| -(p -1)k 0 ) = δ 1 2 |z| 1 - (p -1)k 0 |z| ≥ δ 1 2 p |z| = δ 2 |z|
and (26) holds. Set α = 2|M n 0 (1 p )| (note that by definition of n 0 , we have M n 0 (1 p ) > 0). By superadditivity, we have

P(|Z n 0 | ≥ α|z| | Z 0 = z) ≤ P(|Z n 0 | ≥ α|z| | Z 0 = |z|1 p ). Now Theorem 1 entails that P(|Z n 0 | ≥ αm | Z 0 = m1 p ) = P |Z n 0 | m ≥ 2|M n 0 (1 p )| Z 0 = m1 p -----→ m→+∞ 0.
Using this, we conclude that for all η ∈ (0, 1), there exists k 2 > 0 such that, for all |z| ≥ k 2 ,

P(|Z n 0 | ≥ α|z| | Z 0 = z) ≤ η/2.
and (27) holds, which concludes the proof of the first part of the lemma.

In addition, setting τ ρ = inf{n ≥ 0, |Z n | ≥ ρ}, we have for all starting point in z ∈ N p \ {0} such that |z| ≤ ρ, P(τ ρ ∧ T 0 < +∞ | Z 0 = z) = 1 (by Assumption 1 and because Z n ∈ N p almost surely), we deduce that, for all η ∈ (0, 1), there exists n 1 ≥ 1 such that

P(τ ρ ∧ T 0 ≤ n 1 | Z 0 = z) ≥ 1 -η, ∀|z| ≤ ρ.
Using the strong Markov property at time τ ρ ∧ T 0 , we deduce that

P (τ δ 0 ,r ∧ T 0 ≤ n 0 + n 1 | Z 0 = z) ≥ (1 -η) 2 , ∀z ∈ N p \ {0}.
Since this is true for all η ∈ (0, 1) this concludes the proof of Lemma 22.

We are now in position to conclude the proof of Theorem 4. Consider δ 0 ∈ (0, 1 /p] and n 0 ∈ N * given by Lemma 22. According to Lemma 21, for all ε ∈ (0, 1), for all η ∈ (0, 1), there exists r > 0 such that, for all z ∈ U δ 0 with |z| ≥ r,

P (∀n ≥ 0, Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) ≥ 1 -η/2. ( 28
)
In addition, Lemma 22 entails that there exists ρ > 0 such that, for all |z| ≥ ρ,

P (τ δ 0 ,r ≤ n 0 | Z 0 = z) ≥ 1 -η/2,
and hence, using the strong Markov property, we deduce that for all |z| ≥ ρ,

P (τ δ 0 ,r ≤ n 0 and ∀n ≥ τ δ 0 ,r , Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) ≥ 1 -η.
Finally, this implies that

P (Z n 0 = 0 and ∀n ≥ n 0 , Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) ≥ 1 -η.
This concludes the proof of the first part of Theorem 4.

We now prove Corollary 5, the result holds true trivially if z = 0. Hence, using the Markov property at time τ δ 0 ,r ∧ T 0 , we deduce from (28) and the fact that P (τ δ 0 ,r ∧ T 0 < +∞) = 1, that for all z ∈ N p ,

P (τ δ 0 ,r ∧ T 0 < +∞ and ∀n ≥ τ δ 0 ,r ∧ T 0 , Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) ≥ 1 -η/2.
Hence,

P (∃N ≥ 0 such that, ∀n ≥ N, Z n+1 ∈ |(1 -ε)M(Z n ), (1 + ε)M(Z n )| | Z 0 = z) ≥ 1 -η/2.
Since the left hand term does not depend on η, this conclude the proof of Corollary 5.

For the second part of Theorem 4, we simply observe that, P(• | Z 0 = z)-almost surely, for all ε > 0, there exists N ≥ 0 (random) such that for all m, k ≥ 0, for all n ≥ N ,

Z n+m ∈ |(1 -ε) m M m (Z n ), (1 + ε) m M m (Z n )|, hence Z n+m+k |Z n+m | ∈ (1 -ε) m+k M m+k (Z n ) (1 + ε) m |M m (Z n )| , (1 + ε) m+k M m+k (Z n ) (1 -ε) m |M m (Z n )| ⊂ (1 -ε) m+k M m+k (Z n /|Z n |) (1 + ε) m |M m (Z n /|Z n |)| , (1 + ε) m+k M m+k (Z n /|Z n |) (1 -ε) m |M m (Z n /|Z n |)| ⊂ (1 -ε) m+k (1 + ε) m inf u∈S M m+k (u) |M m (u)| , (1 + ε) m+k (1 -ε) m sup u∈S M m+k (u) |M m (u)| ,
where the infimum and supremum should be understood component-wise. But, according to Theorem 17 (5), M m+k (u) |M m (u)| converges uniformly in u ∈ S toward (λ * ) k z * when m → +∞, and hence, for all ε > 0, choosing m large enough, we have, for all n ≥ N ,

Z n+m+k |Z n+m | ∈ (1 -ε) m+k (1 + ε) m (1 -ε )(λ * ) k z * , (1 + ε) m+k (1 -ε) m (1 + ε )(λ * ) k z * .
We deduce that (again the lim inf and lim sup should be understood component-wise)

(1 -ε) m+k (1 + ε) m (1 -ε )(λ * ) k z * ≤ lim inf n→+∞ Z n+m+k |Z n+m | ≤ lim sup n→+∞ Z n+m+k |Z n+m | ≤ (1 + ε) m+k (1 -ε) m (1 + ε )(λ * ) k z * .
Taking, first the limit when ε → 0 (m depends on ε but not on ε), then the limit when ε → 0, concludes the proof of Theorem 4.

6. Identification of an intrinsic supermartingale and proof of Theorem 6

In this section we prove Theorem 6. In order to do so, we start by introducing the process

C n = P(Z n ) (λ * ) n , ( 29
)
where P is the function defined by (6).

We claim that a convergence (almost surely or in L 1 ) of the process (C n ) n∈N to a non-negative random variable C implies the convergence of the process Zn (λ * ) n n∈N . On the event of extinction, the result is trivial. On the event of survival, as P is positively homogeneous by Theorem 17, we have

Z n (λ * ) n = C n Z n |Z n | 1 P Zn |Zn| .
Applying Theorem 4 and if

C n → C we deduce that Z n (λ * ) n -→ Cz * 1 P(z * ) = Cz * , as n → +∞,
and so the proof is complete. We also remark that the convergence of the sequence Zn (λ * ) n n∈N is of the same type as the convergence of (C n ) n∈N . We divide this section into five parts. The first and second parts are dedicated to the proof of both results in Theorem 6. In the third part, we prove the convergence in L 1 under stronger assumptions and prove that, under these assumptions, the limit is non-degenerate at 0. In the fourth part, we prove Proposition 8 and Proposition 9.

6.1. Almost sure convergence. In this section, we prove the first statement of Theorem 6. The proof comes from the following result.

Lemma 23. The sequence (C n ) n∈N is a supermartingale with respect to (F n ) n∈N , the natural filtration of (Z n ) n∈N .

Proof. First, from the definition of P and Theorem 17, it follows that P • M = λ * P. Then we recall that, by Lemma 13, we have for all n ∈ N,

E(Z n+1 |F n ) ≤ M(Z n ).
Hence, Jensen's inequality together with the fact that P is increasing (since M is) and concave imply that

E(C n+1 |F n ) = E(P(Z n+1 )|F n ) (λ * ) n+1 ≤ P(E(Z n+1 |F n )) (λ * ) n+1 ≤ P(M(Z n )) (λ * ) n+1 = P(Z n ) (λ * ) n = C n .
Since (C n ) n∈N is a non-negative supermartingale, then it exists a non-negative random variable C such that C n a.s.

-----→ n→+∞ C.

We now prove the convergence of the process W n (λ * ) n n∈N in Theorem 6. We follow the proof for the single-type case presented in [START_REF] González | On the limit behaviour of a superadditive bisexual galton-watson branching process[END_REF] Let

N n = Z n (λ * ) n , S n = W n (λ * ) n and Fn = σ(Z 0 , V (k, ) i,j , 1 ≤ i ≤ p, 1 ≤ j ≤ q, k ∈ N * , 1 ≤ ≤ n).
We consider Z 0 = z and define for n ≥ 1, Ŝn = ( Ŝn,1 , . . . , Ŝn,q ) with for j ∈ {1, . . . , q},

Ŝn,j = 1 (λ * ) n p i=1 Z n-1,i k=1 V (k,n) i,j 1 V (k,n) i,j ≤(λ * ) n-1
Note that for j ∈ {1, . . . , q},

E( Ŝn+1,j | Fn ) = 1 λ * p i=1 N n,i E(V i,j 1 V i,j ≤(λ * ) n ). ( 30
)
Since λ * > 1, we have that V i,j 1 V i,j ≤(λ * ) n → V i,j almost surely as n → +∞. Hence, by the Monotone Convergence Theorem,

E(V i,j 1 V i,j ≤(λ * ) n ) → E(V i,j
). Then, taking n → +∞, we have that

E( Ŝn+1 | Fn ) -→ 1 λ * Cz * V a.s. ( 31
)
We consider now the martingale given by m n=1

Ŝn -E( Ŝn | Fn-1 ) m∈N .

We have that for all j ∈ {1, . . . , q} and n ∈ N

Var Ŝn+1,j -E( Ŝn+1,j | Fn ) = Var   1 (λ * ) n+1 p i=1 Zn,i k=1 V (k,n) i,j 1 V (k,n) i,j ≤(λ * ) n -E V (k,n) i,j 1 V (k,n) i,j ≤(λ * ) n   = 1 (λ * ) -2(n+1) p i=1 E(Z n,i )Var V i,j 1 V i,j ≤(λ * ) n ≤ (λ * ) -n-2 p i=1 M n i (z) (λ * ) n Var V i,j 1 V i,j ≤(λ * ) n ≤ (λ * ) -n-2 p i=1 M n i (z) (λ * ) n +∞ 0 x 2 1 x≤(λ * ) n dF i,j (x),
where F i,j (x) = P(V i,j ≤ x).

Since

M n (z) (λ * ) n n∈N is convergent, then it is bounded, and so there exists

C > 0 such that n∈N Var Ŝn+1,j -E( Ŝn+1,j | Fn ) ≤ C p i=1 +∞ 0 x 2 n∈N 1 (λ * ) n 1 x≤(λ * ) n dF i,j (x) = C p i=1 +∞ 0 x 2 O(x -1 ) dF i,j (x) < +∞,
and so applying the Martingale Convergence Theorem, we have that n∈N Ŝn+1,j -E( Ŝn+1,j | Fn ) is convergent almost surely and in L 1 . This implies that Ŝn+1,j -E( Ŝn+1,j | Fn ) → 0 almost surely. Thanks to (31), we have that Ŝn+1 → (λ * ) -1 Cz * V. We finish by proving that (S n ) n∈N is an equivalent sequence. Fix j ∈ {1, . . . , q} n∈N P(S n+1,j = Ŝn+1,j )

= n∈N E(P(∃i ≤ p, ∃k ≤ Z n,i , V (k,n) i,j > (λ * ) n | Fn )) ≤ n∈N p i=1 E   Z n,i k=1 P(V i,j > (λ * ) n )   ≤ C p i=1 n∈N (λ * ) n P(V i,j > (λ * ) n ) ≤ C p i=1 +∞ 0 n∈N (λ * ) n 1 x>(λ * ) n dF i,j (x) = p i=1 +∞ 0 O(x) dF i,j (x) < +∞.
The conclusion then follows by the Borel-Cantelli lemma. 6.2. Extinction vs {C = 0}. In this section, we prove the second part of Theorem 6.

The inclusion {∃n ∈ N, Z n = 0} ⊂ {C = 0} is obvious. We then consider the case λ * > 1 and show that {∀n, Z n = 0) ⊂ {C > 0}.

For all ε > 0, we set

τ ε = inf{n ≥ 0, P(Z n ) (λ * ) n ≤ ε}.
By assumption C is non-degenerate at 0 for all z such that q z < 1. Hence, for all z such that q z < 1, there exists ε z > 0 such that

P(τ εz = +∞ | Z 0 = z) > 0.
In addition, Theorem 4 entails that there exists r 0 ∈ N such that, for all |z| ≥ r 0 , q z < 1. Hence, setting ε 0 = min i=1,...,p ε r 0 e i and using the fact that, by superadditivity, P(τ ε 0 = +∞ | Z 0 = z) is increasing with z, we deduce that, for all |z| ≥ pr 0 ,

P(τ ε 0 = +∞ | Z 0 = z) ≥ a 0 := min i=1,...,p P(τ ε 0 = +∞ | Z 0 = r 0 e i ) ≥ min i=1,...,p P(τ εr 0 e i = +∞ | Z 0 = r 0 e i ) > 0.
We define τ 0 = 0, τ 1 = τ ε 0 and, for all n ≥ 1,

τ n+1 = inf{n ≥ τ n + 1, P(Z n ) (λ * ) n ≤ ε 0 }.
Then we obtain, for all |z| ≥ pr 0 and all n ≥ 1,

P({τ n+1 < +∞} ∩ {|Z m | ≥ pr 0 , ∀m ∈ [0, τ n ]} | Z 0 = z) ≤ E 1 τ n <+∞, |Zm|≥pr 0 , ∀m∈[0,τ n ] P(τ ε 0 < +∞ | Z 0 = z ) |z =Z τ n | Z 0 = z ≤ P({τ n < +∞} ∩ {|Z m | ≥ pr 0 , ∀m ∈ [0, τ n-1 ]} | Z 0 = z) (1 -a 0 ) ≤ . . . ≤ (1 -a 0 ) n+1 . ( 32
)
In addition, according to Theorem 1 and Theorem 4, for all η ∈ (0, 1), there exists r η > 0 such that, for all |z| ≥ r η ,

P(|Z n | ≥ pr 0 , ∀n ≥ 0 | Z 0 = z) ≥ 1 -η.
Using this last inequality and (32), we deduce that, for all |z| ≥ r η and all n ≥ 1,

P(τ n+1 < +∞ | Z 0 = z) ≤ P({τ n+1 < +∞} ∩ {|Z n | ≥ pr 0 , ∀n ∈ [0, τ n ]} | Z 0 = z) + P(∃n ≥ 0 such that |Z n | < pr 0 | Z 0 = z) ≤ (1 -a 0 ) n+1 + η.
Since {C = 0} ⊂ {τ n+1 < +∞, ∀n ≥ 1}, we deduce that, for all |z| ≥ r η ,

P(C = 0 | Z 0 = z) ≤ η. ( 33
)
Denoting T η := inf{n ≥ 0, |Z n | ≥ r η }, we deduce from the transitivity assumption that {T η < +∞} ⊃ {∀n, Z n = 0}. Hence, for all z ∈ N p , using the strong Markov property at time T η and then (33),

P(C = 0 and ∀n, Z n = 0 | Z 0 = z) ≤ P(C = 0 and T η < +∞ | Z 0 = z) = E   1 Tη<+∞ P lim n→+∞ P(Z n ) (λ * ) n+u = 0 | Z 0 = z |u=Tη, z =Z Tη | Z 0 = z   = E 1 Tη<+∞ P C = 0 | Z 0 = z |z =Z Tη | Z 0 = z ≤ η.
Since the last inequality holds true for all η ∈ (0, 1), we deduce that, for all z ∈ N p , P(C = 0 and ∀n, Z n = 0 | Z 0 = z) = 0.

6.3. L 1 convergence and non-degeneracy of the limit. In this section, we prove the convergence of (C n ) n∈N in L 1 to a non-degenerate limit C, which corresponds to Proposition 7. We recall that in this part we assume that there exists a concave monotone increasing function U : R + -→ R + , such that for all y ∈ R + , sup

z∈R p + :P(z)=y E(|P(Z 1 ) -P(M( z ))| | Z 0 = z ) ≤ U (y),
with y → U (y) /y non-increasing and +∞ 1 U (y) y 2 dy < +∞.

The idea behind the proof is to use the following lemma.

Lemma 24.

[See [Kle84] -Lemma 2] Let f : R + -→ R + be a non-increasing function, such that x → xf (x) is non-decreasing and

+∞ n=1 f (n) n < +∞.
Let (a n ) n∈N be a sequence of positive numbers satisfying for some m > 1 and all n ≥ 0

|a n+1 -a n | ≤ a n f (a n m n ). Then • lim n→+∞ a n = a exists,
• there exists a constant b 0 depending only on the function f and m such that if a 0 > b 0 then a > 0.

We start our proof by noticing that, for all z ∈ N p ,

|E(C n+1 | Z 0 = z) -E(C n | Z 0 = z)| ≤ E(|C n+1 -C n | | Z 0 = z) = E(E(|C n+1 -C n | | F n ) | Z 0 = z) = 1 (λ * ) n+1 E(E(|P(Z n+1 ) -λ * P(Z n )| | F n ) | Z 0 = z) ≤ 1 (λ * ) n+1 E(U (P(Z n )) | Z 0 = z),
and so applying Jensen's inequality, since U is concave, we get

E(|C n+1 -C n | | Z 0 = z) ≤ (λ * ) -1 E(C n | Z 0 = z) U (E(P(Z n ) | Z 0 = z)) E(P(Z n ) | Z 0 = z) (34)
From this inequality, we have that if we define

F (x) = U (x) λ * x ,
then, we obtain,

|E(C n+1 | Z 0 = z) -E(C n | Z 0 = z)| ≤ E(C n | Z 0 = z)F (E(C n | Z 0 = z)(λ * ) n ),
and we can apply Lemma 24 with f = F and m = λ * . This implies that, for all z ∈ N p ,

c(z) := lim n→+∞ E(C n |Z 0 = z)
exists. It also implies that there exists b 0 such that if P(z) ≥ b 0 , then c(z) > 0. Since P is homogeneous and lower bounded away from 0 on S, we deduce that there exists r 0 > 0 such that, for all z ∈ N p with |z| ≥ r 0 , c(z) > 0. Since in addition c(z) is increasing with z, we deduce that

c := inf z∈N p ,|z|≥r 0 c(z) > 0.
If we now define

T = inf{n ∈ N : |Z n | ≥ r 0 },
we have that, if z ∈ N p is such that q z < 1, then P(T < +∞ | Z 0 = z) ≥ q z > 0 by transitivity assumption and so, applying the strong Markov property we obtain

c(z) = lim n→∞ E(C n |Z 0 = z) ≥ lim n→+∞ E E(C n | Z 0 = y) |y=Z T (λ * ) -T 1 T <+∞ | Z 0 = z ≥ c E (λ * ) -T 1 T <+∞ | Z 0 = z > 0.
Now fix z ∈ N p such that q z < 1 and take ε such that c(z) -ε > 0. We can find N 0 such that for all

n ≥ N 0 , c(z) -ε ≤ E(C n | Z 0 = z) ≤ c(z) + ε.
Hence, using this in (34), we get that for all n ≥ N 0 , since x → Û (x) /x is non-increasing,

E(|C n+1 -C n | | Z 0 = z) ≤ (λ * ) -1 (c(z) + ε) U ((c(z) + ε)(λ * ) n ) (c(z) -ε)(λ *
) n , and so we can find C > 0 and δ > 0, such that for all n ∈ N,

E(|C n+1 -C n | | Z 0 = z) ≤ C U (δ(λ * ) n ) (λ * ) n .
On the other hand, since the integral +∞ 1 U (y) y 2 dy is finite, we have that the series n∈N U (δ(λ * ) n ) (λ * ) n converges (see [START_REF] Fima | A limit theorem for population-size-dependent branching processes[END_REF]). This implies that (C n ) n∈N is a Cauchy sequence in L 1 , which gives the L 1 convergence.

Finally, we have that if z ∈ N p is such that q z < 1, then

E(C|Z 0 = z) = lim n→+∞ E(C n |Z 0 = z) > 0,
which proves that the limit is non-degenerate at 0. 6.4. Sufficient Conditions for the existence of U . In this section, we prove Proposition 8 and Proposition 9. We start the proof by stating and proving a lemma that is useful for the proof of both results.

Lemma 25. Consider p non-negative independent and integrable random variables X 1 , . . . , X p . Set z ∈ N p and β > 0, then

E p i=1 z i k=1 X (k) i -E(X i ) ≤ |z| β+ 1 /2 + 2|z| p i=1 +∞ |z| β x dF i (x),
where (X

(k) 1 , . . . , X (k)
p ) k∈N are i.i.d. copies of (X 1 , . . . , X p ), and F i (x) = P(X i ≤ x).

Proof. We have

E p i=1 z i k=1 X (k) i -E(X i ) ≤ E p i=1 z i k=1 X (k) i 1 X (k) i ≤|z| β -E(X i 1 X i ≤|z| β ) + E p i=1 z i k=1 X (k) i 1 X (k) i >|z| β -E(X i 1 X i >|z| β )
.

We bound the two expectations above separately. For the first one we have, setting

Y (k) i = X (k) i 1 X (k) i ≤|z| β , E p i=1 z i k=1 Y (k) i -E(Y (k) i ) 2 ≤ E   p i=1 z i k=1 Y (k) i -E(Y (k) i ) 2   = p i=1 z i k=1 E Y (k) i -E(Y (k) i ) 2 = p i=1 z i k=1 Var X (k) i 1 X (k) i ≤|z| β ≤ p i=1 z i k=1 |z| 2β = |z| 2β+1 .
For the second term, we have

E p i=1 z i k=1 X (k) i 1 X (k) i >|z| β -E(X (k) i 1 X (k) i >|z| β ) ≤ 2 p i=1 z i k=1 E X (k) i 1 X (k) i >|z| β = 2 p i=1 z i +∞ 0 x1 x>|z| β dF i (x) ≤ 2|z| p i=1 +∞ |z| β x dF i (x),
and the result follows.

The following lemma is a key ingredient of the proof.

Lemma 26. [See

[Kle85] -Page 52] Let f be a positive function on [1, +∞) such that x → f (x)
x is non-increasing and

+∞ 1 f (x) x 2 dx < +∞.
Then there exists a monotone increasing function f such that for all

x ∈ R + , f (x) ≥ f (x), x → f (x) /x is non-increasing, +∞ 1 ( f (x) /x 2 ) dx < +∞ and f is concave on R + .
Proof of Proposition 8. We recall that for this proof, we have the following additional assumption.

Assumption 2. We assume that (1) The mating function ξ and the function P are Lipschitz.

(2) For all i ∈ {1, . . . p} and j ∈ {1, . . . , q}, we have E(V i,j log V i,j ) < +∞.

(3) There exists α > 0 such that, ∀z ∈ R

p + \ {0}, ξ(zV) |z| - M(z) |z| = O(|z| -α ). ( 35
)
Since P is continuous, positive on R p + \ {0} and positively homogeneous, there exist

L 1 , L 2 > 0 such that, for all z ∈ R p + , L 1 |z| ≤ P(z) ≤ L 2 |z|. (36)
Let us consider first z ∈ N p . Since P is Lipschitz, there exists K 1 > 0 such that

E(|P(Z 1 ) -λ * P(z)| | Z 0 = z) = E(|P(Z 1 ) -P(M(z))| | Z 0 = z) ≤ K 1 E ( |Z 1 -M(z)| | Z 0 = z) ≤ K 1 E (| Z 1 -ξ(zV)| | Z 0 = z) + |z| ξ(zV) |z| - M(z) |z| .
Using that ξ is also Lipschitz, there exists K 2 > 0 such that

E (| Z 1 -ξ(zV)| | Z 0 = z) ≤ K 2 q j=1 E p i=1 z i k=1 V (k) i,j -(zV) j = K 2 q j=1 E p i=1 z i k=1 V (k) i,j -E(V (k) i,j )
Making use of Lemma 25 with β ∈ (0, 1 /2) and applying Assumption 2-3, we obtain that there exists K 3 such that for all z ∈ N p ,

E(|P(Z 1 ) -λ * P(z)| | Z 0 = z) ≤ K 3   |z| β+ 1 /2 + |z| p i=1 q j=1 +∞ |z| β x dF i,j (x) + |z| 1-α    ,
for α > 0 and F i,j (x) = P(V i,j ≤ x).

To extend the previous bound to z /

∈ N p , note that if z ∈ R p + with |z| > 2p, we have 0 < 1 2 |z| ≤ | z | ≤ |z|. Hence there exist K 4 , K 5 > 0 such that for all z ∈ R p + with |z| > 2p, E(|P(Z 1 ) -λ * P( z )| | Z 0 = z ) ≤ K 4   |z| p i=1 q j=1 +∞ K 5 |z| β x dF i,j (x) + |z| β+ 1 /2 + |z| 1-α    .
Finally, applying (36), we get that there exists C 1 , C 2 > 0, such that for all z ∈ R p + with |z| > 2p

E(|P(Z 1 ) -λ * P( z )| | Z 0 = z ) ≤ C 1     P(z) p i=1 q j=1 +∞ C β 2 P(z) β x dF i,j (x) + P(z) β+ 1 /2 + P(z) 1-α     . Now define C 3 := max z∈N p :|z|≤2p E(|P(Z 1 ) -P( z )| | Z 0 = z) < +∞.
Then, for all z ∈ R p + ,

E(|P(Z 1 )-λ * P( z )| | Z 0 = z ) ≤ C 1     P(z) p i=1 q j=1 +∞ C β 2 P(z) β x dF i,j (x) + P(z) β+ 1 /2 + P(z) 1-α     +C 3 .
This implies that for all y ∈ R + , sup

z:P(z)=y E(|P(Z 1 ) -λ * P( z )| | Z 0 = z ) ≤ C 1     y p i=1 q j=1 +∞ C β 2 y β x dF i,j (x) + y β+ 1 /2 + y 1-α     + C 3 . Now set F : R + -→ R + given by F (y) = C 1     y p i=1 q j=1 +∞ C β 2 y β x dF i,j (x) + y β+ 1 /2 + y 1-α     + C 3 .
Then we have on one hand that

F (y) y = C 1     p i=1 q j=1 +∞ C β 2 y β
x dF i,j (x) + y β-1 /2 + y -α Once again, since α > 0 and β ∈ (0, 1 /2), we only need to prove that the first integral is finite. In fact, applying Fubini's Theorem, The proof is then complete.

Proof of Proposition 9. For this proof, we consider the case of the Promiscuous mating of Example 6. We set p = n f and ξ((x 1 , . . . x p ), (y 1 , . . . , y nm )) = (x 1 , . . . x p ) nm j=1 1 y j >0 .

Hence, we have M(z) = zX1 zY>0 . Since Y i,j > 0 for all i ∈ {1, . . . p} and all j ∈ {1, . . . n m }, we have M(z) = zX. We also note that with (X (k) ) k∈N a family of i.i.d copies of X = (X i,j ) 1≤i,j≤p . For the second term, we have for i, j ∈ {1, . . . , p} and k ∈ {1, . . . z i } fixed

E X (k) i,j 1 ∃ ≤nm, M 1, =0 Z 0 = z ≤ nm =1 E X (k) i,j 1 M 1, =0 Z 0 = z = nm =1 E X (k) i,j p i =1 z i k =1 1 Y (k ) i , =0 ≤ nm =1 X i,j p i =1 z i k =1 (i ,k ) =(i,k) P Y (k ) i , = 0 ≤ X i,j n m γ |z|-1 ,
with γ = max i ≤p, ≤nm P(Y i , = 0) ∈ [0, 1). Then, applying Lemma 25 with β ∈ (0, 1 /2), we deduce that there exists C 1 > 0 such that, for all z ∈ N p , The result then follows as in the proof of Proposition 8.

Appendix A. Transitivity condition under strong primitivity

Our aim is to show that the transitivity condition of Assumption 1.1 holds true under the third criterion provided below Assumption 1. We use the following auxiliary lemma, which is a consequence of superadditivity.

Lemma 27. For all z 0 , z0 , z 1 , z1 ∈ N p , P(Z n ≥ z 1 + z1 | Z 0 ≥ z 0 + z0 ) ≥ P(Z n ≥ z 1 + z1 | Z 0 = z 0 + z0 ) ≥ P(Z n ≥ z 1 | Z 0 = z 0 ) × P(Z n ≥ z1 | Z 0 = z0 ) .

Proof. First observe that the first inequality is a direct consequent of the fact that ξ is an increasing function, so we deal only with the second inequality. Consider z 0 , z0 , z 1 , z1 ∈ N p . We first treat the case n = 1, and then proceed by induction. We have Observe that the first inequality in particular entails that if (Z n (z)) n∈N for z ∈ N p denotes a bGWbp with Z 0 = z, then for all z 0 , z0 ∈ N p , Z 1 (z 0 + z0 ) stochastically dominates Z 1 (z 0 ) + Z 1 (z 0 ). Assume now that the result is true for some n ∈ N and we prove it for n + 1. which finishes the proof.

P(Z

We now prove that the third criterion provided below Assumption 1 leads to Assumption 1.1 by contradiction. If this assumption is not satisfied, then there exits a non empty recurrent states class in N p \ {0}. We can chose z = (z 1 , . . . , z p ) ∈ N p \ {0} with the minimal size in its class, so that P(|Z n | ≥ |z| | Z 0 = z) = 1 for all n ≥ 1. We prove that 2z is reachable from z, that is, there exist n ≥ 1 such that P(Z n ≥ 2 z|Z 0 = z) > 0. Once this is proved, we deduce from Lemma 27 that P(|Z n | ≥ 2 |z||Z 0 ≥ 2 z) ≥ P(|Z n | ≥ |z||Z 0 = z) 2 = 1, which implies that P(∃n, Z n = z | Z 0 ≥ 2 z) = 0 and contradicts the fact that z is recurrent. We thus deduce that there is no recurrent state, except the absorbing state {0}, and Assumption 1.1 is satisfied.

Let us prove that 2z is reachable from z (in fact, we prove that every population can be doubled). As the process is strongly primitive, for all i ∈ {1, . . . , p}, there exists n i such that for all m ≥ n i , P(Z m, ≥ 1|Z 0 = e i ) > 0. Let m = max 1≤i≤p n i , then for all i ∈ {1, . . .

F

  -increasing function on (0, +∞) since β ∈ (0, 1 /2) and α > 0. On the other hand +∞ 1

U

  log x) dF i,j (x),where the last integral is finite, since E(V i,j log V i,j ) < +∞ by assumption.Applying Lemma 26, there exists a concave monotone increasing function U : R + -→ R + , with y → U (y) /y non-increasing and +∞ 1 (y) y 2 dy < +∞, and such that for all y ∈ R + , sup z:P(z)=y E(|P(Z 1 ) -λ * P( z )| | Z 0 = z ) ≤ F (y) ≤ U (y).

L

  (z) = lim n→+∞ M n (z) (λ * ) n = u * , z z * ,where u * is the positive right eigenvector of X such that u * , z * = 1 and •, • stands for the Euclidean product. In particular, P(z) = u * , z .Consider z ∈ N p , thenE(|P(Z 1 ) -λ * P(z)| | Z 0 = z) = E(| u * , Z 1 -u * , zX | | Z 0 = z) = E(| u * , Z 1 -zX | | Z 0 = z) 1.j -(zX) j | | Z 0 = z),for some constant C 0 > 0 and henceE(|P(Z 1 ) -λ * P(z)| | Z 0 = z) ≤ C 0 i,j 1 ∀ ≤nm, M 1, >0 -1 ∃ ≤nm, M 1, =0 Z 0 = z .

Ex

  (|P(Z 1 ) -λ * P(z)| | Z 0 = z) ≤ C 1 dF i,j (x) + |z| β+ 1 /2 + |z|γ |z|-1

P(Z 1

 1 ≥ z 1 + z1 | Z 0 = z 0 + z0 ) = P Z 1 ≥ z 1 | Z 0 = z 0 ) × P(Z 1 ≥ z1 | Z 0 = z0 ).

V

  n+1 ≥ z 1 + z1 | Z 0 = z 0 + z0 ) = P i (z 0 )+Z n,i (z 0 ) k=Z n,i (z 0 )+1 i (z 0 )+Z n,i (z 0 ) k=Z n,i (z 0 )+1 V Z n+1 ≥ z 1 | Z 0 = z 0 ) × P(Z n+1 ≥ z1 | Z 0 = z0 ).

  , p}, P(Z m ≥ e | Z 0 = e i ) > 0.(37)Moreover, as P(|Z 1 | = 2|Z 0 = e ) > 0, there exist j 1 , j 2 ∈ {1, . . . , p} such thatP(Z 1 ≥ e j 1 + e j 2 | Z 0 = e ) > 0. (38)By strongly primitive assumption, for n = max{n j 1 , n j 2 },P(Z n ≥ e i | Z 0 = e j 1 ) > 0 and P(Z n ≥ e i | Z 0 = e j 2 ) > 0,hence, by Lemma 27,P(Z n ≥ 2e i | Z 0 = e j 1 + e j 2 ) ≥ P(Z n ≥ e i | Z 0 = e j 1 ) P(Z n ≥ e i | Z 0 = e j 2 ) n+m+1 ≥ 2e i | Z 0 = e i ) ≥ P(Z n+m+1 ≥ 2e i | Z m+1 ≥ e j 1 + e j 2 ) × P(Z m+1 ≥ e j 1 + e j 2 | Z m ≥ e ) × P(Z m ≥ e | Z 0 = e i )then, from (37), (38), (39) and the Markov property, P(Z n+m+1 ≥ 2e i | Z 0 = e i ) > 0 for all i ∈ {1, . . . , p}, and we conclude by Lemma 27 that P(Z n+m+1 ≥ 2z | Z 0 = z) > 0.

  Assume that Assumption 1 holds, that M is finite over S and that λ * > 1. If there exists a concave monotone increasing function U : R + -→ R + , such that for all y ∈ R + ,

	Proposition 7. sup
	z∈R p + :P(z)=y
	)
	which is well defined, according to Theorem 17 in Section 4.1 below. The condition we present
	is inspired by the work of González and Molina in the single-type case [GM96] and Klebaner's
	article [Kle85].