and Jean-Franc ¸ois Delmas

https://doi.org/10.1051/ps/2019026 www.esaim-ps.org

VERY FAT GEOMETRIC GALTON-WATSON TREES

Romain Abraham

, Aymen Bouaziz

and Jean-Franc ¸ois Delmas

Abstract. Let τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on{Zn=an}whereZn is the size of thenth generation and (an, n∈N^∗) is a deterministic positive sequence. We study the local limit of these trees τn as n→ ∞ and observe three distinct regimes: if (an, n∈N^∗) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈N^∗) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.

Mathematics Subject Classification.60J80; 60F15, 05C05.

Received October 5, 2018. Accepted November 12, 2019.

1. Introduction

A Galton-Watson (GW for short) process (Zn, n≥0) describes the size of an evolving population where, at each generation, every extant individual reproduces according to the same offspring distributionpindependently of the rest of the population. The associated genealogical tree τ is called a GW tree. Letµ denote the mean number of offspring per individual, that is the mean of p. Whenpis non degenerate, a classical result states that ifµ <1 (sub-critical case) orµ= 1 (critical case), then the population becomes a.s. extinct (i.e.Zn = 0 for some n≥0 a.s.) whereas if µ >1 (super-critical case), the population has a positive probability of non extinction.

Another classical result from Kesten’s work [7] describes the local limit in distribution of a critical or sub- critical GW tree conditioned on {Zn >0} as n→ ∞, which can be seen as a critical or sub-critical GW tree conditioned on non-extinction. The limiting tree is the so-called sized-biased tree or Kesten tree, and it can also be viewed as a two-type GW tree.

There are other ways of conditioning the tree of being large: conditioning on having a large total population size, or a large number of leaves... In the critical case, all these conditionings lead to the same local limit, see [2]

and the references therein. In the sub-critical case, a condensation phenomenon (i.e.a vertex with an infinite number of offspring at the limit) may happen. This phenomenon has been pointed out first in [6], see also [5],

Keywords and phrases: Galton-Watson tree, random discrete tree, local limit, non extinction, branching process; geometric Galton-Watson process.

1 Institut Denis Poisson, Universit´e d’Orl´eans, Universit´e de Tours, CNRS, B.P. 6759, 45067 Orl´eans cedex 2, France.

2 Institut Pr´eparatoire aux ´Etudes Scientifiques et Techniques, La Marsa, 2070-Tunis.

3 Universit´e Paris-Est, ´Ecole des Ponts, CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vall´ee, France.

* Corresponding author:[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2020

[1] or [8]. But even there, there can be only two different limiting trees, a size-biased GW tree or a condensation tree.

In order to have different limits, an idea is to condition the tree to be even bigger,i.e.to consider conditionings of the form {Zn =an} for some positive deterministic sequence (an, n∈N^∗) possibly converging to infinity.

Some results on branching processes conditioned on their limit behaviour already appeared in previous works, see for instance [10] where the distributions of the conditioned Yule process (which corresponds to a super- critical branching process) or a critical binary branching process are described via an infinitesimal generator and a martingale problem. The first study of local limits for GW trees with such a conditioning appears in [2]

where it is proven that, if pis a critical offspring distribution with finite variance, then the tree conditioned on {Z_n=a_n} converges in distribution to the associated sized-biased tree if and only if lim_n→∞a_nn⁻²= 0.

The goal of this paper is to study what happens beyond that condition and to consider the sub-critical and super-critical cases. We give a complete description of all the cases when the offspring distribution is a geometric distribution with a Dirac mass at 0 (in that case, the distribution of Zn is explicit). We observe three regimes according to the speed of growth of (an, n∈N^∗). We set:

cn=







µ⁻ⁿ ifµ <1 (sub-critical case), n² ifµ= 1 (critical case), µⁿ ifµ >1 (super-critical case), and we shall consider that:

n→∞lim an

cn

=θ∈[0,+∞].

Let τ^0,0 denote the GW tree τ conditioned on the extinction event E =S

n∈N^∗{Z_n = 0}. Notice that τ^0,0 is distributed asτ in the sub-critical and critical cases.

– In theKesten regime(θ= 0), the limiting tree,τ⁰, is the Kesten tree, which is a two-type GW tree, with an infinite spine corresponding to the individuals having an infinite progeny (called the survivor type), on which are grafted independent GW trees distributed asτ^0,0 corresponding to individuals having a finite progeny (called extinction type).

– In thePoisson regime (θ∈(0,+∞)), the limiting tree, τ^θ, is no more a GW tree, but it still has two types, with a backbone without leaves corresponding to individuals having an infinite progeny (also called the survivor type), on which are grafted independent GW trees distributed asτ^0,0. However, the backbone can not be seen as a GW tree, as it lacks the branching property. This is more like a random tree with a Poissonian immigration at each generation with rates depending on θ and with all the configurations having the same probability.

– In the condensation regime (θ = +∞), the limiting tree τ^∞ is again a two-type GW tree, with a backbone without leaves corresponding to individuals having an infinite progeny (also called the survivor type), on which are grafted independent GW trees distributed asτ^0,0. The backbone can be seen as an inhomogeneous GW tree with the root having an infinite number of children (condensation regime), and super-critical offspring distribution at levelh >0 with finite meanµh which decreases to 1 as hgoes to infinity.

We also prove that the family (τ^θ, θ∈[0,+∞]) is continuous in distribution (the most interesting cases are the continuity at 0 and +∞), see Remark5.2 and Proposition6.3.

Remark 1.1. The main ingredient of the proofs is equation (2.3) and hence is the limit of the ratio

n→+∞lim

Pk(Z_n−h=an) P(Z_n=a_n)

which is closely related to the extremal space-time harmonic functions associated with the GW process, see [10]. This limit is computed in the Kesten regime at the end of the proof of Proposition4.2, and at the end of the proof of Proposition 5.3 in the Poisson regime. In the condensation regime, this limit is 0. Notice that in this regime, the conditioned Galton-Watson process converges to a trivial process which is always equal to +∞

(except atn= 0) but considering the genealogical tree gives a non-trivial limit.

Partial results in a more general setting for super-critical and some sub-critical cases are given in [3]: conver- gence of τn in the Kesten and the Poisson regimes for general offspring distributions, and in the condensation regime in the Harris case (offspring distribution with bounded support), the continuity in distribution of the family of limiting trees at θ= 0 and some partial results atθ= +∞. Some similar results can also be derived for sub-critical offspring distributions under strong additional assumptions.

The rest of the paper is organized as follows: Section 2 introduces the framework of discrete trees with the notion of local convergence for sequences of trees, the GW trees and some properties of the geometric distribution. Section3describes the GW tree with geometric offspring distribution with some technical lemmas that are used in the proofs of the main theorems. Section4studies the Kesten regime, where the Kesten treeτ⁰ is defined and the convergence in distribution ofτntoτ⁰is stated (Prop.4.2). In Section5, the family of random trees (τ^θ, θ∈(0,+∞)) is introduced and a convergence result is obtained for the Poisson regime (Prop. 5.3) as well as the continuity in distribution of (τ^θ, θ∈(0,+∞)) at θ= 0 (Rem.5.2). Finally, Section 6 introduces the condensation tree τ^∞, proves the convergence ofτn to τ^∞ in the condensation regime (Prop.6.4) and the continuity in distribution of (τ^θ, θ∈(0,+∞)) atθ= +∞(Prop.6.3).

2. Notations

We denote byN={0,1,2, . . .}the set of non-negative integers, byN^∗={1,2, . . .}the set of positive integers and ¯N=N∪ {+∞}. For any finite setE, we denote by]Eits cardinal.

2.1. The set of discrete trees

We recall Neveu’s formalism [9] for ordered rooted trees. LetU =S

n≥0(N^∗)ⁿ be the set of finite sequences of positive integers with the convention (N^∗)⁰={∅}. We also setU^∗=S

n≥1(N^∗)ⁿ=U \{∅}.

For u∈ U, let |u| be the length or the generation of u defined as the integer n such that u∈(N^∗)ⁿ. If u and v are two sequences of U, we denote by uv the concatenation of two sequences, with the convention that uv=vu=uifv=∅.

The set of strict ancestors of u∈ U^∗ is defined by:

Anc(u) ={v∈ U, ∃w∈ U^∗, u=vw}, and forS ⊂ U^∗, being non-empty, we set Anc(S) =S

u∈S Anc(u).

A treetis a subset ofU that satisfies : – ∅ ∈t.

– Ifu∈t, then Anc(u)⊂t.

– For everyu∈t, there existsku(t)∈N¯ such that, for every positive integeri,ui∈t ⇐⇒ 1≤i≤ku(t).

We denote by T∞ the set of trees. Let t∈T∞ be a tree. The vertex ∅ is called the root of the tree tand we denote by t^∗=t\{∅}the tree without its root. For a vertexu∈t, the integer ku(t) represents the number of offspring (also called the out-degree) of the vertexu∈t. By convention, we shall writeku(t) =−1 ifu6∈t.

The heightH(t) of the treetis defined by:

H(t) = sup{|u|, u∈t} ∈N¯.

Forn∈N, the size of then-th generation oftis defined by:

zn(t) =]{u∈t,|u|=n}.

We denote byT^∗f the subset of trees with finite out-degrees except the root’s:

T^∗f ={t∈T∞;∀u∈t^∗, ku(t)<+∞}

and byTf ={t∈T^∗f;k_∅(t)<+∞} the subset of trees with finite out-degrees.

Leth, k∈N^∗. We defineT^(h)f the subset of finite trees with heighth:

T^(h)f ={t∈Tf;H(t) =h}

and T^(h)k ={t∈T^(h)f ; k_∅(t) =k} the subset of finite trees with height equal to hand out-degree of the root equal tok. We also define the restriction operators rhandrh,k, for everyt∈T∞, by:

rh(t) ={u∈t; |u| ≤h} and rh,k(t) ={∅} ∪ {u∈rh(t)^∗;u1≤k},

where u1 represents the first term of the sequence u if u 6=∅. In other words, rh(t) represents the tree t truncated at heighthandr_h,k(t) represents the subtree ofr_h(t) where only thek-first offspring of the root are kept. Remark that, fort∈Tf, ifH(t)≥hthenrh(t)∈T^(h)f and if furthermorek_∅(t)≥k thenrh,k(t)∈T^(h)k .

2.2. Convergence of trees

SetN1={−1} ∪N¯, endowed with the usual topology of the one-point compactification of the discrete space {−1} ∪N. For a treet∈T∞, recall that by convention the out-degreeku(t) ofuis set to -1 ifudoes not belong to t. Thus a tree t∈T∞ is uniquely determined by the sequence (ku(t), u∈ U) and then T∞ is a subset of N^U1. By Tychonoff theorem, the setN^U1 endowed with the product topology is compact. SinceT∞ is closed it is thus compact. In fact, the setT∞ is a Polish space (but we don’t need any precise metric at this point). The convergence of sequences of trees is then characterized as follows. Let (tn, n∈N) andtbe trees inT∞. We say that lim_n→∞t_n=tif and only if lim_n→∞k_u(t_n) =k_u(t) for allu∈ U. It is easy to see that:

– If (t_n, n∈N) andtare trees inTf, then we have lim_n→∞t_n =tif and only if lim_n→∞r_h(t_n) =r_h(t) for allh∈N^∗.

– If (tn, n∈N) andtare trees inT^∗f, then we have lim_n→∞tn =tif and only if lim_n→∞rh,k(tn) =rh,k(t) for allh, k∈N^∗.

LetT be aTf-valued (resp.T^∗f-valued) random variable. It is easy to get that if a.s.H(T) = +∞(resp. a.s.

H(T) = +∞andk_∅(T) = +∞), then the distribution ofT is characterized by

P(rh(T) =t);h∈N^∗,t∈T^(h)f

(resp.

P(r_h,k(T) =t);h, k∈N^∗,t∈T^(h)k

). Using the Portmanteau theorem, we deduce the following results:

– Let (T_n, n∈N) andT be Tf-valued random variables. Then we have the following characterization of the convergence in distribution if a.s.H(T) = +∞:

Tn

−−−−→(d)

n→∞ T ⇐⇒ lim

n→∞P(rh(Tn) =t) =P(rh(T) =t) for allh∈N^∗,t∈T^(h)f . (2.1)

– Let (Tn, n∈N) andT beT^∗f-valued random variables. Then we have the following characterization of the convergence in distribution if a.s.H(T) = +∞,k_∅(T) = +∞:

Tn

−−−−→(d)

n→∞ T ⇐⇒ lim

n→∞P(rh,k(Tn) =t) =P(rh,k(T) =t) for allh, k∈N^∗,t∈T^(h)k . (2.2) 2.3. GW trees

Letp= (p(n), n∈N) be a probability distribution onN. ATf-valued random variableτ is called a GW tree with offspring distributionpif for allh∈N^∗andt∈Tf withH(t)≤h:

P(r_h(τ) =t) = Y

u∈rh−1(t)

p(k_u(t)).

The generation size process defined by (Z_n=z_n(τ), n∈N) is the so called GW process. We refer to [4] for a general study of GW processes. We setPk the probability under which the GW process (Z_n, n∈N) starts with Z₀=kindividuals and write PforP1 so that:

Pk(Zn =a) =P(Z_n⁽¹⁾+· · ·+Z_n^(k)=a), where the (Z⁽ⁱ⁾,1≤i≤k) are independent copies ofZ underP.

We consider a sequence (an, n∈N^∗) of elements in N^∗ and, when P(Zn =an)>0, τn a random tree dis- tributed as the GW treeτ conditionally on{Zn=an}. Letn≥h≥1 andt∈T^(h)_f . We have by the branching property of GW-trees at heighth, settingk=zh(t):

P(rh(τn) =t) =P(rh(τ) =t)Pk(Z_n−h=an)

P(Z_n =a_n) · (2.3)

2.4. Geometric distribution

Letη∈(0,1] andq∈(0,1). We define the geometric G(η, q) distributionp= (p(k), k∈N) by (p(0) = 1−η,

p(k) =ηq(1−q)^k−1 fork∈N^∗. (2.4)

We shall always consider thatτ is a GW tree with geometric offspring distributionG(η, q).

The mean ofG(η, q) is given byµ=η/qand its generating functionf is given by:

f(s) = (1−η)−s(1−q−η)

1−s(1−q) , s∈[0,1/(1−q)).

We set:

γ= 1

1−q and κ= 1−η

1−q (2.5)

where γ is the radius of convergence of f and κ and 1 are the only fixed points of f on [0, γ). If µ= 1 then there is only one fixed point asκ= 1. We shall use frequently the following relations:

γ−κ=µ(γ−1) and, ifµ6= 1, γ−1 = κ−1

1−µ· (2.6)

Notice that κ∈[0,+∞) andγ∈(1,+∞) allow to recoverη andq as:

η= 1−κ

γ and q= 1−1

γ· (2.7)

For this reason, we shall also write G[κ, γ] forG(η, q). Notice that if µ <1, then q > η and γ > κ >1; and if µ >1, thenη > qandγ >1> κ≥0.

Sincef is an homography, we get fors∈[0, γ)\{1}:

f(s)−κ f(s)−1 = 1

µ s−κ

s−1· (2.8)

We set f1=f and, for n∈N^∗, fn+1=f◦fn. Notice thatκis a fixed point of fn as it is a fixed point off. We deduce from (2.8) and the second equality of (2.6) ifµ6= 1 and by direct recurrence ifµ= 1, thatfn, for n∈N^∗, is the generating function of the geometric distributionG[κ, γn] =G(ηn, qn) with meanµn =µⁿ and, thanks to (2.7):

η_n = 1− κ

γ_n, q_n = 1− 1

γ_n withγ_n=





 κ−µⁿ

1−µⁿ = 1 + (γ−1)qⁿ⁻¹(q−η)

qⁿ−ηⁿ ifµ6= 1,

1 + (γ−1)_n¹ ifµ= 1.

(2.9)

By convention, we set f0 the identity function defined on [0,+∞) andγ0 = +∞so that for all n∈N, we have γn= limr→+∞f_n⁻¹(r) that is in shortγn=f_n⁻¹(∞). We deduce that for alln≥`≥0:

f_{`</sub>(γ<sub>n</sub>) =γ<sub>n−`}. (2.10)

We derive some asymptotics forγ_n for large n. It is easy to deduce from (2.9) that:

n→∞lim γn = max(1, κ) =

(κ ifµ≤1,

1 ifµ≥1. (2.11)

Using (2.6), we get for large n:

(γ_n−κ)(γ_n−1) =







µⁿ(κ−1)²+O(µ²ⁿ) ifµ <1, (γ−1)²n⁻² ifµ= 1, µ⁻ⁿ(κ−1)²+O(µ⁻²ⁿ) ifµ >1.

(2.12)

We derive from (2.9) the logarithm asymptotics of γn/γn−hfor givenh∈N^∗ and largen:

log(γ_n−h/γn) = log(γ_n−h)−log(γn) =







µ^n−h 1−µ^h

(κ−1)/κ+O(µ²ⁿ) ifµ <1, (γ−1)hn⁻²+O(n⁻³) ifµ= 1, µ⁻ⁿ µ^h−1

(1−κ) +O(µ⁻²ⁿ) ifµ >1.

(2.13)

We recall the following well-known equality which holds for all k∈N^∗ andr∈(0,1):

X

`≥k

`−1 k−1

r^`= r

1−r k

. (2.14)

And we end this section with an elementary lemma.

Lemma 2.1. Let (X`, ` ∈ N^∗) be independent random variables with distribution G(η, q) = G[κ, γ].

Fora≥k≥1:

P

k

X

`=1

X_`=a

!

=

k

X

i=1

k i

a−1 i−1

κ^k−i(γ−κ)ⁱ(γ−1)ⁱγ^−a−k.

Proof. We have:

P

k

X

`=1

X`=a

!

=

k

X

i=1

k i

P(X1= 0)^k−iP

i

X

`=1

X`=a, X`≥1 for`∈ {1, . . . , i}

!

(2.15)

=

k

X

i=1

k i

(1−η)^k−i a−1

i−1

(ηq)ⁱ(1−q)^a−i.

Then use (2.7) to conclude.

3. The geometric GW tree

Letτbe a GW tree with geometricG(η, q) offspring distributionpgiven by (2.4), withη∈(0,1] andq∈(0,1).

Recall that (Zn, n∈N) is the associated GW process.

For k∈N^∗, we denote byPk the distribution of the geometric GW forest composed ofk independent GW trees with geometric offspring distributionG(η, q), and writePforP1. For convenience, we shall underPdenote byZ^(k)= (Zn^(k), n∈N) a GW process distributed asZ= (Zn, n∈N) underPk. Forn∈N^∗, we set:

M_n=γ₁^−Z1γ_n^Zn. (3.1)

Since Zn has generating functionfn under P, we deduce from (2.10) that (Mn, n∈N^∗) is a martingale with M1= 1.

Forn > h≥1, we set:

b_n,h= γ_n

γn−h

an

. (3.2)

We shall use the following formula when lim_n→∞bn,h exists and belongs to (0,∞).

Lemma 3.1. Letn > h≥1 andk∈N^∗. We have:

Pk(Z_n−h=a_n) P(Zn=an) =bn,h

k

X

i=1

k i

κ^k−iGn,h(k, i), (3.3)

with

Gn,h(k, i) =

an−1 i−1

γn

γ_n−h^k

(γ_n−h−κ)ⁱ(γ_n−h−1)ⁱ

(γ_n−κ)(γ_n−1) · (3.4)

Proof. Letn > h≥1. SinceZn has distribution G[κ, γn], we obtain thanks to (2.5):

P(Zn=an) =ηnqn(1−qn)^an−1= (γn−κ)(γn−1)γ_n^−an−1.

Using that Z_n−h is under Pk distributed as the sum of k independent random variables with distribution G[κ, γ_n−h], we deduce from Lemma2.1that:

Pk(Zn−h=an) P(Zn=an) =

k

X

i=1

k i

an−1 i−1

κ^k−i (γn−h−κ)ⁱ(γn−h−1)ⁱ γ_n−h^an+k

γ_n^an+1 (γn−κ)(γn−1)

=b_n,h

k

X

i=1

k i

κ^k−iG_n,h(k, i).

This gives the result.

We shall use the following formula when lim_n→∞b_n,h= 0 and lim_n→∞a_n = +∞.

Lemma 3.2. Letn > h≥1,k0∈N^∗ andt∈T^(h)_k0 . We have, withan ≥k=zh(t):

P(r_h,k0(τ_n) =t) = 1−q

ηq P(r_h(τ) =t) γ_h^k−R¹_n,h(k)−R²_n,h(k)

, (3.5)

with αn = (γn−h−κ)(γn−h−1),xn =γn/γn−h and:

0≤R¹_n,h(k)≤b_n,h αn

1−x_nmax(1, κ)^k−12^2k−1 2 + αn

1−x_n k−1

+ (α_na_n)^k−1

!

, (3.6)

R²_n,h(k) = (κ+ 1−γ)Pk(Z_n−h=an)

P(Z_n=a_n) · (3.7)

Proof. Letn > h≥1,k₀∈N^∗andt∈T^(h)k₀. We setk=z_h(t). For every 1≤j≤k₀, we denote byt_jthe subtree rooted at thej-th offspring of the rooti.e.

u∈tj ⇐⇒ ju∈t.

In what follows, we denote by ˜Z⁽ⁱ⁾a process distributed as Z⁽ⁱ⁾and independent ofZ^(k). We have:

P(rh,k₀(τn) =t) =

+∞

X

i=0

p(i+k0)





k0

Y

j=1

P(rh−1(τ) =tj)





P(Z_n−h^(k) + ˜Z_n−1⁽ⁱ⁾ =an) P(Zn =an)

=P(rh(τ) =t)

+∞

X

i=0

(1−q)ⁱP(Z_n−h^(k) + ˜Z_n−1⁽ⁱ⁾ =a_n) P(Z_n =a_n) by the branching property

=P(rh(τ) =t)

+∞

X

i=1

1−q ηq p(i)

a_n

X

`=0

P(Z_n−h^(k) =`)P(Z<sub>n−1</sub><sup>(i)</sup> =a<sub>n</sub>−`) P(Z_n=a_n)

+P(rh(τ) =t)P(Z_n−h^(k) =an) P(Zn=an)

= 1−q

ηq P(r_h(τ) =t)(A+B),

where

A=p(0)P(Z_n−h^(k) =a_n) P(Z_n=a_n) +

a_n

X

`=0

P(Z_n−h^(k) =`)

+∞

X

i=1

p(i)P(Z_n−1⁽ⁱ⁾ =an−`) P(Z_n=a_n)

=

a_n

X

`=0

P(Z_n−h^(k) =`)

+∞

X

i=0

p(i)P(Z_n−1⁽ⁱ⁾ =an−`) P(Z_n =a_n) and

B= ηq

1−q−p(0)

P(Z_n−h^(k) =an) P(Z_n=a_n) · We have:

A=

an

X

`=0

P(Z_n−h^(k) =`)P(Zn=an−`) P(Zn =an) =

an

X

`=0

P(Z_n−h^(k) =`)γ<sup>`</sup>_n=

fn−h(γn)^k−R¹_n,h(k) ,

where we used thatk_∅(τ) has distributionpfor the first equality, thatZ_nhas distributionG[κ, γ_n] for the second one and thus P(Z_n=k) =η_nq_nγn^−(k−1), and for the last one that:

R¹_n,h(k) =X

`>0

P(Z_n−h^(k) =`+an)γ<sub>n</sub><sup>`+an</sup>. We have, with αn = (γ_n−h−κ)(γ_n−h−1) and xn =γn/γ_n−h:

P(Z_n−h^(k) =`+a<sub>n</sub>)γ<sub>n</sub><sup>`+an</sup>=b_n,h

k

X

i=1

k i

`+a_n−1 i−1

κ^k−i(γ_n−h−κ)ⁱ(γ_n−h−1)ⁱγ_n−h^{−`−k</sup>γ<sub>n</sub><sup>`}

≤bn,hx^`_n max(1, κ)^k−1

k

X

i=1

k i

`+an−1 i−1

αⁱ_n,

where we used Lemma 2.1 for the first equality and γ_n−h ≥ max(1, κ) for the last. Using that (x+y)^j ≤ 2^j−1(x^j+y^j) forj∈N^∗ andx, y∈(0,+∞), we deduce that:

`+a_n−1 i−1

≤ 2ⁱ⁻¹

(i−1)! `ⁱ⁻¹+aⁱ⁻¹_n .

We have the following rough bounds:

0≤R¹_n,h(k)≤b_n,hmax(1, κ)^k−12^k−1

k

X

i=1

αⁱ_n k

i

X

`>0

`ⁱ⁻¹

(i−1)!x^{`</sup><sub>n</sub>+a<sup>i−1</sup><sub>n</sub> x<sup>`}_n

≤bn,h

xnαn

1−x_nmax(1, κ)^k−12^k−1

k

X

i=1

k i

αn

1−x_n i−1

+ (αnan)ⁱ⁻¹

!

≤b_n,h α_n 1−xn

max(1, κ)^k−12^2k−1 2 + α_n

1−xn

k−1

+ (α_na_n)^k−1

!

where we used thatx_n ∈(0,1) as the sequence (γ_m, m∈N^∗) is non-increasing and thatP

`>0`ⁱ⁻¹x^`/(i−1)!≤ x(1−x)ⁱ⁻¹for the last inequality but one. Then use (2.10), which givesfn−h(γn) =γh, to getA=γ_h^k−R¹_n,h(k) as well as (3.6).

We can rewrite the constant in B as ηq

1−q −p(0) = ηq

1−q−(1−η) =−(κ+ 1−γ),

so thatB=−R²_n,h(k), see (3.7), and thusA+B=γ_h^k−R¹_n,h(k)−R²_n,h(k). This ends the proof.

4. The Kesten regime or the not so fat case

In this section, we denote byτ a GW tree with geometricp=G(η, q) withη, q∈(0,1). It is well known that the extinction eventE ={H(τ)<+∞} has probability c= min(1, κ). Moreover, as we assumeη <1, we have c>0. We define the probability distributionp= (p(n), n∈N) by:

p(n) =cⁿ⁻¹p(n) forn∈N. (4.1)

We denote byτ^0,0a random tree distributed asτ conditionally on the extinction eventE, that is a GW tree with offspring distribution p. We denote by m the mean ofp. Ifµ≤1, then we havep=p,m=µ, c= 1 and that τ^0,0 is distributed as τ. Ifµ >1, then we have thatpis the geometric distribution G(q, η), m= 1/µ and c=κ.

Letk∈N^∗. We define thek-th order size-biased probability distribution ofpasp_[k]= (p_[k](n), n∈N) defined by:

p_[k](n) = n!

(n−k)!f^(k)(1)p(n) forn∈Nandn≥k. (4.2) The generating function of p[k] is f[k](s) =s^kf^(k)(s)/f^(k)(1). The probability distribution p[1] is the so-called size-biased probability distribution ofp.

For the distribution G(η, q), we have f^(k)(1) =k!ηq^−k(1−q)^k−1, so the k-th order size-biased probability distribution ofpis given by:

p_[k](n) = n

k

q^k+1(1−q)^n−k forn∈Nandn≥k. (4.3) We now define the so-called Kesten tree ˆτ⁰ associated with the offspring distributionpas a two-type GW tree where the vertices are either of type s (for survivor) or of type e (for extinction). It is then characterized as follows.

– The number of offspring of a vertex depends, conditionally on the vertices of lower or same height, only on its own type (branching property).

– The root is of type s.

– A vertex of type e produces only vertices of type e with offspring distributionp.

– The random number of children of a vertex of type s has the size-biased distribution of p that is p[1]

defined by (4.2) with k= 1. Furthermore, all of the children are of type e but one, uniformly chosen at random, which is of type s.

Informally the individuals of type s in ˆτ⁰ form an infinite spine on which are grafted independent GW trees distributed asτ^0,0.

We define τ⁰= Ske(ˆτ⁰) as the tree ˆτ⁰ when one forgets the types of the vertices. The distribution of τ⁰ is given in the following classical result.

Lemma 4.1. Letp=G(η, q)withη, q∈(0,1). The distribution ofτ⁰ is characterized by: for alln≥h≥1 and t∈T^(h)f withk=zh(t):

P(rh(τ⁰)) =t) =kc^k−1m^−hP(rh(τ) =t). (4.4) We give a short proof of this well-known result.

Proof. Since τ⁰ belongs to Tf and has infinite height, its distribution is indeed characterized by (4.4) for all n≥h≥1 andt∈T^(h)f withk=z_h(t).

Letn≥h≥1,t∈T^(h)_f andv∈tsuch that|v|=h. LetV be the vertex of type s at levelhin ˆτ⁰. We have, withk=zh(t):

P(r_h(τ⁰) =t, V =v) = Y

u∈t\Anc({v});|u|<h

p(k_u(t)) Y

u∈Anc({v})

1

ku(t) p_[1](k_u(t))

=m^−hc

P

u∈rh−1 (t)(ku(t)−1) Y

u∈rh−1(t)

p(ku(t))

=m^−hc^k−1P(rh(τ) =t),

where we used (4.2) (with k= 1, n=k_u(t) and p replaced by p) and (4.1) (with n=k_u(t)) for the second equality and thatP

u∈rh−1(t)(k_u(t)−1) =k−1 for the last one. Summing over allv∈tsuch that|v|=hgives the result.

4.2. Convergence of the not so fat geometric GW tree

We consider a sequence (an, n∈N^∗) withan∈N^∗ and a random tree τn distributed as the GW treeτ with offspring distributionp=G(η, q) conditionally on{Zn =an}. We have the following result.

Proposition 4.2. Letη ∈(0,1)andq∈(0,1). Assume that lim_n→∞anµⁿ= 0 ifµ <1,lim_n→∞ann⁻²= 0 if µ= 1orlim_n→∞anµ⁻ⁿ= 0 ifµ >1. Then we have the following convergence in distribution:

τn

−−−−→(d) n→∞ τ⁰.

The critical case,µ= 1, appears in Corollary 6.2 of [2] for general offspring distribution with second moment.

Proof. Leth∈N^∗andk∈N^∗. Recall the definitions ofbn,hin (3.2) and ofGn,hin (3.4). According to Lemma3.1, we have for n > h≥1 andk∈N^∗:

Pk(Z_n−h=a_n) P(Zn=an) =b_n,h

k

X

i=1

k i

κ^k−iG_n,h(k, i).

According to (3.2), we have bn,h= exp (−anlog(γn−h/γn)). We deduce from (2.13) and the hypothesis on (an, n∈N^∗) that limn→∞anlog(γn−h/γn) = 0 and thus limn→∞bn,h= 1. We deduce from (3.4), (2.11) and

(2.12) that, fork≥i >1, lim_n→∞Gn,h(k, i) = 0 and fork≥1:

n→∞lim Gn,h(k,1) =







κ^1−kµ^−h ifµ <1,

1 ifµ= 1,

µ^h ifµ >1.

We deduce that:

n→∞lim

Pk(Z_n−h=an) P(Z_n=a_n) =







kµ^−h ifµ <1 k ifµ= 1 kκ^k−1µ^h ifµ >1







=kc^k−1m^−h.

Then, as a.s.H(τ⁰) = +∞, we can use the characterization (2.1) of the convergence inTf, as well as (2.3) and Lemma 4.1to conclude.

5. The Poisson regime or the fat case

Letθ∈(0,+∞). We consider a two-type random tree ˆτ^θwhere the vertices are either of type s (for survivor) or of type e (for extinction). We defineτ^θ= Ske(ˆτ^θ) as the tree ˆτ^θwhen one forgets the types of the vertices of ˆ

τ^θ. We denote by Sh={u∈τ^θ;|u|=handuis of type s in ˆτ^θ} the set of vertices of ˆτ^θ with type s at level h∈N. Notice that (S`,0≤` < h) = Anc(Sh) and that ˆτ^θ is completely characterized byτ^θ and (Sh, h∈N).

Recallp defined by (4.1) and thek-th order size-biased distribution,p_[k], defined by (4.2). The random tree ˆτ^θ is defined as follows.

– The root is of type s (i.e.S0={∅}).

– The number of offspring of a vertex of type e does not depend on the vertices of lower or same height (branching property only for individuals of type e).

– A vertex of type e produces only vertices of type e with offspring distributionp (as in the Kesten tree).

– Forh∈N, let ∆h=]Sh+1−]Sh be the increase of number of vertices of type s between generations h andh+ 1. Conditionally on rh(τ^θ) and (S`,0≤`≤h), ∆h is distributed as a Poisson random variable with meanθζh, where:

ζ_h=







µ^−h−1(1−µ)(κ−1)/κ ifµ <1,

(γ−1) ifµ= 1,

µ^h(µ−1)(1−κ) ifµ >1.

(5.1)

We denote byκ^s(u) the number of children ofuof type s. Conditionally givenr_h(τ^θ), (S`,0≤`≤h) and

∆h, the vector (κ^s(u), u∈Sh) is uniformly distributed on the set of vectors of positive integers (ni,1≤ i≤]Sh) that sum to]Sh+ ∆h, each configuration having hence probability 1/ ^]S_]S^h+1−1

h−1

= 1/ ^]Sh+1_∆ ⁻¹

h

. (This breaks the branching property for the tree and the population process since the number of offspring of type s of a vertex depends on the size of the whole population of type s at the same level). Furthermore, conditionally onrh(τ^θ), Sh and (κ^s(v) =sv ≥1, v∈Sh), the vertexu∈Sh has κ^e(u) vertices of type e such that ku(τ^θ) =κ^s(u) +κ^e(u) has distribution p_[su] and the su individuals of type s are chosen uniformly at random among thek_u(τ^θ) children.

More precisely, forh∈N,n∈N,u∈Sh,k_u≥s_u≥1,A_u⊂ {1, . . . , k_u}with]A_u=s_u andP

u∈Shs_u= n+]Sh, we have withk=P

u∈Shk_u:

P κ^s(u) +κ^e(u) =ku andSh+1∩ {u1, . . . , uku}=uAu ∀u∈Sh|rh(τ^θ),Sh

= (θζh)ⁿ

n! e^−θζh 1

]Sh+n−1 n

Y

u∈Sh

1

k_u s_u

p_[su](ku)

= (]Sh−1)!

(]Sh+n−1)!(θ(γ−1)ζh)ⁿe^−θζh Y

u∈Sh

p(ku)

(µ^−]Sh ifµ≤1, µ^{]Sh µ}_κⁿ

ifµ >1,

(5.2)

where we used (4.3) and (4.1) as well as (2.7) for the last equality.

By construction, a.s. individuals of type s have a progeny which does not suffer extinction whereas individuals of type e have a progeny which suffers extinction. Since the individuals of type s do not satisfy the branching property, the random tree ˆτ^θ is not a multi-type GW tree. We stress out that ˆτ^θ truncated at level hcan be recovered fromrh(τ^θ) andSh as all the ancestors of a vertex of type s are also of type s and a vertex of type s has at least one child of type s.

We have the following result.

Lemma 5.1. Let η ∈ (0,1] and q ∈(0,1). Let θ ∈ (0,+∞). Let n ≥h ≥1 and t∈ T^(h)f . We have, with k=z_h(t):

P(rh(τ^θ) =t) =H(h, k, θ)P(rh(τ) =t), where H(h, k, θ)is equal to

µ^−he^{−θ(µ−h−1)(κ−1)/κ}

k

X

i=1

k i

θµ^−h(κ−1)²/κⁱ⁻¹

(i−1)! ifµ <1,

e^{−θ(γ−1)h}

k

X

i=1

k i

θ(γ−1)²i−1

(i−1)! ifµ= 1,

µ^he^{−θ(µh−1)(1−κ)}

k

X

i=1

k i

κ^k−i θµ^h(1−κ)²i−1

(i−1)! ifµ >1.

Remark 5.2. We deduce from Lemma4.1thatτ^θ −−−→^(d)

θ→0 τ⁰. Therefore the treesτ^θ appear as a generalization of the Kesten tree. We will also prove in Proposition6.3that a limit also exists whenθ→+∞.

Proof. We consider only the super-critical case. The sub-critical case and the critical case can be handled in a similar way.

Let h∈N^∗, t∈T^(h)f and S_h ⊂ {u∈t;|u|=h} be non empty. In order to shorten the notations, we set A= Anc(Sh). Notice that Ais tree-like. We set, for `∈ {0, . . . , h−1}, S` ={u∈ A,|u|=`} the vertices at level `which have at least one descendant inSh and ∆`=]S`+1−]S`. We recall that ˆτ^θ truncated at level h can be recovered from rh(τ^θ) andSh. We compute CS_h =P(rh(τ^θ) =t,Sh =Sh). We have, using (5.2) and (5.1):

CSh =





Y

u∈rh−1(t),u6∈A

p(k_u(t))





h−1

Y

`=0

"

(]S`−1)!

(]S<sub>`+1</sub>−1)!(θ(γ−1)ζ`)^{∆`</sup>e<sup>−θζ`}

"

Y

u∈S_`

p(ku(t))

#

µ^]S`µ κ

∆`

#

=



 Y

u∈rh−1(t)

p(ku(t))





θ(γ−1)(µ−1)(1−κ) κ

^Ph−1<sub>`=0</sub>∆`

(]Sh−1)! e^{−θPh−1`=1ζ`}

h−1

Y

`=0

µ^{(`+1)∆`+]S`}

=



 Y

u∈rh−1(t)

κ^ku(t)−1







 Y

u∈rh−1(t)

p(ku(t))





_θ(1−κ)2

κ

^]Sh−1

(]S_h−1)! e^{−θ(µh−1)(1−κ)} µ^h]Sh

=κ^zh(t)−]ShP(rh(τ) =t) µ^h θµ^h(1−κ)²]Sh−1

(]Sh−1)! e^{−θ(µh−1)(1−κ)}, where we used for the third equality that Ph−1

`=0∆` =]Sh−1, Ph−1

`=1ζ` = (µ^h−1)(1−κ) and Ph−1

`=0(`+ 1)∆`+]S`=Ph−1

`=0(`+ 1)]S`+1−`]S`=h]Sh. SinceCS_h depends only of]Sh, we shall writeC]S_h forCS_h. Set k=zh(t) =]{u∈t;|u|=h}. Since]Sh≥1 as the root if of type s, we obtain:

P(rh(˜τ^θ) =t) =

k

X

i=1

X

S_h⊂{u∈t;|u|=h}

1_{]Sh=i}CS_h =

k

X

i=1

k i

Ci=P(rh(τ) =t)H(h, k, θ),

where we used the definition ofHfor the last equality.

5.2. Convergence of the fat geometric GW tree

We consider a sequence (an, n∈N^∗), with an ∈N^∗ and τn a random tree distributed as the GW tree τ conditionally on{Zn=an}. We have the following result.

Proposition 5.3. Let η ∈(0,1], q ∈ (0,1) and θ ∈ (0,+∞). Assume that lim_n→∞anµⁿ =θ if µ < 1 or lim_n→∞ann⁻² =θ if µ = 1 or lim_n→∞anµ⁻ⁿ = θ if µ >1. Then we have the following convergence in distribution:

τn

−−−−→(d) n→∞ τ^θ.

Proof. Recall the definitions ofb_n,hin (3.2) and ofG_n,hin (3.4). According to Lemma3.1, we have forn > h≥1 andk∈N^∗:

Pk(Z_n−h=an) P(Z_n=a_n) =bn,h

k

X

i=1

k i

κ^k−iGn,h(k, i).

According to Definition (3.2), we have bn,h = exp (−anlog(γ_n−h/γn)). We deduce from (2.13) and the hypothesis on (an, n∈N^∗) that

n→∞lim −log(b_n,h) =







θ(µ^−h−1)(κ−1)/κ ifµ <1, θ(γ−1)h ifµ= 1, θ(µ^h−1)(1−κ) ifµ >1.

We deduce from (3.4), (2.11) and (2.12), that forh∈N^∗, k≥i≥1:

n→∞lim(i−1)!G_n,h(k, i) =







θµ^−h(κ−1)²i−1

µ^−hκ^1−k ifµ <1, θ(γ−1)²i−1

ifµ= 1, θµ^h(1−κ)²i−1

µ^h ifµ >1.

Using definition of Hin Lemma5.1, we obtain that:

n→∞lim

Pk(Z_n−h=a_n)

P(Zn =an) =H(h, k, θ).

Then use the characterization of the convergence inTf, (2.3) and Lemma 5.1to conclude.

6. The condensation regime or the very fat case

Recallγn defined in (2.9). Forn∈N^∗, we define the probability ˜pn= (˜pn(k), k∈N) by:

˜

p_n(k) =γ_n+1^k γn

p(k).

Thanks to (2.10), we get P

k∈Np˜n(k) =f(γn+1)γ_n⁻¹ = 1, so that ˜p is indeed a probability distribution onN. Forn= 0, we set ˜p0the Dirac mass at +∞, which is a degenerate probability measure on ¯N.

We define τ^∞ as an inhomogeneous GW tree with reproduction distribution ˜ph at generation h∈N. In particular the root has an infinite number of children, whereas all the other individuals have a finite number of children. More precisely, for all h∈N^∗,k0∈N^∗ andt∈T^(h)k₀ , we have:

P(rh,k₀(τ^∞) =t) = Y

u∈rh−1(t)^∗

˜

p_|u|(ku(t)), (6.1)

where we recall that t^∗=t\ {∅}. Remark that a.s.τ^∞∈T^∗f.

We give a representation of the distribution ofτ^∞ as the distribution ofτ with a martingale weight.

Lemma 6.1. Let η∈(0,1]andq∈(0,1). For all h∈N^∗,k0∈N^∗ and F a non-negative function on T∞, we have:

E[F(r_h,k0(τ^∞))] = E

F(rh(τ))Mh1_{{k∅(τ)=k0}} P(k_∅(τ) =k₀) ,

where (Mh, h∈N^∗) is the martingale defined by (3.1). Equivalently, for allh∈N^∗, k0∈N^∗ andt∈T^(h)k0 , we have with k=zh(t):

P(rh,k₀(τ^∞) =t) = 1−q

ηq γ_h^k P(rh(τ) =t). (6.2)

Proof. Leth∈N^∗,k₀∈N^∗ andt∈T^(h)k0 . Setk=z_h(t). We have:

1−q

ηq γ_h^kP(r_h(τ) =t) =1−q ηq



 Y

u∈t,|u|=h−1

γ_h^ku(t)







 Y

u∈rh−1(t)

p(k_u(t))





=1−q ηq γ₁^k0



 Y

u∈rh−1(t)^∗

γ_|u|⁻¹γ_|u|+1^ku(t)







 Y

u∈rh−1(t)

p(ku(t))





=1−q

ηq γ₁^k0p(k0)



 Y

u∈rh−1(t)^∗

˜

p_|u|(ku(t))





=P(rh,k₀(τ^∞) =t), where we used that P

u∈t,|u|=`ku(t) =P

u∈t,|u|=`+11 for the second equality and the definition ofp(k0) and γ₁=γas well as (6.1) for the last one. To conclude, notice also that thanks to the definition ofp(k₀) andγ₁=γ as well as (3.1), we have on{k_∅(τ) =k₀}:

1−q

ηq γ_h^zh(τ)= Mh

p(k0)·

We give an alternative description ofτ^∞ as the skeleton of a two-type GW tree. We set forn∈N:

ν_n= 1−γn+1−1 γ₁−1 =

(µ(1−µⁿ) (1−µⁿ⁺¹)⁻¹ ifµ6= 1,

n(n+ 1)⁻¹ ifµ= 1.

We have νn ∈[0,1). It is easy to check (using the first expression ofνn−1 for the first equality and the second expression forν_n−1 andν_n for the second equality) that for alln∈N^∗:

1−qν_n−1

1−q =γ_n and 1

µ(1−ν_n−1) ν_n 1−νn

= 1. (6.3)

We consider an inhomogeneous two-type GW tree ˆτ^∞ where the vertices are either of type s (for survivor) or of type e (for extinction). We define Ske(ˆτ^∞) as the tree ˆτ^∞ when one forgets the types of the vertices of ˆ

τ^∞. We denote bySh={u∈Ske(ˆτ^∞);|u|=hand uis of type s in ˆτ^∞}the set of vertices of ˆτ with type s at levelh∈N. The random tree ˆτ^∞ is defined as follows:

– The number of offspring of a vertex depends, conditionally on the vertices of lower or same height, only on its own type (branching property).

– The root is of type s (i.e.S0={∅}).

– A vertex of type e produces only vertices of type e with offspring distributionp defined by (4.1).

– A vertexu∈τˆ^∞at levelhof type s producesκ^s(u) vertices of type s with probability distributionG(1, νh) (with the convention that if h= 0, then κ^s(∅) = +∞) and κ^e(u) vertices of type e such that the type of the vertices (ui,1≤i≤κ^s(u) +κ^e(u)) is a sequence of heads (type s) and tails (type e) where the probability to get an head is q∨η and a tail is 1−q∨η, stopped just before the (κ^s(u) + 1)-th head.

Equivalently, for|u| ≥1, conditionally on κ^s(u) =su≥1, the vertexuhasκ^e(u) vertices of type e such that ku(Ske(ˆτ^∞)) =κ^s(u) +κ^e(u) has distributionp[s_u], defined in (4.3), and thesu individuals of type