Romain Abraham
email: romain.abraham@univ-orleans.fr

AND Jean-Franc ¸ois Delmas
email: delmas@cermics.enpc.fr

Hongsong Guo
email: hsguo@mail.bnu.edu.cn

CRITICAL MULTI-TYPE GALTON-WATSON TREES CONDITIONED TO BE LARGE

Keywords: Mathematics Subject Classification. 60J80, 60B10 Galton-Watson, random tree, local-limit, strong ratio theorem, branching process

published or not. The documents may come

Introduction

In [START_REF] Kesten | Subdiffusive behavior of random walk on a random cluster[END_REF], Kesten shows that the local limit of a critical or subcritical Galton-Watson (GW) tree conditioned on having a large height is an infinite GW tree (in fact a multi-type GW tree with one special individual per generation) with a unique infinite spine, which we shall call Kesten's tree in the present paper. In Abraham and Delmas [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] a sufficient and necessary condition is given for a wide class of conditionings for a critical GW tree to converge locally to Kesten's tree under minimal hypotheses on the offspring distribution. Notice that condensation may arise when considering sub-critical GW trees, see Janson [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF], Jonnson and Stefansson [START_REF] Jonnson | Condensation in nongeneric trees[END_REF], He [START_REF] He | Conditioning Galton-Watson trees on large maximal out-degree[END_REF] or Abraham and Delmas [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the condensation case[END_REF] for results in this direction. When scaling limits of multi-type GW tree are considered, one obtains as a limit a continuous GW tree, see Miermont [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF] or Gorostiza and Lopez-Mimbela [START_REF] Luis | The multitype measure branching process[END_REF] (when the probability to give birth to different types goes down to 0). In this latter case see Delmas and Hénard [START_REF] Delmas | A Williams decomposition for spatially dependent superprocesses[END_REF] for the limit on the conditioned random tree to have a large height.

In the multi-type case, Pénisson [START_REF] Pénisson | Beyond q-process: Various ways of conditioning the multitype Galton-Watson process[END_REF] has proved that a critical d-types GW process conditioned on the total progeny to be large with a given asymptotic proportion of types converges locally to a multi-type GW process (with a special individual per generation) under the condition that the branching process admits moments of order d + 1. Stephenson [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF] gave, under an exponential moments condition, the local convergence of a multi-type GW tree, conditioned on a linear combination of population sizes of each type to be large, towards the multi-type Kesten's tree introduced by Kurtz, Lyons, Pemantle and Peres [START_REF] Kurtz | A conceptual proof of the Kesten-Stigum theorem for multitype branching processes[END_REF]. The aim of this paper is to give minimal hypotheses to ensure the local convergence of a critical multi-type GW tree conditioned on the total progeny to be large towards the associated multi-type Kesten's tree, see Theorem 3.1. When the offspring distribution is aperiodic, the minimal hypotheses is the existence of the mean matrix which is assumed to be primitive. Furthermore, we exactly condition on the asymptotic proportion of types for the total progeny of the GW tree to be given by the (normalized) left eigenvector associated with the Perron-Frobenius eigenvalue of the mean matrix.

If the asymptotic proportion of types is not equal to the (normalized) left eigenvector associated with the Perron-Frobenius eigenvalue of the mean matrix, then under an exponential moments condition for the offspring distribution, it is possible to get a Kesten's tree as local limit, see [START_REF] Pénisson | Beyond q-process: Various ways of conditioning the multitype Galton-Watson process[END_REF]. However, without an exponential moments condition for the offspring distribution no results are known, and results in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the condensation case[END_REF] for the mono-type case suggests a condensation phenomenon (at least in the sub-critical case). Conditioning large multi-type (or even mono-type) continuous GW tree to have a large population in the spirit of [START_REF] Delmas | A Williams decomposition for spatially dependent superprocesses[END_REF] is also an open question.

The proof of Theorem 3.1 relies on two arguments. The first one is a generalization of the Dwass formula for multi-type GW processes given by Chaumont and Liu [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF] which encodes critical or sub-critical d-multi-type GW forests using d random walks of dimension d. The second one is the strong ratio theorem for random walks in Z d , see Theorem 4.7, which generalizes a result by Neveu [START_REF] Neveu | Sur le théorème ergodique de Chung-Erdős[END_REF] in dimension one. The proof of the strong ratio theorem relies on a uniform version of the d-dimensional local theorem of Gnedenko [START_REF] Gnedenko | On a local limit theorem of the theory of probability[END_REF], see also Gnedenko and Kolmogorov [START_REF] Gnedenko | Limit distributions for sums of independent random variables[END_REF] (for the sum of independent random variables), Rvaceva [START_REF] Rvaceva | On domains of attraction of multi-dimensional distributions[END_REF] (for the sum of d-dimensional i.i.d. random variables) or Stone [START_REF] Stone | On local and ratio limit theorems[END_REF] (for the sum of d-dimensional i.i.d. lattice or non lattice random variables), which is given in Section 4.2, and properties of the Legendre-Laplace transform of a probability distribution. As we were unable to find those latter properties in the literature, we give them in a general framework in Section 4.1, as we believe they might be interesting by themselves.

The paper is organized as follows. We present in Section 2 the topology on the set of the multitype trees and a sufficient and necessary condition for the local convergence of random multi-type trees, see Corollary 2.2, the definition of a multi-type GW tree with a given offspring distribution and the aperiodicity condition on the offspring distribution, see Definition 2.5. Section 3 is devoted to the main result, Theorem 3.1, and its proof. The Appendix collects results on the Legendre-Laplace transform in a general framework in Section 4.1, Gnedenko's d-dimensional local theorem in Section 4.2, and the strong ratio limit theorem for d-dimensional random walks in Section 4.3.

Multi-type trees

2.1. General notations. We denote by N = {0, 1, 2, . . .} the set of non-negative integers and by N * = {1, 2, . . .} the set of positive integers. For d ∈ N * , we set [d] = {1, . . . , d}.

Let d ≥ 1. We say x = (x i , i ∈ [d]) ∈ R d is a column vector in R d . We write 1 = (1, . . . , 1) ∈ R d , 0 = (0, . . . , 0) ∈ R d and denote by e i the vector such that the i-th element is 1 and others are 0. For vectors x = (x i , i ∈ [d]) ∈ R d and y = (y i , i ∈ [d]) ∈ R d , we denote by x, y the usual scalar product of x and y, by x y the product d i=1 x y i i , by |x| = d i=1 |x i | and x = x, x the ℓ 1 and ℓ 2 norms of x, and we write x ≤ y (resp.

x < y) if x i ≤ y i (resp. x i < y i ) for all i ∈ [d].
For any nonempty set A ⊂ R d , we define span A as the linear sub-space generated by A (that is span

A = { n i=1 α i y i ; α i ∈ R, y i ∈ A, i ∈ [n], n ∈ N * }) and for x ∈ R d , we denote x + A = {x + y; y ∈ A}. For A and B nonempty subsets of R d , we denote A -B = {x -y; x ∈ A, y ∈ B}.
For a random variable X and an event A, we write

E[X; A] for E [X1 A ].
(uv) = M(v) if |v| ≥ 1; M(uv) = M(u) if |v| = 0. Let u, v ∈ U .
We say that v (resp. v) is an ancestor of u (resp. û) and write v u (resp. v û) if there exists w ∈ U such that u = vw (resp. ŵ ∈ Û such that û = v ŵ).

A tree t is a subset of Û such that:

• ∅ ∈ t. • If û ∈ t, then {v; v û} ⊂ t.
• For every û ∈ t, there exists k û[ t] ∈ N such that, for every positive integer ℓ,

ûℓ ∈ t iff 1 ≤ ℓ ≤ k û[ t].
A marked tree t is a subset of U such that:

(a) The set t = {û; u ∈ t} of (unmarked) individuals of t is a tree. (b) There is only one type per individual: for u, v ∈ t, û = v implies M(u) = M(v) and thus u = v. Thanks to (b), the number of offsprings of the marked individual

u ∈ t, k u [t], corresponds to k û[ t].
In what follows we will deal only with marked trees and simply call them trees.

Denote by ∅ t = ( ∅, M(∅ t )) ∈ U the root of the tree t and write ∅ instead of

∅ t when the context is clear. The parent of v ∈ t \ ∅ t in t, denoted by Pa v (t) is the only u ∈ t such that |u| = |v| -1 and u v. The set of the children of u ∈ t is C u (t) = {v ∈ t, Pa v (t) = u}. Notice that k u [t] = Card (C u (t)) for u ∈ t. We set k u (t) = (k (i) u [t], i ∈ [d]), where for i ∈ [d] k (i) u [t] = Card ({v ∈ C u (t); M(v) = i})
is the number of offsprings of type i of u ∈ t. We have i∈

[d] k (i) u [t] = k u [t]. The vertex u ∈ t is called a leaf if k u [t] = 0 and let L 0 (t) = {u ∈ t, k u [t] = 0} be the set of leaves of t.
We denote by T the set of marked trees. For t ∈ T, we define |t| = (|t

(i) |, i ∈ [d]) with |t (i) | = Card ({u ∈ t, M(u) = i})
the number of individuals in t of type i. Let us denote by T 0 = {t ∈ T : Card (t) < ∞} the subset of finite trees. We say that a sequence v = (v n , n ∈ N) ⊂ U is an infinite spine if v n v n+1 and |v n | = n for all n ∈ N. We denote by T 1 the subset of trees which have one and only one infinite spine. For t ∈ T 1 , denote by v t the infinite spine of the tree t. Let T ′ 1 be the subset of T 1 such that the infinite spine features each type infinitely many times:

T ′ 1 = {t ∈ T 1 ; ∀i ∈ [d], Card ({v ∈ v t ; M(v) = i}) = ∞}.
The height of a tree t is defined by H(t) = sup{|u|, u ∈ t}. For h ∈ N, we denote by T (h) = {t ∈ T; H(t) ≤ h} the subset of marked trees with height less than or equal to h.

Convergence determining class.

For h ∈ N, the restriction function r h from T to T is defined by r h (t) = {u ∈ t, |u| ≤ h}. We endow the set T with the ultra-metric distance d(t, t ′ ) = 2 -max{h∈N,r h (t)=r h (t ′ )} . The Borel σ-field associated with the distance d is the smallest σ-field containing the singletons for which the restrictions (r h , h ∈ N) are measurable. With this distance, the restriction functions are continuous. Since T 0 is dense in T and (T, d) is complete, we get that (T, d) is a Polish metric space.

Let t, t ′ ∈ T and x ∈ L 0 (t). If the type of the root of t ′ is M(x), we denote by

t ⊗ (t ′ , x) = t ∪ {xv, v ∈ t ′ }
the tree obtained by grafting the tree t ′ on the leaf x of the tree t; otherwise, let t ⊗ (t ′ , x) = t.

Then we consider T(t, x) = {t ⊗ (t ′ , x), t ′ ∈ T} the set of trees obtained by grafting a tree on the leaf x of t. For t ∈ T 0 , it is easy to see that T(t, x) is closed and also open.

Set F = {T(t, x); t ∈ T 0 , x ∈ L 0 (t) and M(∅ t ) = M(x)} ∪ {{t}; t ∈ T 0 }. Following the proof of Lemma 2.1 in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF], it is easy to get the following result.

Lemma 2.1. The family F is a convergence determining class on T 0 ∪ T ′ 1 . We deduce the following corollary.

Corollary 2.2. Let (T n , n ∈ N * ) and T be random variables taking values in T 0 T ′ 1 . Then the sequence (T n , n ∈ N * ) converges in distribution towards T if and only if we have for all t ∈ T 0 lim n→+∞ P(T n = t) = P(T = t) and for all x ∈ L 0 (t) such that M(∅ t ) = M(x):

lim n→+∞ P(T n ∈ T(t, x)) = P(T ∈ T(t, x)).

Aperiodic distribution.

Let us consider a probability distribution F = (F (x), x ∈ Z d ) on Z d . In order to avoid degenerate cases, we assume that there exists x 0 ∈ Z d such that:

(1) 0 < F (x 0 ) < 1.
Denote by supp (F ) = {x ∈ Z d , F (x) > 0} the support set of F and by R 0 the smallest subgroup of Z d which contains the set supp (F )supp (F ).

Definition 2.3. A distribution F on Z d is called aperiodic if R 0 = Z d .
For x ∈ Z d , let G x be the smallest subgroup of Z d that contains -x + supp (F ). According to the next lemma, an aperiodic distribution is called strongly aperiodic in [23, p.42].

Lemma 2.4. If x ∈ supp (F ), then G x = R 0 . The distribution F is aperiodic if and only if G x = Z d for some x ∈ supp (F ) or equivalently if and only if

G x = Z d for all x ∈ Z d . Proof. Let x ∈ Z d . Let z ∈ R 0 . There exists n, n ′ ∈ N and x i , x ′ i , y i , y ′ i ∈ supp (F ) for all i ∈ N * such that n i=1 (y i -x i ) -n ′ i=1 (y ′ i -x ′ i ) = z. This implies that n i=1 (y i -x) + n ′ i=1 (x ′ i -x) - n i=1 (x i -x) -n ′ i=1 (y ′ i -x) = z and thus z ∈ G x .
This gives R 0 ⊂ G x . For x ∈ supp (F ), we get G x ⊂ R 0 and thus G x = R 0 . The end of the lemma is obvious.

2.5. Multi-type offspring distribution. We define a multi-type offspring distribution p of d types as a sequence of probability distributions: (1) , . . . , f (d) ) the generating function of the offspring distribution p, i.e. for i ∈

p = (p (i) , i ∈ [d]), with p (i) = (p (i) (k), k ∈ N d ) a probability distribution on N d . Denote by f = (f
[d] and s ∈ [0, 1] d : (2) f (i) (s) = E[s X i ],
with

X i = (X (j) i , j ∈ [d]) a random variable on N d with distribution p (i) . Denote by m ij = ∂ s j f (i) (1) = E[X (j)
i ] ∈ [0, +∞] the expected number of offsprings with type j of a single individual of type i. Denote by M the mean matrix M = (m ij ; i, j ∈ [d]) and set (m [3, p.184], we say that:

(n) ij ; i, j ∈ [d]) = M n for n ∈ N * . Following
-

p is non-singular if f (s) = M s. -M is finite if m ij < +∞ for all i, j ∈ [d].
-M is primitive if M is finite and there exists n ∈ N * such that for all i, j

∈ [d], m (n) ij > 0.
By the Frobenius theorem, see [3, p.185], if M is primitive, then M has a unique maximal (for the modulus in C) eigenvalue ρ. Furthermore ρ is simple, positive (ρ ∈ (0, +∞)), and the corresponding right and left eigenvectors can be chosen to be positive. If ρ = 1 (resp. ρ > 1, ρ < 1), we say that the offspring distribution and the associated multi-type GW tree are critical (resp. supercritical, subcritical).

Recall the definition of an aperiodic distribution given in Definition 2.3.

Definition 2.5. Let p = (p (i) , i ∈ [d]) be an offspring distribution. We say that p is aperiodic, if the smallest subgroup of Z d that contains d i=1 supp (p (i) )supp (p (i) ) is Z d . For an offspring distribution p, we shall consider the following assumptions: (H 1 ) The mean matrix M of p is primitive, and p is critical and non-singular. (H 2 ) The offspring distribution p is aperiodic.

2.6. Multi-type Galton-Watson tree and Kesten's tree. We define the multi-type GW tree τ with offspring distribution p. Definition 2.6. Let p be an offspring distribution of d types and α a probability distribution on [d]. A T-valued random variable τ is a multi-type GW tree with offspring distribution p and root type distribution α, if for all h ∈ N, t ∈ T (h) , we have:

P α (r h (τ ) = t) = α(M(∅ t )) u∈t,|u|<h k (1) u [t]! • • • k (d) u [t]! k u [t]! p (M(u)) (k u (t)).
We deduce from the definition that for t ∈ T 0 , we have

P α (τ = t) = α(M(∅ t )) u∈t k (1) u [t]! • • • k (d) u [t]! k u [t]! p (M(u)) (k u (t)).
The multi-type GW tree enjoys the branching property: an individual of type i generates children according to p (i) independently of any born individual, for i ∈ [d].

Let p be an offspring distribution of d types such that (H 1 ) holds. Denote by a * (resp. a) the right (resp. left) positive normalized eigenvector of M such that a, 1 = a, a * = 1. Those eigenvectors correspond to the eigenvalue ρ = 1. Notice that a is a probability distribution on [d]. The corresponding size-biased offspring distribution p = (p (i) , i ∈ [d]) is defined by: for

i ∈ [d] and k ∈ N d , (3) p(i) (k) = k, a * a * i p (i) (k).
For α a probability distribution on [d], we also define the corresponding size-biased distribution α = (α(i

), i ∈ [d]) by, for i ∈ [d]: (4) α(i) = α(i) a * i α, a * • Definition 2.7.
Let p be an offspring distribution of d types whose mean matrix is primitive and let α be a probability distribution on [d]. A multi-type Kesten's tree τ * associated with the offspring distribution p and with the root type distribution α is defined as follows:

-Marked individuals are normal or special.

-The root of τ * is special and its type has distribution α.

-A normal individual of type i ∈ [d] produces only normal individuals according to p (i) .

-A special individual of type i ∈ [d] produces children according to p(i) . One of those children, chosen with probability proportional to a * j where j is its type, is special. The others (if any) are normal.

Notice that the multi-type Kesten's tree is a multi-type GW tree (with 2d types). The individuals which are special in τ * form an infinite spine, say v * , of τ * ; and the individuals of τ * \v * are normal.

Let r ∈ [d]. We shall write P r (dτ ), resp. P r (dτ * ), for the distribution of τ , resp. τ * , when the type of its root is r (that is α = δ r the Dirac mass at r). From [START_REF] Kurtz | A conceptual proof of the Kesten-Stigum theorem for multitype branching processes[END_REF], we get that for h ∈ N, t ∈ T (h) with M(∅ t ) = r, and x ∈ L 0 (t) with |x| = h and M(x) = i:

(5) P r (r h (τ * ) = t, v * h = x) = a * i a * r P r (r h (τ ) = t).
Notice that if M is primitive and p is critical or sub-critical, then a.s. Kesten's tree τ * belongs to T 1 . The next lemma asserts that there are infinitely many individuals of all types on the infinite spine. Proof.

Recall that a * = (a * i , i ∈ [d]) is the normalized right eigenvalue of M such that a * , a = 1. By construction, the sequence (M(v * n ), n ∈ N) is a Markov chain on [d] and transition matrix Q = (Q i,j , i, j ∈ [d]) given by Q i,j = P(M(v * 1 ) = j| M(v * 0 ) = i) = k=(k 1 ,...,k d )∈N d k j a * j k, a * p(i) (k) = a * j a * i m i,j ,
where we used (3) for the definition of p and the definition of the mean matrix M for the last equality. Since a * is positive and M is primitive, we deduce that Q is also primitive. This implies that the Markov chain (M(v * n ), n ∈ N) is recurrent on [d] and hence it visits a.s. infinitely many times all the states of [d].

The next lemma will be used in the proof of Theorem 3.1. In the next lemma, we shall consider a leaf x of a finite tree t with type i and the root of type r. However, we will only use the case i = r in the proof of Theorem 3.1. Lemma 2.9. Let p be an offspring distribution of d types satisfying (H 1 ) and r ∈ [d]. Let τ be a GW tree with offspring distribution p and τ * be a Kesten's tree associated with p.

For all t ∈ T 0 with M(∅ t ) = r, x ∈ L 0 (t) with M(x) = i ∈ [d],
and k ∈ N d such that k ≥ |t|, we have:

(6) P r (τ ∈ T(t, x) |τ | = k) = a * r a * i P i (|τ | = k -|t| + e i ) P r (|τ | = k) P r (τ * ∈ T(t, x)).
Proof. Since τ * has a unique infinite spine v * and t ∈ T 0 , we deduce that τ * ∈ T(t, x) implies that x belongs to v * and we get in the same spirit of (5) that:

(7) P r (τ * ∈ T(t, x)) = a * i a * r P r (τ ∈ T(t, x)).
We have, following the ideas of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF]:

P r (τ ∈ T(t, x), |τ | = k) = t ′ ∈T 0 P r (τ = t ⊗ (t ′ , x))1 {|t⊗(t ′ ,x)|=k} = t ′ ∈T 0 P r (τ ∈ T(t, x)) P i (τ = t ′ )1 {|t⊗(t ′ ,x)|=k} = P r (τ ∈ T(t, x)) t ′ ∈T 0 P i (τ = t ′ )1 {|t ′ |=k-|t|+e i } = P r (τ ∈ T(t, x))P i (|τ | = k -|t| + e i ),
where we used the branching property of the multi-type GW tree for the second equality. Use [START_REF] Gnedenko | On a local limit theorem of the theory of probability[END_REF] to deduce (6).

Main results

3.1. Conditioning on the total population size. Recall that under (H 1 ), we denote by

a = (a ℓ , ℓ ∈ [d]) and a * = (a * ℓ , ℓ ∈ [d]
) the positive normalized left and right eigenvectors of the mean matrix M associated with the eigenvalue ρ = 1 such that a, a * = a i = 1. The proof of the following main theorem is given in Section 3.3.

Theorem 3.1. Assume that (H 1 ) and (H 2 ) hold. Let (k(n), n ∈ N * ) be a sequence of N d satisfying lim n→∞ |k(n)| = +∞ and lim n→∞ k(n)/|k(n)| = a.
Let τ be a random GW tree with critical offspring distribution p and root type distribution α, and τ n be distributed as τ conditionally on {|τ | = k(n)}. Then the sequence (τ n , n ∈ N * ) converges in distribution to the Kesten's tree τ * associated with p and α. Remark 3.2. Let τ be a critical GW tree with offspring distribution p satisfying (H 1 ). We can consider τ conditionally on the event that the population of type i, |τ (i) |, is large. According to Proposition 4 in [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF], the random variable |τ (i) | is distributed as the total number of vertices of a critical mono-type GW tree under M τ (∅) = i, or as the total number of vertices of a random number of independent mono-type critical GW trees with the same distribution under M τ (∅) = i. In particular, we deduce from [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] that, if p (i) is aperiodic, the key equality lim n→+∞ P(|τ

(i) | = n -b)/P r (|τ (i) | = n) = 1 holds for any b ∈ Z.
And following the proof of Theorem 3.1 after Equation ( 19), we easily get that τ conditioned on |τ (i) | being large converges locally to Kesten's tree. See [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF] for a detailed proof.

Remark 3.3. The local convergence of a multi-type critical GW tree τ conditioned on the number of vertices of one fixed type being large to a Kesten's tree has been proved in [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF]. It would be easy to extend Theorem 3.1, with the same minimal conditions (H 1 ) and (H 2 ) to a conditioning on an asymptotic proportion per types for d ′ types, with d ′ < d by using the constructions from [START_REF] Rizzolo | Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set[END_REF] or from [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF]. The idea is to map a multi-type GW tree τ with d types onto another GW tree τ ′ with d ′ < d types and offspring distribution p ′ so that the size of the population of types 1 to d ′ of τ and τ ′ are the same. Then the key Equation ( 19) is now replaced by the one for τ ′ which holds if the offspring distribution p ′ satisfies (H 1 ) and (H 2 ). Then the proof follows as in the proof of Theorem 3.1 after Equation [START_REF] Pénisson | Beyond q-process: Various ways of conditioning the multitype Galton-Watson process[END_REF].

Remark 3.4. The change of offspring distribution given in Section 1.4 of [START_REF] Pénisson | Beyond q-process: Various ways of conditioning the multitype Galton-Watson process[END_REF], when it exists, allows to extend Theorem 3.1 to sub-critical multi-type GW trees. In order to consider an asymptotic proportion of types different from the one given by the (positive normalized) left eigenvector associated with the Perron-Frobenius eigenvalue, one has to change the offspring distribution, see Theorem 3 of [START_REF] Pénisson | Beyond q-process: Various ways of conditioning the multitype Galton-Watson process[END_REF]. However, this requires exponential moments for the offspring distribution.

We end this Section by using Theorem 3.1 to extend results of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the condensation case[END_REF] on mono-type GW tree in the following sense. Let τ be a mono-type GW tree (that is d = 1) with critical aperiodic offspring distribution q = (q(ℓ), ℓ ∈ N). Let f q denote the generating function of q and Q = {γ > 0; f q (γ) < +∞} its domain on (0, +∞).

Let d ≥ 2 and assume that Card (supp q) ≥ d + 1. Since q is critical we have 0 ∈ supp q. Let A 1 , . . . , A d be a partition of supp q such that 0 ∈ A 1 and Card (A 1 ) > 1. We set α(i) = ℓ∈A i q(ℓ) for all i ∈ [d]. Notice that α is a positive probability distribution and α(1) > q(0). We set

|τ | = (|τ (i) |, i ∈ [d]
) where |τ (i) | be the number of individuals of τ whose number of offsprings belongs to

A i . For x = (x i , i ∈ [d]) we set m x := i∈[d] x i inf A i and for γ ∈ Q: h x (γ) = i∈[d] x i f ′ A i (γ) f A i (γ) with f A i (γ) = ℓ∈A i γ ℓ q(ℓ) for all i ∈ [d].
Corollary 3.5. Let q be a critical aperiodic offspring distribution. Let α = (α(i), i ∈ [d]) ∈ R d be such that α > 0 and α, 1 = 1, so that α is a non-degenerate proportion. Assume that:

(8) m α < 1 and (9)
there exists a (unique

) γ ∈ Q such that h α(γ) = 1. Let (k(n), n ∈ N * ) be a sequence of N d satisfying lim n→∞ |k(n)| = +∞ and lim n→∞ k(n)/|k(n)| = α.
Let τ be a random mono-type GW tree with offspring distribution q, and τ n be distributed as τ conditionally on {|τ | = k(n)}. Then the sequence (τ n , n ∈ N * ) converges in distribution to the Kesten's tree τ * associated to the offspring distribution q = (q(ℓ), ℓ ∈ N) where for i ∈ [d], ℓ ∈ A i :

q(ℓ) = α(i) f A i (γ) γ ℓ q(ℓ).
Notice that if α = α, then condition (8) holds as Card (A 1 ) > 1 and q is critical; and condition (9) also holds with γ = 1 as q is critical. We deduce that if α = α, then q = q and τ * = τ * is simply the Kesten's tree associated to the offspring distribution q. We now comment on the conditions (8) and (9). Remark 3.6. One can see that condition (8) is almost optimal. This is easy to check in the binary case. Assume q(0) + q(1) + q(2) = 1, q(0)q(1)q(2) > 0, A 1 = {0, 1} and A 2 = {2}. Since we always have |τ (1) | > |τ (2) |, then any asymptotic proportion has to satisfies α(1

) ≥ α(2) that is m α = 2α(2) ≤ 1. Remark 3.7. For i ∈ [d], let P i,γ denote the distribution of a random variable Z taking values in A i such that P i,γ (Z = ℓ) = 1 {ℓ∈A i } γ ℓ q(ℓ)/f A i (γ). In particular, we have h α(γ) = i∈[d] α(i)E i,γ [Z]. An elementary computation gives that ∂ γ E i,γ [Z] = γ -1 Var i,γ (Z). Using that Card (A 1 ) > 1,
we get Var 1,γ (Z) > 0 and thus h ′ α is positive on Q. We deduce that, if it exists, the root of the equation ( 9) is then unique. Since lim γ→0 h α(γ) = m α, we deduce that condition (8) implies h α(0+) < 1. A necessary and sufficient condition for the existence of a root to [START_REF] He | Conditioning Galton-Watson trees on large maximal out-degree[END_REF] is that lim γ↑R h α(γ) ≥ 1, with R = sup Q the radius of convergence of the series f q . A sufficient condition to get this latter condition is for example lim γ↑R f ′ q (γ) = +∞ or even the stronger condition R = +∞.

As noticed earlier, for α = α, as q is critical, we get that h α (1) = 1. So in this case no further hypothesis are needed. We also deduce that, if α(i) ≥ α(i) for all i ∈ [d] such that f A i (1) ≥ 1, then we have h α(1) ≥ h α (1) = 1. And thus, in this particular case also, the root of ( 9) exists without further assumptions on q.

Proof of Corollary 3.5. We consider artificially that τ is a d dimensional multi-type GW tree, by saying that an individual u ∈ τ is of type i if the number of offsprings of u belongs to A i . The corresponding root type distribution is α and the corresponding offspring distribution p = (p (i) , i ∈ [d]) is defined as follows: for k = (k 1 , . . . , k d ) ∈ N d , ( 10)

p (i) (k) = 1 {|k|∈A i } q(|k|) α(i) |k| k α k ,
where we recall that

α k = j∈[d] α(j) k j and |k| = i∈[d]
k i , and we use the following notation for

the multinomial coefficient |k| k = |k|!/ i∈[d] k i !.
For simplicity we shall still denote the corresponding multi-type GW tree by τ . We define α * = (α

* (i), i ∈ [d]) with α * (i) = ℓ∈A i ℓq(ℓ)/α(i) so that α, 1 = α, α * = 1.
Notice that α * is positive as α(1) > q(0) and 0 ∈ A 1 . It is easy to check that the mean matrix is given by M = (α * ) T α. Its only non zero eigenvalue is 1 and α and α * are the non-negative associated left and right eigenvectors. The mean matrix M is primitive as all its entries are positive. We get that condition (H 1 ) holds. Notice (H 2 ) holds as we assumed q is aperiodic.

We first consider the case α = α. (As noticed just after Corollary 3.5, condition (8) holds and condition (9) also holds with γ = 1.) We easily deduce from Theorem 3.

1 that if (k(n), n ∈ N * ) is a sequence of N d satisfying lim n→∞ k(n)/|k(n)| = α with lim n→+∞ |k(n)| = +∞, then τ n , which is distributed as τ conditionally on {|τ | = k(n)}, converges in distribution to the (d-type)
Kesten's tree τ * associated to the offspring distribution p and type root distribution α.

Let τ * be the mono-type Kesten's tree associated to q. We shall check that τ * is distributed as τ * seen as a multi-type GW tree, where an individual u ∈ τ * is of type i if the number of offsprings of u belongs to A i and that u is normal if it has a finite number of descendants (that is Card ({v ∈ τ * ; u ≺ v}) < +∞) and special otherwise. Let t ∈ T 0 and x ∈ L 0 (t). For i ∈ [d], we set x i = {x, i} and t i = {x i } (t\{x}), the tree which is equal to t except for the leaf x which is of type i instead of M(x). We denote by T(t, x) the set of trees obtained by grafting trees on the leaf x of t with possibly changing the type of x, that is T(t, x) = i∈[d] T(t i , x i ). We write P α(dτ * ) = r∈[d] α(r)P r (dτ * ). We have for i ∈ [d]:

P α(τ * ∈ T(t i , x i )) = α * (i) r∈[d] α(r) α * (r) P r (τ ∈ T(t i , x i )) = α * (i) P(τ ∈ T(t i , x i )),
where we used [START_REF] Gnedenko | On a local limit theorem of the theory of probability[END_REF] for the first equality and (4) as well as α, α * = 1 for the second. Using that P(τ ∈ T(t i , x i )) = α(i)P(τ ∈ T(t, x)) and α, α * = 1, we deduce that:

P α(τ * ∈ T(t, x)) = i∈[d] α * (i) P(τ ∈ T(t i , x i )) = P(τ ∈ T(t, x)).
Using [START_REF] Gnedenko | On a local limit theorem of the theory of probability[END_REF] in the mono-type case, we get P(τ * ∈ T(t, x)) = P(τ ∈ T(t, x)) and thus P α(τ * ∈ T(t, x)) = P(τ * ∈ T(t, x)). It is left to the reader to check that F ′ = {T(t, x); t ∈ T 0 , x ∈ L 0 (t)} is a separating class on T ′ 1 . Hence τ * is distributed as τ * , and can thus be seen as the (monotype) Kesten tree associated with the offspring distribution q.

We now shall condition on a general asymptotic proportion α ∈ R d satisfying condition [START_REF] Gnedenko | Limit distributions for sums of independent random variables[END_REF] and condition [START_REF] He | Conditioning Galton-Watson trees on large maximal out-degree[END_REF]. We assume that there exists a root to the equation ( 9), say γ. This root is unique according to Remark 3.7. The probability q defined in Corollary 3.5 is a critical (as γ is a root of ( 9)) and aperiodic (as q is aperiodic) and that α(i) = ℓ∈A i q(ℓ) for all i ∈ [d]. Let τ be a mono-type GW tree with offspring distribution q (which can also be seen as a multi-type GW tree where the type of an individual is A i if the number of its offspring lies in A i ). We deduce that for all t ∈ T 0 , we have:

P α(τ = t) = u∈t q(k u [t]) = γ |t|,1 -1 Γ |t| P α (τ = t), where Γ = (α(i)/f A i (γ), i ∈ [d]
). In particular, for all k ∈ N d , the random tree τ n , which is distributed as τ conditionally on {|τ | = k}, has the same distribution as the random tree τn , which is distributed as τ conditionally on {|τ | = k}. According to the first part, since q is critical aperiodic, we deduce that if

(k(n), n ∈ N * ) is a sequence of N d satisfying lim n→∞ k(n)/|k(n)| = α with lim n→+∞ |k(n)| = +∞,
then τn , and thus τ n , converges in distribution to the mono-type Kesten's tree τ * associated to q.

3.2. Around the Dwass formula. Let τ be a random GW tree with critical offspring distribution p. We have no assumption on p for the moment. For i, j ∈ [d], we define the total number of individuals of type i whose parent is of type j:

B ij = Card ({u ∈ τ, M(u) = i and M(Pa(u)) = j}) .

And we set

B = (B ij ; i, j ∈ [d]). Notice that j∈[d] B ij = |τ (i) |. Let (X i,ℓ ; ℓ ∈ N * ) for i ∈ [d] be d independent families of independent random variables in N d with X i,ℓ having probability distribution p (i) . For i ∈ [d], we consider the random walk S i,n = n ℓ=1 X i,ℓ for n ∈ N * with S i,0 = 0. For k = (k 1 , . . . , k d ) ∈ N d , we set S k = i∈[d] S i,k i .
We adopt the following convention for a d-dimensional random variable X to write X = (X (j) , j ∈ [d]), so that we have in particular S

(11) S ij (k, r) = -S (j) i,k i + (S (j)
k + 1 {r=i} )1 {i=j} . The following lemma is a direct consequence of the representation of Chaumont and Liu [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF] for multi-type GW process, which generalizes the Dwass formula to the multi-type case. Lemma 3.8. Let τ be a random GW tree with critical offspring distribution p. For r ∈ [d] and k ∈ (N * ) d , we have:

P r (|τ | = k) = 1 i∈[d] k i E [det(S(k, r)); S k + e r = k] . Proof. For κ = (κ ij ; i, j ∈ [d]) ∈ N d×d , we denote, for j ∈ [d], by κ j the column vector (κ ij , i ∈ [d]). We deduce from Theorem 1.2 in [5] that, for r ∈ [d], k = (k 1 , . . . , k d ) ∈ (N * ) d , κ = (κ ij ; i, j ∈ [d]) ∈ N d×d such that (12) k = e r + j∈[d] κ j ,
we have:

(13) P r (B = κ) = det(∆(k) -κ) j∈[d] P(S j,k j = κ j ) k j ,
where ∆(k) is the d × d diagonal matrix with diagonal k. Notice that additional hypotheses on the offspring distribution p were required in Theorem 1.2 from [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF]. However, for fixed κ, (13) is a finite algebraic expression of p. According to [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF], it holds in particular for all p such that there exists a finite constant c ≥ 2 and

p (i) (k) > 0 if |k| ≤ c and p (i) (k) = 0 if |k| > c for all i ∈ [d].
This gives that ( 13) holds for all p.

Because of ( 12), we have:

(14) P r (|τ | = k, B = κ) = P r (B = κ).
Thanks to the definition of S(k, r), we have that ∆(k)κ is equal to the transpose of S(k, r) on j∈[d] S j,k j = κ j . By summing [START_REF] Kesten | Subdiffusive behavior of random walk on a random cluster[END_REF] and thus ( 13) over all the possible values of κ such that (12) holds, we get:

P r (|τ | = k) = κ P r (B = κ)1 {k=er + j∈[d] κ j } = 1 j∈[d] k j κ det(∆(k) -κ) 1 {k=er + j∈[d] κ j } P(∀j ∈ [d], S j,k j = κ j ) = 1 i∈[d] k i E [det(S(k, r)); e r + S k = k] .
In order to compute the determinant det(S(k, r)), instead of using a development based on permutations, we shall use a development based on elementary forests, see Lemma 4.5 in [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF] and Formula (15) below. (As we are interested in computing the determinant of a matrix whose all columns but one sum up to 0, we shall only consider forests reduced to one tree.)

Recall 1 = (1, . . . , 1) ∈ R d . For r ∈ [d], we consider T r the subset of T 0 of trees with root of type r, and having exactly d individuals, all of them with a distinct type:

T r = {t ∈ T 0 ; |t| = 1, and M(∅ t ) = r}.

For t ∈ T r and j ∈ [d] \ {r}, let j t denote the type of the parent of the individual of type j: j t = M(Pa(u j )), where u j is the only element of t such that M(u j ) = j. We shall use the following formula to give asymptotics on det(S(k, r)). Lemma 3.9. For r ∈ [d] and k ∈ (N * ) d , we have:

det(S(k, r)) = t∈Tr j∈[d]\{r} S (j) j t ,k j t .
Proof. We follow the presentation of [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF]. We say that a collection of trees is a forest. A forest f = (t j , j ∈ J) is called elementary if the trees are pairwise disjoint and if the forest contains exactly one individual of each type, that is j∈J |t j | = 1. Let F denote the set of elementary forests. For f ∈ F, set u i the individual in f of type i, which belongs to a tree of f say t j , and write i f = M(v) for the type of the parent

v = Pa u i (t j ) of u i if |u i | > 0 and i f = 0 if |u i | = 0.
According to Lemma 4.5 in [START_REF] Chaumont | Coding multitype forests: application to the law of the total population of branching forests[END_REF], we have for κ

= (κ ij ; i, j ∈ [d]) ∈ R d×d (15) det(κ) = (-1) d f ∈F j∈[d] κ j f ,j ,
with the convention that κ 0,j = -i∈[d] κ ij .

Thanks to Definition (11) of S(k, r), this implies that for r ∈ [d] and k ∈ (N * ) d , we have:

(16) det(S(k, r)) = f ∈F j∈[d] S (j) j f ,k j f , with the convention that if j f = 0, then S (j) j f ,k j f = 1 {j=r} . Notice that j∈[d] S (j) j f ,k j f
= 0 if the forest f is not reduced to a single tree whose root is of type r. To conclude, use that j f = j t if the forest f is reduced to a single tree t.

Let ( Xi,ℓ ; ℓ ∈ N * , i ∈ [d]) be a sequence of random variables independent of (X i,ℓ ; ℓ ∈ N * , i ∈ [d]) with the same distribution.

For a finite subset K of N, we shall consider partitions

A (ℓ,K) = (A K 1 , . . . , A K ℓ ) of K such that inf A K 1 < • • • < inf A K ℓ . For t ∈ T r , i ∈ [d],
recall that u i is the individual in t of type i. Denote by C i (t) = {j ∈ [d]; j t = i} the set of types of the children of u i in t. Let A t be the family of all A = (m, (A

(m i ,C i (t)) ), i ∈ [d]), with m = (m 1 , . . . , m d ) ∈ N d such that, for all i ∈ [d], m i = 0 if Card (C i (t)) = 0 and 1 ≤ m i ≤ Card (C i (t)) if Card (C i (t)) > 0.
For convenience, we may write m A for m. With this notation, we set:

Sm A = i∈[d] m i ℓ=1 Xi,ℓ , G(A) = i∈[d] m i ℓ=1 j∈A C i (t) ℓ X(j) i,ℓ ,
with the convention that ∅ = 0 and ∅ = 1, and for k

= (k 1 , . . . , k d ) ∈ N d such that k i ≥ d for all i ∈ [d]: B k (m A ) = i∈[d] k i ! (k i -m i )! • Since Xi,ℓ for i ∈ [d], ℓ ∈ N * takes values in N d and i∈[d] m i ℓ=1 Card (A C i (t) ℓ ) = d -1, we deduce that: (17) 0 ≤ G(A) ≤ Sm A d-1
.

We have the following result.

Corollary 3.10. For r ∈ [d] and b, k ∈ (N * ) d such that k ≥ d1, we have:

E [det(S(k, r)); S k = b] = t∈Tr A∈A t B k (m A ) E G(A); Sm A + S k-m A = b .
Proof. For r ∈ [d], t ∈ T r , and k ∈ (N * ) d , we have:

j∈[d]\{r} S (j) j t ,k j t = i∈[d] j∈C i (t) k i ℓ=1 X (j) i,ℓ .
Using the exchangeability of (X i,ℓ ; ℓ ∈ N * ) for all i ∈ [d], we easily get for b, k ∈ (N * ) d such that k ≥ d1:

E   j∈[d]\{r} S (j) j t ,k j t ; S k = b   = A∈A t B k (m A ) E G(A); Sm A + S k-m A = b .
Then use Lemma 3.9 to conclude.

(18) i∈[d] (k i -b i ) i∈[d] k i P r (|τ | = k -b) P r (|τ | = k) = E [det(S(k -b, r)); S k-b + e r = k -b] E [det(S(k, r)); S k + e r = k] = t∈Tr A∈A t B k-b (m A ) E G(A); Sm A + S k-b-m A = k -b -e r t∈Tr A∈A t B k (m A ) E G(A); Sm A + S k-m A = k -e r •
The next lemma is an extension of the strong ratio limit theorem given in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the condensation case[END_REF]. Its proof is postponed to Section 3.4. Recall that a is the positive normalized left eigenvector of the mean matrix M . (Notice that no moment condition is assumed for G or H.) Lemma 3.11. Assume that (H 1 ) and (H 2 ) hold. Let G and H be two random variables in N and N d respectively, independent of (X i,ℓ ; ℓ ∈ N * , i ∈ [d]) and such that P(G = 0) < 1 and a.s.

G ≤ |H| c for some c ≥ 1. Set (k(n), n ∈ N * ) and (s n , n ∈ N * ) be two sequences in N d satisfying lim n→∞ |k(n)| = +∞ and lim n→∞ k(n)/|k(n)| = lim n→∞ s n /|k(n)| = a.
Then for any given m, b ∈ N d , we have:

lim n→∞ E[G; H + S k(n)-m = s n -b] E[G; H + S k(n) = s n ] = 1.
Let (k(n), n ∈ N * ) be a sequence of elements in N d such that lim n→∞ |k(n)| = +∞ and lim n→∞ k(n)/|k(n)| = a. Since P(G(A) = 0) < 1 and thanks to [START_REF] Miermont | Invariance principles for spatial multitype Galton-Watson trees[END_REF], we deduce from Lemma 3.11 that:

lim n→+∞ E G(A); Sm A + S k(n)-b-m A = k(n) -b -e r E G(A); Sm A + S k(n)-m A = k(n) -e r = 1.
We also have:

lim n→+∞ B k(n)-b (m A ) B k(n) (m A ) = 1.
Since all the terms in ( 18) are non-negative, and lim n→+∞ i∈

[d] (k i (n)-b i )/k i (n) = 1, we deduce that: (19) lim n→+∞ P r (|τ | = k(n) -b) P r (|τ | = k(n)) = 1.
Then, using Lemmas 2.9 (with i = r in ( 6)), we obtain that, for all r ∈ [d], t ∈ T 0 and x ∈ L 0 (t) such that M(x) = r, lim n→+∞ P r (τ n ∈ T(t, x)) = P r (τ * ∈ T(t, x)). Of course we have for t ∈ T 0 and n large enough that P r (τ n = t) = 0 = P r (τ * = t). We deduce from Corollary 2.2 that (τ n , n ∈ N * ) converges in distribution towards τ * under P r for all r ∈ [d].

Let α be a probability distribution on [d]. Let t ∈ T 0 and x ∈ L 0 (t). Set r = M(∅ t ) and i = M(x). We have using (6) that:

P α (τ n ∈ T(t, x)) = α(r) P r (τ ∈ T(t, x), |τ | = k(n)) j∈[d] α(j) P j (|τ | = k(n)) = α(r)a * r j∈[d] α(j)a * i Γ j (n) P r (τ * ∈ T(t, x)),
where

Γ j (n) = P j (|τ | = k(n)) P i (|τ | = k(n) -|t| + e i )
,

with the convention that Γ j (n) = +∞ if P i (|τ | = k(n) -|t| + e i ) = 0. Let t ′ ∈ T 0 and
x ′ ∈ t ′ such that M(x) = i and P j (τ * ∈ T(t ′ , x ′ )) > 0 (which is possible thanks to Lemma 2.8). Using ( 6) and the convergence of (τ n , n ∈ N * ) towards τ * under P j , we deduce that lim n→+∞ P j (|τ |=k(n))

P i (|τ |=k(n)-|t ′ |+e i ) = a * j /a * i . Then use (19) (with r = i) to deduce that lim n→+∞ Γ j (n) = a * j /a * i for all j ∈ [d].
Using the definition (4) of α, we deduce that:

lim n→+∞ P α (τ n ∈ T(t, x)) = α(r)a * r j∈[d] α(j)a * j P r (τ * ∈ T(t, x)) = P α(τ * ∈ T(t, x)),
where P α(dτ * ) = r∈[d] α(r)P r (dτ * ).

3.4. Proof of Lemma 3.11. We assume (H 1 ). In particular, this implies that P(X i,1 = 0) > 0 for some i ∈ [d]. Without loss of generality, we can assume this holds for i = d:

P(X d,1 = 0) > 0.
Recall that a is the normalized left positive eigenvector of the mean matrix M such that |a| = 1. In particular a is a probability on

[d]. Set v d = 0 ∈ N d-1 and for i ∈ [d -1], set v i = (v (1) i , . . . , v (d-1) i ) ∈ N d-1 such that v (j) i = 1 {j=i} for j ∈ [d -1]. Let Y = (U, V ) be a random variable in N d × N d-1 such that for i ∈ [d], P(V = v i ) = a i , and the distribution of U conditionally on {V = v i } is p (i) .
Recall Definition 2.3 of an aperiodic probability distribution.

Lemma 3.12. Under (H 2 ), the distribution of Y on Z 2d-1 is aperiodic.

Proof. Recall v d = 0 ∈ N d-1
. Let H be the smallest subgroup of Z 2d-1 that contains supp (F )supp (F ), with F be the probability distribution of Y . In particular, we have that H contains (supp (p (i) )supp (p (i) )) × {v d } for all i ∈ [d] and thus their union. Since (H 2 ) holds, we deduce that H contains Z d × {v d }. This implies also that (0, v i ) belongs to H for all i ∈ [d], and thus

H = Z 2d-1 .
For x ∈ R d and z = (z 1 , . . . , z d ) ∈ R d , we set δ(x, z) = (x, z 1 , . . . , z d-1 ). By definition of Y and since a is the left eigenvector of the mean matrix with eigenvalue 1, we have E[Y ] = δ(a, a).

We consider (Y ℓ , ℓ ∈ N * ) independent random variables distributed as Y . We set

W n = n ℓ=1 Y ℓ . Let s ∈ N d and k ∈ (N * ) d .
We have:

(20)

P W |k| = δ(s, k) = D(k)P(S k = s) with D(k) = |k|! i∈[d] k i ! i∈[d] a i k i .
Recall G and H given in Lemma 3.11. We set

H ′ = δ(H, 0) ∈ N 2d-1 . We get for k, m, s and b in N d : (21) E[G; H + S k-m = s -b] E[G; H + S k = s] = D(k) D(k -m) E[G; H ′ + W |k|-|m| = δ(s, k) -δ(b, m)] E[G; H ′ + W |k| = δ(s, k)] •
Thanks to Lemma 3.12 and (H 2 ), the distribution of

Y on Z 2d-1 is aperiodic. Since 0 ≤ G ≤ |H| c , we also have 0 ≤ G ≤ |H ′ | c and P(G = 0) < 1. Let (k(n), n ∈ N * ) and (s n , n ∈ N * ) be two sequences in N d satisfying lim n→∞ |k(n)| = +∞ and lim n→∞ k(n)/|k(n)| = lim n→∞ s n /|k(n)| = a. Notice, this implies that lim n→∞ δ(s n , k(n))/|k(n)| = E[Y 1 ]
. We deduce from Lemma 4.11 that:

lim n→+∞ E[G; H ′ + W |k(n)|-|m| = δ(s n , k(n)) -δ(b, m)] E[G; H ′ + W |k(n)| = δ(s n , k(n))] = 1. Then notice that lim n→+∞ D(k(n))/D(k(n) -m) = 1 as lim n→+∞ k(n)/|k(n)| = a.
And use [START_REF] Rockafellar | Convex analysis[END_REF] to get:

lim n→+∞ E[G; H + S k(n)-m = s n -b] E[G; H + S k(n) = s n ] = 1.
This ends the proof of Lemma 3.11.

ri A = {x ∈ A; aff A B(x, δ) ⊂ A for some δ > 0}.
Notice that, for A convex, we have int

A = ri A if and only if aff A = R d . For a function f on R d taking its values in R {+∞}, its domain is defined by dom(f ) = {x ∈ R d : f (x) < ∞}.
Let F be a probability distribution on R d and X be a random variable on R d with distribution F . Denote by supp (F ) the closed support of F : x ∈ supp (F ) if and only if P(X ∈ B(x, δ)) = 0 for some δ > 0. Denote also by cv (F ) the convex hull of its support, aff (F ) and ri (F ) the affine hull and the relative interior of cv (F ). We define ϕ the log-Laplace of X taking values in (-∞, +∞] as:

(22) ϕ(θ) = log E e θ,X , θ ∈ R d .
The function ϕ is convex, ϕ(0) = 0 (which implies that ϕ is proper), and lower-semicontinuous (thanks to Fatou's lemma). Its conjugate, ψ, is defined by: ( 23)

ψ(x) = sup θ∈dom(ϕ) ( θ, x -ϕ(θ)) , x ∈ R d .
We recall that ψ is a lower-semicontinuous (proper) convex function. Since ϕ(0) = 0, we deduce that ψ is non-negative. We first give a general lemma on the domain of ψ.

Lemma 4.1. Let F be a probability distribution on R d . We have ri (F ) = ri dom(ψ). If ψ(x) = 0, then we have x ∈ ri dom(ψ).

Proof. Let x ∈ cl ri (F ) = cl cv (F ). According to the separation theorem, there exists θ ∈ R d and ε > 0 such that a.s. θ, Xx ≤ -ε. This gives that for all t > 0, ϕ(tθ)t θ, x ≤ -tε and thus ψ(x) ≥ sup t>0 tε = +∞. This implies that dom(ψ) ⊂ cl ri (F ). Let x ∈ ri (F ). By translation invariance, we can assume that x = 0. We set h(θ) = E[max(0, min(1, θ, X ))]. The function h is continuous and, since 0 ∈ ri (F ), it is non zero on A = {θ ∈ aff (F ), |θ| = 1}. Thus h has a strictly positive minimum on A. Since P( θ, X > 0) ≥ h(θ/|θ|) for θ = 0, we deduce that a = inf θ∈aff (F )\{0} log(P( θ, X > 0)) is finite. For θ ∈ R d , let θ F denote its orthogonal projection on aff (F ). If θ F = 0, then ϕ(θ) = 0, otherwise we have ϕ(θ) = ϕ(θ F ) ≥ log(P( θ F , X > 0)) ≥ a. We deduce that ϕ ≥ a and we get ψ(x) = ψ(0) ≤ -a. We deduce that x ∈ dom(ψ). This implies that ri (F ) ⊂ dom(ψ).

We deduce that ri (F ) ⊂ dom(ψ) ⊂ cl ri (F ), which gives that ri (F ) = ri dom(ψ). We denote by ∂(F ) = cl ri (F ) \ ri F the relative boundary of dom(ψ). Let x ∈ ∂(F ). Let X be a random variable with probability distribution F . According to the separation theorem, there exists q ∈ R d such that a.s. q, Xx ≤ 0 and P( q, Xx < 0) > 0. This implies that ϕ(q) < q, x and thus ψ(x) ≥ q, xϕ(q) > 0. This gives that ψ(x) = 0 implies x ∈ ri dom(ψ).

We have the following corollary. Corollary 4.2. Let X be a random variable on R d with probability distribution F . If X is integrable then E[X] belongs to ri dom(ψ) and ψ(E[X]) = 0.

Proof. Jensen's inequality implies that ϕ(θ) ≥ θ, E[X] . This gives θ, E[X]ϕ(θ) ≤ 0. Then use [START_REF] Spitzer | Principles of random walk[END_REF] and that ψ is non-negative to deduce that ψ(E[X]) = 0. Use Lemma 4.1 to conclude.

For θ ∈ dom(ψ), we define a probability measure on R d by: [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF] dP θ (X ∈ dx) = e θ,X -ϕ(θ) dP(X ∈ dx).

We denote by m θ and Σ θ the corresponding mean vector and covariance matrix if they exist, i.e:

(25)

m θ = E θ [X] = E[X e θ,X -ϕ(θ) ] = ∇ϕ(θ) and Σ θ = Cov θ (X, X).
We set I F = int dom(ϕ) the interior of the domain of the log-Laplace of F . Notice that X under P θ has small exponential moment for θ ∈ I F and its mean and covariance matrix are thus well-defined for θ ∈ I F . For a symmetric positive semi-definite matrix Σ, we denote by |Σ| its determinant. The elementary proof of the next lemma is left to the reader.

Lemma 4.3. Let F be a probability distribution on R d . For any compact set K ⊂ I F , we have:

(26) sup θ∈K |Σ θ | < +∞ and sup θ∈K E θ |X -m θ | 3 < +∞.
We set O F = int cv (F ) the interior of the convex hull of the support of F .

Lemma 4.4. Assume O F is non-empty and bounded. Then the application θ → m θ is one-toone from R d onto O F and continuous as well as its inverse. In particular, for any compact set K ⊂ O F , there exists r such that K ⊂ {m θ ; |θ| ≤ r}.

Proof. It is easy to check, using Hölder's inequality, that if O F is non-empty then ϕ is strongly convex on its domain. If O F is bounded, then X is also bounded and the function ϕ is finite on R d , so that dom(ϕ) = R d , as well as differentiable throughout R d . This implies that ϕ is smooth on R d in the sense of [START_REF] Rockafellar | Convex analysis[END_REF] Section 26. According to Theorem 26.5 in [START_REF] Rockafellar | Convex analysis[END_REF], this implies that ∇ϕ is oneto-one from R Lemma 4.5. Assume F is an aperiodic probability distribution on Z d . Then, we have that O F is non-empty and that for any compact set

K ⊂ I F , (27)
inf θ∈K |Σ θ | > 0.
Proof. Since F is aperiodic, we have aff (F ) = R d . This implies the first part of the lemma. Let θ ∈ I F be such that |Σ θ | = 0. Then there exists h ∈ R d \ {0} such that h, Σ θ h = 0. This implies that P θ -a.s. h, X = c with c = h, m θ . This equality also holds P-a.s. as the two probability measures P and P θ are equivalent. Since aff (F ) = R d , we get h = 0. Since this is absurd, we deduce that |Σ θ | > 0 for all θ ∈ I F . Then use the continuity of θ → |Σ θ | on I F to get the second part of the lemma. 4.2. Gnedenko's d-dimensional local theorem. Recall the definitions of ϕ, P θ , m θ and Σ θ given by ( 22), ( 24) and [START_REF] Stone | On local and ratio limit theorems[END_REF] and that I F = int dom(ϕ). The next theorem is an extension of the one-dimensional theorem of Gnedenko [START_REF] Gnedenko | On a local limit theorem of the theory of probability[END_REF], see also [START_REF] Rvaceva | On domains of attraction of multi-dimensional distributions[END_REF][START_REF] Stone | On local and ratio limit theorems[END_REF].

Theorem 4.6. Let F be an aperiodic probability distribution on Z d such that I F is non-empty. Let (X ℓ , ℓ ∈ N * ) be independent random variables with distribution F and set S n = n ℓ=1 X ℓ for n ∈ N * . Then for any compact subset K of I F , we have:

(28) lim n→∞ sup θ∈K sup s∈Z d n d/2 |Σ θ | 1/2 P θ (S n = s) -(2π) -d/2 e -zn(θ,s) 2 /2 = 0, with z n (θ, s) = n -1/2 Σ -1/2 θ (s -nm θ ).
The end of this section is devoted to the proof of Theorem 4.6.

Let K ⊂ I F be compact. Thanks to Lemmas 4.3 and 4.5, we have |Σ θ | > 0 and Σ -1/2 θ is well defined. We define:

(29) Y = n -1/2 Σ -1/2 θ (X 1 -m θ ) and f θ (t) = E θ e i t, Y .
By the inversion formula, we know that for s ∈ Z d :

(2π

) d P θ (S n = s) = (-π,π) d E θ e i u,Sn-s du = (-π,π) d E θ e i n 1/2 Σ 1/2 θ u, n -1/2 Σ -1/2 θ (Sn-s) du = (-π,π) d E θ e i n 1/2 Σ 1/2 θ u, Y n e -i n 1/2 Σ 1/2 θ u, zn(θ,s) du.
In order to simplify the notation, we shall write z for z n (θ, s). By considering the change of variable t = n 1/2 Σ 1/2 θ u, we obtain:

(2π) d P θ (S n = s) = n -d/2 |Σ θ | -1/2 J θ f θ (t) n e -i t, z dt,
where

J θ = {t ∈ R d : n -1/2 Σ -1/2 θ t ∈ (-π, π) d }.
We set:

I n (θ) = n d/2 |Σ θ | 1/2 P θ (S n = s) -(2π) -d/2 e -z 2 /2 . Notice that (2π) d/2 e -z 2 /2 = R d e -t 2 /2-i t,z dt.
We obtain:

(2π) d I n (θ) = R d 1 J θ (t)f θ (t) n -e -t 2
/2 e -i t,z dt.

Let (C n , n ∈ N * ) be a sequence of positive numbers such that:

(30) lim n→∞ C n = ∞ and lim n→∞ n -1/(12+6d) C n = 0.
We deduce, using the expression of Σ -1 θ based on the cofactors, that θ → Σ -1 θ is continuous on

I F . This implies that Σ -1/2 θ t 2 = t, Σ -1 θ t is continuous in (θ, t) on I F × R d . We deduce that: (31) c 1 := sup θ∈K, t =1 t, Σ -1 θ t < ∞. Set J 1 = {t ∈ R d ; t ≤ C n }, so that t ∈ J 1 implies n -1/2 Σ -1/2 θ t 2 ≤ n -1 c 1 t 2 ≤ n -1 c 1 C 2 n .
Thanks to (30), we get there exists n 1 finite, such that J 1 ⊂ J θ for all n ≥ n 1 and all θ ∈ K.

For ε ∈ (0, 1) and n ≥ n 1 , we obtain:

(32) (2π) d |I n (θ)| ≤ R d 1 J θ (t)f θ (t) n -e -t 2 /2 dt ≤ I n,1 (θ) + I n,2 (θ) + I n,3 (θ) + I n,4 , with I n,1 (θ) = J 1 |f θ (t) n -e -t 2 /2 |dt, I n,2 (θ) = J 2,θ |f θ (t)| n dt, I n,3 (θ) = J 3,θ |f θ (t)| n dt,
and

I n,4 = J c 1 e -t 2 /2 dt as well as J 2,θ = {t ∈ R d ; t > C n and n -1/2 Σ -1/2 θ t < ε}, J 3,θ = {t ∈ J θ ; n -1/2 Σ -1/2 θ t ≥ ε}.
The proof of the Theorem will be complete as soon as we prove the converge of the terms I n,i to 0 for i ∈ {1, . . . , 4} uniformly for θ ∈ K (notice the terms I n,i do not depend on s ∈ Z d ).

4.2.1. Convergence of I n,4 . Notice that I n,4 does not depend on θ. And we deduce from (30) that lim n→∞ I n,4 = 0.

4.2.2. Convergence of I n,3 . Set h(θ, u) = |E θ [e i u,X 1 ]| for u ∈ R d and L = {u ∈ [-2π + ε, 2π - ε] d ; u ≥ ε}.
Since F is aperiodic, we deduce from Proposition P8 in [23, p.75], that h(θ, u) < 1 for u ∈ L. Since h is continuous in (θ, t) on the compact set K × L, there exists δ < 1 such that h(θ, u) ≤ δ on K × L. We get for θ ∈ K:

I n,3 (θ) ≤ n d/2 |Σ θ | 1/2 (-π,π) d h(θ, u) n 1 { u ≥ε} du ≤ n d/2 |Σ θ | 1/2 (2π) d δ n ,
where we used that |f θ (t)| = h(θ, u) with t = n 1/2 Σ 1/2 θ u for the first inequality and that h is bounded by δ on {u ∈ (-π, π) d ; u ≥ ε}. Thanks to (26) we have sup θ∈K |Σ θ | < ∞ and since δ < 1, we get lim n→∞ sup θ∈K I n,3 (θ) = 0. 4.2.3. Convergence of I n,2 . From (26), we have (33)

a 2 := sup θ∈K E θ [ X 1 -m θ 2 ] < ∞ and a 3 := sup θ∈K E θ [ X 1 -m θ 3 ] < ∞.
Using c 1 defined in (31), we can choose ε small enough such that (34)

ε 2 a 2 + εa 3 c 1 < 1. Recall Y = n -1/2 Σ -1/2 θ (X 1 -m θ )
. By the symmetry of Σ θ , we get that

(35) E θ Y 2 = 1 n E θ X 1 -m θ , Σ -1 θ (X 1 -m θ ) = 1 n d j=1 d ℓ=1 Σ -1 θ (j, ℓ)Σ θ (ℓ, j) = d n •
Using similar computations, we obtain:

(36) E θ t, Y 2 = t 2 n •
Recall notations a 3 in (33) and c 1 in (31). For t ∈ J 2,θ , we get:

(37) E θ | t, Y | 3 ≤ n -3/2 Σ -1/2 θ t 3 E θ [ X 1 -m θ 3 ] ≤ t 2 n εa 3 c 1 ≤ t 2 n ,
where we used n -1/2 Σ -1/2 θ t < ε, (33) and (31) for the second inequality and (34) for the last. Recall a 2 given in (33). From (34) and since t ∈ J 2,θ , we get:

(38) E θ t, Y 2 ≤ n -1/2 Σ -1/2 θ t 2 E θ [ X 1 -m θ 2 ] ≤ ε 2 a 2 < 1.
We deduce that, for all θ ∈ K and t ∈ J 2,θ ,

|f θ (t)| = |E θ [e i t,Y ]| = 1 - E θ [| t, Y | 2 ] 2 -iE θ t,Y 0 v 0 s 0 e iu dudsdv ≤ 1 - E θ [| t, Y | 2 ] 2 + E θ | t,Y | 0 v 0 s 0 dudsdv = 1 - E θ [| t, Y | 2 ] 2 + E θ [| t, Y | 3 ] 6 ≤ 1 - t 2 2n + t 2 6n = 1 - t 2 3n ,
where we used that E θ [Y ] = 0 for the first equality, that E θ [ t, Y 2 ] ≤ 1 for the first inequality (see (38)) and (36) as well as (37) for the last inequality. Therefore, we get that: Since 0 ≤ x + e -x -1 ≤ x 2 /2 for x ≥ 0, we get for t ∈ J 1 :

I n,2 (θ) ≤ J 2,θ |f θ (t)| n dt ≤ J 2,θ
(40) ng(n, t) ≤ t 4 8n ≤ n -1 C 4 n .
Since E θ [Y ] = 0 and E θ t, Y 2 = t 2 /n, see (36), we deduce that:

h θ (n, t) = E θ e i t,Y -1 + i t, Y + t, Y 2 2 .
Let L n = n 1 4 . We have:

|h θ (n, t)| ≤ E θ e i t,Y -1 + i t, Y + t, Y 2 2 = E θ e i t,Y -1 + i t, Y + t, Y 2 2 ; X 1 -m θ < L n + E θ e i t,Y -1 + i t, Y + t, Y 2 2 ; X 1 -m θ ≥ L n ≤ 1 6 E θ | t, Y | 3 ; |X 1 -m θ | < L n + E θ t, Y 2 ; X 1 -m θ ≥ L n ,
where we used | e iα -1iα + α 2 2 | ≤ min(|α| 3 /6, α 2 ) for α ∈ R for the second inequality. We have:

E θ [| t, Y | 3 ; X 1 -m θ < L n ] = E θ t, Y 2 | t, n -1/2 Σ -1/2 θ (X 1 -m θ ) |; X 1 -m θ < L n ≤ n -1/2 t √ c 1 L n E θ t, Y 2 = n -3/2 t 3 √ c 1 L n ,
where we used c 1 defined in (31) for the inequality and (36) for the last equality. Hölder's inequality gives:

E θ t, Y 2 ; X 1 -m θ ≥ L n ≤ E θ | t, Y | 3 2/3 P θ ( X 1 -m θ ≥ L n ) 1/3 .
Using a 3 defined in (33), we get:

E θ | t, Y | 3 ≤ n -3/2 Σ -1/2 θ t 3 E θ X 1 -m θ 3 ≤ n -3/2 c 3/2 1 t 3 a 3 .
Using Tchebychev's inequality and a 2 defined in (33), we get:

P θ ( X 1 -m θ ≥ L n ) ≤ E θ X 1 -m θ 2 L -2 n ≤ a 2 L -2 n .
This gives:

E θ t, Y 2 ; X 1 -m θ ≥ L n ≤ n -1 c 1 t 2 a 2/3 3 a 1/3 2 L -2/3 n . For t ∈ J 1 , that is t ≤ C n , we get: n|h θ (n, t)| ≤ 1 6 n -1/4 C 3 n √ c 1 + n -1/6 c 1 C 2 n a 2/3 3 a 1/3 2 .
Using (39) and (40), we deduce there exists a constant c which does not depend on t, θ and n such that for t ∈ J 1 , θ ∈ K, we have:

|f θ (t) n -e -t 2 /2 | ≤ c(n -1/4 C 3 n + n -1/6 C 2 n + n -1 C 4 n ).
We deduce that for θ ∈ K:

I n,1 (θ) = J 1 |f θ (t) n -e -t 2 /2 | ≤ c(n -1/4 C 3 n + n -1/6 C 2 n + n -1 C 4 n )2 d C d n .
Recall that lim n→∞ n -1/(12+6d) C n = 0. This implies lim n→∞ sup θ∈K I n,1 (θ) = 0.

4.3.

Strong ratio limit theorem. Recall Definition 2.3 for an aperiodic probability distribution. Consider an aperiodic distribution F on Z d . Let X be a random variable with distribution F . Recall the function ϕ(θ) = log E[e θ,X ] defined in [START_REF] Rvaceva | On domains of attraction of multi-dimensional distributions[END_REF] and its conjugate ψ defined in [START_REF] Spitzer | Principles of random walk[END_REF]. We state the following strong ratio theorem, which is of interest by itself. However, in this paper we used the extension of the strong ratio theorem given in Section 4.5. We adapt the proof of Neveu [START_REF] Neveu | Sur le théorème ergodique de Chung-Erdős[END_REF]. We first state a preliminary lemma.

Lemma 4.9. Let F be an aperiodic probability distribution on Z d . Let (s n , n ∈ N * ) be elements of Z d satisfying (a) and (b) of Theorem 4.7. Then, for all b ∈ Z d and m ∈ Z, we have lim n→∞ ψ( sn+b n+m ) = 0. Proof. Assume that (a) and (b) of Theorem 4.7 hold. Let x be a limit of a converging subsequence of (s n /n, n ∈ N * ). Since ψ is lower-semicontinuous and non-negative, we deduce from (b) that ψ(x) = 0. Thus, the possible limits of sub-sequences of ((s n + b)/(n + m), n + m ≥ 1), which are also the the possible limits of sub-sequences of (s n /n, n ∈ N * ), are zeros of ψ. Then, using the second part of Lemma 4.1 and the continuity of ψ on the interior of its domain, we deduce that lim n→∞ ψ( sn+b n+m ) = 0. Since F is aperiodic, using elementary arithmetic consideration and Lemma 4.9, we see it is enough to prove (41) for m = 1 and b ∈ Z d satisfying p := P(X 1 = b) > 0.

We set N n = Card ({ℓ ≤ n; X ℓ = b}). Since for a ∈ Z d the conditional probability P(X ℓ = b|S n = a) does not depend on ℓ (when 1 ≤ ℓ ≤ n), we get:

E N n n S n = a = P(X n = b|S n = a) = p P(S n-1 = a -b) P(S n = a) •
For ε > 0, we have:

(42) P(S n-1 = a -b) P(S n = a) -1 = E Nn n ; S n = a pP(S n = a) -1 ≤ E[| Nn n -p|; S n = a] pP(S n = a) ≤ ε p + R n (a) p , with R n (a) = P(| Nn n -p| > ε) P(S n = a) •
Thus, the proof will be complete as soon as we prove that for all ε > 0, lim n→∞ R n (s n ) = 0. By Hoeffding's inequality, see Theorem 1 in [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF], since N n is binomial with parameter (n, p), we get:

(43) P N n n -p > ε ≤ 2 e -2nε 2 .
We give a lower bound of P(S n = s n ) in the next lemma, whose proof is postponed to the end of this section. Lemma 4.10. Let F be an aperiodic probability distribution on Z d . Let (X ℓ , ℓ ∈ N * ) be independent random variables with the same distribution F . Let S n = n ℓ=1 X ℓ for n ∈ N * . Then for 0 < η < 1, K 0 compact subset of O F , (s n , n ∈ N * ) a sequence of elements of Z d such that s n /n ∈ K 0 , there exists some n 0 ≥ 1 such that for n ≥ n 0 we have:

P(S n = s n ) e nψ(sn/n) ≥ (1 -η) n .
Using (43) and Lemma 4.10 with 1η = e -ε 2 , we get:

R n (s n ) = P Nn n -p > ε P(S n = s n )
≤ 2 e -nε 2 +nψ(sn/n) .

Since lim n→∞ ψ(s n /n) = 0 by assumption, we get the result.

Proof of Lemma 4.10. Since F is aperiodic, Lemma 4.5 implies that O F is non-empty. We first assume that the support of F is bounded. In particular the domain of ϕ defined by ( 22) is R d . Recall notation [START_REF] Stephenson | Local convergence of large critical multi-type Galton-Watson trees and applications to random maps[END_REF] as well as m θ = E θ [X] and Σ θ = Cov θ (X, X). Let K 0 be a compact subset of O F . According to Lemma 4.4, there exists a compact set K ⊂ R d such that K 0 ⊂ {m θ , θ ∈ K}. According to Theorem 4.6, we have that for all ε > 0, there exists n 0 such that for all n ≥ n 0 : Since s n /n belongs to {m θ ; θ ∈ K}, we get that inf θ∈K u n (θ, s n ) = 0. Thanks to (26) and Lemma 4.5, we can also choose ε > 0 and δ > 0 both small enough so that (2π) -d/2 sup q∈K |Σ q | -1/2 -(inf q∈K |Σ q |) -1/2 ε > δ. Then we deduce that for all n ≥ n 0 : This gives, for some δ > 0, for all n ≥ n 0 :

(44) P(S n = s n ) e nψ(sn/n) ≥ δn -d/2 > 0.

This gives Lemma 4.10 when the support of F is bounded.

Let F be a general aperiodic probability distribution on Z d , and X a random variable with distribution F . Let M > 0 so that δ M = P(|X| > M ) < 1. Let X M be distributed as X conditionally on {|X| ≤ M }. Let (X M ℓ , ℓ ∈ N) be independent random variables distributed as X M , and set S M n = n ℓ=1 X M ℓ . We have:

P(S M n = s n ) = P(S n = s n , |X ℓ | ≤ M for 1 ≤ ℓ ≤ n) P(|X| ≤ M ) n ≤ P(S n = s n ) (1 -δ M ) n •
Let F M be the probability distribution of X M and ϕ M defined by [START_REF] Rvaceva | On domains of attraction of multi-dimensional distributions[END_REF] with F replaced by F M and ψ M defined by [START_REF] Spitzer | Principles of random walk[END_REF] with ϕ replaced by ϕ M . Since F is aperiodic, we get that F M is aperiodic for M large enough. We get:

P(S n = s n ) e nψ(sn/n) ≥ P(S M n = s n ) e nψ(sn/n) (1δ M ) n = P(S M n = s n ) e nψ M (sn/n) e n∆ M (sn/n) , where we define ∆ M (s) = ψ(s) -ψM (s) and ψM (x) = sup θ∈R d ( θ, x -φM (θ)) with φM (θ) = log E e θ,X 1 {|X|≤M } so that ψM (x) = ψ M (x)log(1δ M ).

Notice that the sequence of continuous finite convex functions ( φM , M ∈ N * ) is non-decreasing and converges point-wise to the convex function ϕ (which is not identically +∞ as ϕ(0) = 0). By definition, the sequence of convex functions ( ψM , M ∈ N * ) is non-increasing and ψM ≥ ψ. Therefore the sequence converges to a function say ψ such that ψ ≥ ψ. Thanks to Theorem B.3.1.4 in [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF] or Theorem II.10.8 of [START_REF] Rockafellar | Convex analysis[END_REF], ψ is convex and ( ψM , M ∈ N * ) converges to ψ uniformly on any compact subset of ri dom( ψ). Theorem E.2.4.4 in [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF] gives that the closure of ψ (defined in Definition B.1.2.4 in [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF]) is equal to ψ. Thanks to Proposition 1.2.5 in [START_REF] Auslender | Asymptotic cones and functions in optimization and variational inequalities[END_REF], we get that ri dom( ψ) = ri dom(ψ) and on this set we have ψ = ψ. Since ri dom(ψ) = ri (F ) = O F , see Lemmas 4.1 and 4.5, this implies that lim M →+∞ ∆ M = 0 uniformly on any compact subset of O F .

Notice that ∆ M ≤ 0. Therefore for any γ > 0, K 0 compact subset of O F , there exists M 0 such that for M ≥ M 0 , 0 ≥ ∆ M ≥ -γ on K 0 . We deduce from (44) with S n and ψ replaced by S M n and ψ M that for some δ > 0 and γ > 0, there exists n 0 ≥ 1 such that for all n ≥ n 0 : P(S n = s n ) e nψ(sn/n) ≥ δn -d/2 e -γn . This completes the proof. -1 = 0, which gives the result.

2. 2 .

 2 Notations for marked trees. Let d ∈ N * . Denote by [d] the set of types or marks, by U = n≥0 (N * ) n the set of finite sequences of positive integers with the convention (N * ) 0 = { ∅} and by U = n≥0 (N * ) n × [d] the set of finite sequences of positive integers with a type. For a marked individual u ∈ U , we write u = (û, M(u)) with û ∈ U the individual and M(u) ∈ [d] its type or mark. Let |u| = |û| be the length or height of u defined as the integer n such that û = (u 1 , . . . , u n ) ∈ (N * ) n . If û and v are two sequences in U, we denote by ûv the concatenation of the two sequences, with the convention that ûv = û if v = ∅ and ûv = v if û = ∅. For u, v ∈ U , we denote by uv the concatenation of u and v such that uv = ûv and M

Lemma 2 . 8 .

 28 Let p be an offspring distribution of d types satisfying (H 1 ) and α a probability distribution on [d]. Then a.s. the multi-type Kesten tree τ * belongs to T ′ 1 .

  . For k ∈ N d and r ∈ [d], we define the matrix S(k, r) = (S ij (k, r); i, j ∈ [d]) of size d × d by:

3. 3 .

 3 Proof of Theorem 3.1. We assume that (H 1 ) and (H 2 ) hold. Let r ∈ [d]. Let b ∈ N d . We have, using Lemma 3.8 and Corollary 3.10, for every k ≥ b + 1 such that P r (|τ | = k) > 0:

4 . Appendix 4 . 1 .

 441 Preliminary results. For x ∈ R d and δ ≥ 0, let B(x, δ) be the open ball of R d centered at x with radius δ. For any non-empty subset A of R d , denote: cv A the convex hull of A, cl A the closure of A, int A the interior of A, aff A = x 0 + span (Ax 0 ) the affine hull of A where x 0 ∈ A and, if A is convex, ri A the relative interior of A:

  d onto the open set D = ∇ϕ(R d ), continuous, as is ∇ϕ -1 . Furthermore, according to Corollary 26.4.1 in [21], we have ri dom(ψ) ⊂ D ⊂ dom(ψ). Since D is open, we deduce that D = ri dom(ψ) = int dom(ψ). Then, use Lemma 4.1 to get that D = ri (F ) = O F . Recall Definition 2.3 for an aperiodic probability distribution.

Theorem 4 . 7 . 4 . 4 .

 4744 Let F be an aperiodic probability distribution on Z d . Let (X ℓ , ℓ ∈ N * ) be independent random variables with the same distribution F . Let S n = n ℓ=1 X ℓ for n ∈ N * . For all m ∈ N and b ∈ Z d , we have:(41) lim n→∞ P(S n-m = s nb) P(S n = s n ) = 1,where the sequence (s n , n ∈ N * ) of elements of Z d satisfies the following conditions:(a) sup n∈N * | sn n | < ∞, (b) lim n→∞ ψ( sn n ) = 0.Remark 4.8. Assume that X, with distribution F , is integrable. Thanks to Corollary 4.2, E[X] belongs to ri dom(ψ), the relative interior of the domain of ψ and ψ(E[X]) = 0. According to Theorem 1.2.3 in [4], the function ψ is relatively continuous on ri dom(ψ). Therefore if the sequence (s n , n ∈ N * ) of elements of dom(ψ) satisfies lim n→∞ s n /n = E[X], then (a) and (b) of Theorem 4.7 are satisfied. Notice also that if F is aperiodic (as assumed in Theorem 4.7), then Lemmas 4.5 and 4.1 imply ri dom(ψ) is the (non-empty) interior of dom(ψ) which is also equal to O F = int cv (F ). Proof of Theorem 4.7.

- 1 / 2 e- 1 / 2 e

 1212 sup θ∈K sup s∈Z d n d/2 |Σ θ | 1/2 P θ (S n = s) -(2π) -d/2 e -un(θ,s) < ε, with u n (θ, s) = snm θ , Σ -1 θ (snm θ ) 2n •So we get that for all n ≥ n 0 , θ ∈ K:P θ (S n = s n ) ≥ (2πn) -d/2 |Σ θ | -1/2 e -un(θ,sn) -n -d/2 |Σ θ | -1/2 ε ≥ (2πn) -d/2 sup q∈K |Σ q | -un(θ,sn) -n -d/2 inf q∈KWe deduce that for all n ≥ n 0 :sup θ∈K P θ (S n = s n ) ≥ (2πn) -d/2 sup q∈K |Σ q | -inf θ∈K un(θ,sn) -n -d/2 inf q∈K

  θ∈R d P θ (S n = s n ) ≥ sup θ∈K P θ (S n = s n ) ≥ n -d/2 δ > 0.Using[START_REF] Spitzer | Principles of random walk[END_REF], we get: supθ∈R d P θ (S n = s n ) = sup θ∈R d P(S n = s n ) e θ,sn -nϕ(θ) = P(S n = s n ) e nψ(sn/n) .

4. 5 . 1 .

 51 An extension of Theorem 4.7. We shall need the following extension of Theorem 4.7. Lemma 4.11. Let F be a probability distribution on N d ′ which is aperiodic on Z d ′ . Let (Y n , n ∈ N * ) be independent random variables distributed according to F and set W n = n ℓ=1 Y ℓ for n ∈ N * . Assume that E[|Y 1 |] < +∞. Let G and H ′ be two random variables in N and N d ′ respectively and independent of(Y n , n ∈ N * ) such that P(G = 0) < 1 and a.s. G ≤ |H ′ | c for some c ≥ 1. Let (w n , n ∈ N * ) be a sequence of N d ′ such that lim n→+∞ w n /n = E[Y 1 ].Then for any given ℓ ∈ N and b ∈ N d ′ , we have:(45) lim n→∞ E[G; H ′ + W n-ℓ = w nb] E[G; H ′ + W n = w n ] =Proof. Since F is aperiodic and by elementary arithmetic consideration, it is enough to prove (45) for ℓ = 1 and b ∈ N d ′ satisfying p = P(Y 1 = b) > 0. Let ε > 0. Using similar arguments as in (42), we get:E[G; H ′ + W n-1 = w nb] E[G; H ′ + W n = w n ] Nn n -p| > ε, H ′ + W n = w n E[G; H ′ + W n = w n ] ,withN n = n ℓ=1 1 {Y ℓ =b} . Choose g ∈ N * and h ∈ N d ′ such that q = P(G = g, H ′ = h) > 0. We have: R n ≤ |w n | c P Nn np > ε gqP(W n = w nh) ≤ |w n | c 2 e -2nε2 gqP(W n = w nh) , where we used G ≤ |H ′ | c a.s. and that H ′ +W n = w n implies H ′ ≤ w n for the first inequality, and inequality (43) in the Appendix for the second. Notice that for all ε ′ > 0 we have |w n | c ≤ exp(ε ′ n) for n large enough. Then use Lemma 4.10 and Remark 4.8 to conclude that if lim n→+∞ w n /n = E[Y 1 ], then lim n→+∞ R n = 0. Since ε > 0 is arbitrary, we get lim n→+∞ E[G; H ′ +W n-1 =wn-b] E[G; H ′ +Wn=wn]

  = ∞, we deduce that lim n→∞ sup θ∈K I n,2 (θ) = 0. 4.2.4. Convergence of I n,1 . Since |f θ (t)| ≤ 1, we have: (39) |f θ (t) ne -t 2 /2 | ≤ n|f θ (t)e -t 2 /(2n) | ≤ n|h θ (n, t)| + ng(n, t),

				1 -	t 2 3n	n	dt ≤	t >Cn	e -t 2 /3 dt.
	Since lim
	where	h θ (n, t) = f θ (t) -1 +	t 2 2n	and g(n, t) = e -t 2 /(2n) -1 +	t 2 2n	.

n→∞

C n

Acknowledgements. The authors would like to thank Jean-Philippe Chancelier for pointing out the references on convex analysis and his valuable advice as well as the two anonymous referees for their comments and suggestions. H. Guo would like to express her gratitude to J.-F. Delmas for his help during her stay at CERMICS.

The research has also been supported by the ANR-14-CE25-0014 (ANR GRAAL).