Pruning Galton-Watson trees and tree-valued Markov processes

www.imstat.org/aihp 2012, Vol. 48, No. 3, 688–705

DOI:10.1214/11-AIHP423

© Association des Publications de l’Institut Henri Poincaré, 2012

Pruning Galton–Watson trees and tree-valued Markov processes ¹

Romain Abraham

, Jean-François Delmas

and Hui He

aMAPMO, CNRS UMR 6628, Fédération Denis Poisson FR 2964, Université d’Orléans, B.P. 6759, 45067 Orléans cedex 2, France.

E-mail:romain.abraham@univ-orleans.fr

bCERMICS, Université Paris-Est, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France. E-mail:delmas@cermics.enpc.fr cMAPMO, CNRS UMR 6628, Fédération Denis Poisson FR 2964, Université d’Orléans, B.P. 6759, 45067 Orléans cedex 2, France

dSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China. E-mail:hehui@bnu.edu.cn Received 2 July 2010; revised 14 February 2011; accepted 23 February 2011

Abstract. We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process{G(u)}by pruning Galton–Watson trees and an analogous process{G^∗(u)}by pruning a critical or subcritical Galton–Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process{G(u)}run until its ascension time has a representation in terms of{G^∗(u)}. A similar result was obtained by Aldous and Pitman (Ann. Inst. H. Poincaré Probab. Statist.34(1998) 637–686) in the special case of Poisson offspring distributions where they considered uniform pruning of Galton–Watson trees by adding marks on the edges of trees.

Résumé. Nous présentons une nouvelle procédure d’élagage d’arbres discrets en ajoutant des marques sur les noeuds de l’arbre.

Cette procédure nous permet de définir un processus de Markov{G(u)}à valeurs arbres en élaguant un arbre de Galton–Watson.

Nous définissons également de manière analogue un processus{G^∗(u)}en élaguant un arbre de Galton–Watson critique ou sous- critique conditionné à être infini. Sous de faibles hypothèses sur la loi de reproduction, nous montrons que le processus{G(u)} arrêté en son temps d’ascension admet une représentation en terme du processus{G^∗(u)}. Un résultat similaire a été obtenu par Aldous et Pitman (Ann. Inst. H. Poincaré Probab. Statist.34(1998) 637–686) dans le cas particulier de lois de reproductions poissoniennes en considérant un élagage uniforme sur les branches de l’arbre.

MSC:05C05; 60J80; 60J27

Keywords:Pruning; Branching process; Galton–Watson process; Random tree; Ascension process

1. Introduction

Using percolation on the branches of a Galton–Watson tree, Aldous and Pitman constructed by time-reversal in [4]

an inhomogeneous tree-valued Markov process that starts from the trivial tree consisting only of the root and ends at time 1 at the initial Galton–Watson tree. When the final Galton–Watson tree is infinite, they define the ascension timeAas the first time where the tree becomes infinite. They also define another process by pruning at branches the tree conditioned on non-extinction and they show that, in the special case of Poisson offspring distribution, some connections exist between the first process up to the ascension time and the second process.

Using the same kind of ideas, continuum-tree-valued Markov processes are constructed in [2] and an analogous relation is exhibited between the process obtained by pruning the tree and the other one obtained by pruning the tree conditioned on non-extinction. However, in that continuous framework, such results hold under very general assumptions on the branching mechanism.

1This work is partially supported by the “Agence Nationale de la Recherche,” ANR-08-BLAN-0190, and by NSFC (10525103 and 10721091).

Using the ideas of the pruning procedure [2] (which first appeared in [8] for a different purpose), we propose here to prune a Galton–Watson tree on the nodes instead of the branches so that the connections pointed out in [4] hold for any offspring distribution.

Let us first explain the pruning procedure. Given a probability distribution p= {p_n, n=0,1, . . .}, let Gp be a Galton–Watson tree with offspring distributionp. Let 0< u <1 be a constant. Then, ifνis an inner node ofGpthat hasnoffsprings, we cut it (and discard all the sub-trees attached at this node) with probabilityuⁿ⁻¹independently of the other nodes. The resulting tree is still a Galton–Watson tree with offspring distributionp^(u)defined by:

p_n^(u)=uⁿ⁻¹p_n forn≥1 (1.1)

and

p₀^(u)=1− ∞ n=1

p^(u)_n . (1.2)

This particular pruning is motivated by the following lemma whose proof is postponed to Section5.

Lemma 1.1. Letpandq be two offspring distributions.LetGp andGqbe the associated Galton–Watson trees and let#Lpand#Lqdenote the number of leaves ofGpandGq,respectively.Then we have that

∀N≥1, P(Gp∈ ·|#Lp=N )=P(Gq∈ ·|#Lq=N ) (1.3) if and only if

∃u >0,∀n≥1 q_n=uⁿ⁻¹p_n.

This lemma can be viewed as the discrete analogue of Lemma 1.6 of [1] that explains the choice of the pruning parameters for the continuous case. In [3], a similar result for Poisson–Galton–Watson trees was obtained when conditioning by the total number of vertices, which explains why Poisson–Galton–Watson trees play a key role in [4].

Using the pruning at nodes procedure, given a critical offspring distributionp, we construct in Section4a tree- valued (inhomogeneous) Markov processes(G(u),0≤u≤ ¯u)for someu¯≥1, such that

• the process is non-decreasing,

• for everyu,G(u)is a Galton–Watson tree with offspring distributionp^(u),

• the tree is critical foru=1, sub-critical foru <1 and super-critical for 1< u≤ ¯u.

Let us state the main properties that we prove for that process and let us compare them with the results of [4]. We write(G^AP(u))for the tree-valued Markov process defined in [4].

In Section3, we compute the forward transition probabilities and the forward transition rates for that process and exhibit a martingale that will appear several times (see Corollary3.4). For a treet, we set

M(u,t)=(1−μ(u))#L(t) p^(u)₀

, (1.4)

where #L(t)denotes the number of leaves oftandμ(u)if the mean of offsprings inG(u). Then, the process M

u,G(u)

,0≤u <1

is a martingale with respect to the filtration generated by G. In [4], the martingale that appears (Corollary 23) for Poisson–Galton–Watson trees is(1−μ(u))#G^AP(u).

When the treeG(u)is super-critical, it may be infinite. We define the ascension timeAby:

A=inf

u∈ [0,u¯],#G(u)= ∞

with the convention inf∅= ¯u. We can then compute the joint law ofAandGA−(i.e. the tree just before it becomes infinite), see Proposition4.6: we setF (u)the extinction probability ofG(u)and we have foru∈ [0,u)¯

P(A≤u)=1−F (u), P

G(A−)=t|A=u

=M(u,ˆ t)P

G(u)ˆ =t

withuˆ=uF (u). These results are quite similar with those of Lemma 22 of [4]. They must also be compared to the continuous framework, Theorems 6.5 and 6.7 of [2].

When we havep₀^(u)¯ =0, then the final treeG(u)¯ is a.s. infinite. We consider the tree G^∗(1)which is distributed as the treeG(1)conditioned on non-extinction. From this tree, by the same pruning procedure, we construct a non- decreasing tree-valued process(G^∗(u),0≤u≤1). We then prove the following representation formula (Proposi- tion4.7):

G(u),0≤u < A d

=

G^∗(uγ ),0≤u <F¯⁻¹(1−γ ) ,

whereγ is a r.v., uniformly distributed on(0,1), independent of{G^∗(α): 0≤α≤1}andF (u)¯ =1−F (u). This result must also be compared to a similar result in [4], Proposition 26:

G^AP(u),0≤u < Ad

=

G^AP∗(uγ ),0≤u < −logγ (1−γ )

,

or to its continuous analogue, Corollary 8.2 of [2].

Let us stress again that, although the results are very similar, those in [4] only hold for Poisson–Galton–Watson trees whereas the results presented here hold for any offspring distribution.

The paper is organized as follows. In the next section, we recall some notation for trees and define the pruning procedure at nodes. In Section3, we define the processesG andG^∗and in Section 4we state and prove the main results of the paper. Finally, we prove Lemma1.1in Section5.

2. Trees and pruning

2.1. Notation for trees

We present the framework developed in [9] for trees, see also [7] or [4] for more notation and terminology. Introduce the set of labels

W=

∞ n=0

N^∗n

,

whereN^∗= {1,2, . . .}and by convention(N^∗)⁰= {∅}.

An element ofWis thus a sequencew=(w¹, . . . , wⁿ)of elements ofN, and we set|w| =n, so that|w|represents the generation of w or the height ofw. If w=(w¹, . . . , w^m)andv=(v¹, . . . , vⁿ)belong toW, we writewv= (w¹, . . . , w^m, v¹, . . . , vⁿ)for the concatenation of w and v. In particular w∅=∅w=w. The mapping π:W\ {∅} −→Wis defined byπ((w¹, . . . , wⁿ))=(w¹, . . . , wⁿ⁻¹)ifn≥1 andπ((w¹))=∅, and we say thatπ(w)is the father ofw. We setπ⁰(w)=wandπⁿ(w)=πⁿ⁻¹(π(w))for 1≤n≤ |w|. In particular,π^|w|(w)=∅.

A (finite or infinite) rooted ordered treetis a subset ofWsuch that (1) ∅∈t.

(2) w∈t\ {∅} ⇒π(w)∈t.

(3) For everyw∈t, there exists a finite integerk_wt≥0 such that, for everyj∈N,wj ∈tif and only if 0≤j≤k_wt (kwtis the number of children ofw∈t).

LetT^∞denote the set of all such treest. Given a treet, we call an element in the sett⊂Wa node oft. Denote the height of a treetby|t| :=max{|ν|: ν∈t}. Forh≥0, there exists a natural restriction maprh:T^∞→T^hsuch that r_ht= {ν∈t: |ν| ≤h}, whereT^h:= {t∈T^∞:|t| ≤h}. In particular,T⁰= {∅}.

We denote by #tthe number of nodes oft. Let T:=

t∈T^∞: #t<∞

be the set of all finite trees. ThenT=_∞

h=1T^h. We define the shifted subtree oftaboveνby

Tνt:= {w: νw∈t}.

Forn≥0, let gen(n,t)be thenth generation of individuals int. That is gen(n,t):=

ν∈t: |ν| =n .

We say thatw∈tis a leaf oftifk_wt=0 and set L(t):= {w∈t:k_wt=0}.

SoL(t)denotes the set of leaves oftand #L(t)is the number of leaves oft.

We say thatw∈tis an inner node oftif it is not a leaf (i.e.k_wt>0) and we denote bytⁱ the set of inner nodes of ti.e.

tⁱ=t\L(t).

Given a probability distributionp= {pn, n=0,1, . . .}withp1<1, following [4], call a random treeGpa Galton–

Watson tree with offspring distributionpif the number of children of∅has distributionp:

P(k_∅Gp=n)=p_n ∀n≥0

and for eachh=1,2, . . . ,conditionally givenr_hG=t^h∈T^h, for ν∈gen(h,t^h),k_νGp are i.i.d. random variables distributed according top. That means

P(rh+1G=t|rhG=rht)=

ν∈r_ht\r_h−1t

pk_νt, t∈T^h+1,

where the product is over all nodesνoftof heighth. We have then P(G=t)=

ν∈t

p_kνt, t∈T, (2.1)

where the product is over all nodesνoft.

2.2. Pruning at nodes

LetT be a tree inT^∞. For 0≤u≤1, a random tree T(u)is called anode pruningof T with parameteruif it is constructed as follows: conditionally givenT =t,t∈T^∞, for 0≤u≤1, we consider a family of independent random variables (ξ_ν^u, ν∈t)such that

P ξ_ν^u=1

=1−P ξ_ν^u=0

=

u^kνt−1, ifk_νt≥1, 1, ifk_νt=0, and define

T(u):= {∅} ∪

ν∈t\ {∅} :

|ν|

n=1

ξ_π^un(ν)=1

. (2.2)

This means that if a nodeνbelongs toT(u)andξ_ν^u=1, thenνj, j=0,1, . . . , kν(t), all belong toT(u)and ifξ_ν^u=0, then all subsequent offsprings ofνwill be removed with the subtrees attached to these nodes. ThusT(u)is a random tree,T(u)⊂T and we have for everyν∈W,

P

kνT(u)=n|ν∈T(u),T =t

=

uⁿ⁻¹1_{kνt=n}, n≥1, 1_{kνt=0}+

1−u^kνt−1

1_{kνt≥1}, n=0. (2.3)

We also have that forh≥1 andt∈T^∞, P

r_hT(u)=r_ht|T =t

=P

ν∈n(h,t)

ξ_ν^u=1

=u

ν∈n(h,t)(kνt−1), (2.4)

wheren(h,t):= {ν∈t: k_νt≥1 and|ν|< h}.

IfT is a Galton–Watson tree, we have the following proposition.

Proposition 2.1. IfT is a Galton–Watson tree with offspring distribution{pn, n≥0},thenT(u)is also a Galton–

Watson tree with offspring distribution{p_n^(u), n≥0}defined by(1.1)and(1.2).

Proof. By (2.3), P

k_∅T(u)=0

=P

T = {∅}

+^∞

n=1

1−uⁿ⁻¹

P(k_∅T =n)=p0+^∞

n=1

1−uⁿ⁻¹ p_n,

which is equal top^(u)₀ . Forn≥1, P

k_∅T(u)=n

=uⁿ⁻¹P(k_∅T =n)=uⁿ⁻¹p_n.

The fact that{ξ_ν^u}are, conditionally onT independent random variables, gives that for eachh=1,2, . . . ,condition- ally givenr_hT(u)=t^h∈T^h, forν∈t^hwith|ν| =h,k_νT(u)are independent. Meanwhile, again by (2.3),

P

k_νT(u)=n|r_hT(u)=t^h

=

uⁿ⁻¹P(k_νT =n)=p^(u)_n , ifn≥1, P(k_νT =0)+

k≥1

1−u^k−1

P (k_νT =k)=p₀^(u), ifn=0.

Then the desired result follows readily.

3. A tree-valued Markov process

3.1. A tree-valued process given the terminal tree

LetT be a tree inT^∞. We want to construct aT^∞-valued stochastic process{T(u): 0≤u≤1}such that

• T(1)=T,

• for every 0≤u1< u2≤1,T(u1)is a node pruning ofT(u2)with pruning parameteru1/u2.

Recall thatTⁱ is the set of the inner nodes ofT. Let(ξ_ν, ν∈Tⁱ)be a family of independent random variables such that, for everyν∈Gⁱ,

P(ξ_ν≤u)=u^kνT−1.

Then, for everyu∈ [0,1], we set T(u)=

ν∈T,∀1≤n≤ |ν|, ξπⁿ(ν)≤u .

We call the process(T(u),0≤u≤1)a pruning process associated withT. Let us remark that, contrary to the process of [4], the treeT(0)may not be reduced to the root as the nodes with one offspring are never pruned. More precisely, if we denote by(1)ⁿthen-uple(1,1, . . . ,1)∈(N^∗)ⁿwith the convention(1)⁰=∅, we have

T(0)=

(1)ⁿ, n≤sup{k,∀l < k, k₍₁₎lG=1}

with the convention sup∅=0.

We deduce from Formula (2.4) the following proposition:

Proposition 3.1. We have that

ulim→1T(u)=T a.s., (3.1)

where the limit means that for almost everyωin the basic probability space,for eachhthere exists au(h, ω) <1such thatrhT(u, ω)=rhT(ω)for allu(h, ω) < u≤1.

3.2. Pruning Galton–Watson trees

Letp= {p_n, n=0,1, . . .}be an offspring distribution. LetGbe a Galton–Watson tree with offspring distributionp.

Then we consider the process (G(u),0≤u≤1)such that, conditionally on G, the process is a pruning process associated withG.

Then for eachu∈ [0,1],G(u)is a Galton–Watson tree with offspring distributionp^(u). Letg(s)denote the gener- ating function ofp. Then the distribution ofG(u)is determined by the following generating function

gu(s)=1−g(u)/u+g(us)/u, 0< u≤1. (3.2)

3.3. Forward transition probabilities

LetL(u)be the set of leaves ofG(u). Fixαandβwith 0≤α≤β≤1. Let us define p_α,β(k)=(1−(α/β)^k−1)p^(β)_k

p₀^(α) fork≥1 and p_α,β(0)=p₀^(β) p^(α)₀ ·

We define a modified Galton–Watson tree in which the size of the first generation has distributionp_α,β, while these and all subsequent individuals have offspring distributionp^(β). More precisely, letN be a random variable with law p_α,β and let(Tk, k∈N^∗)be a sequence of i.i.d. Galton–Watson trees with offspring distributionp^(β) independent ofN. Then we define the modified Galton–Watson treeGα,βby

Gα,β= {∅} ∪ {k,1≤k≤N} ∪

N

k=1

{kw, w∈Tk}. (3.3)

Let(G^ν_α,β, ν∈L(α))be, conditionally givenG(α), i.i.d. modified Galton–Watson trees. We set Gˆ(β)=G(α)∪

ν∈L(α)

νw: w∈G_α,β^ν

. (3.4)

That isGˆ(β)is a random tree obtained by adding a modified Galton–Watson treeG_α,β^ν on each leafν ofG(α). The following proposition, which implies the Markov property of{G(u),0≤u≤1}, describes the transition probabilities of that tree-valued process.

Proposition 3.2. For every0≤α≤β≤1,(G(α),G(β))=^d (G(α),Gˆ(β)).

Proof. Letα < β, leth∈N^∗and letsandtbe two trees ofT^hsuch thatscan be obtained fromtby pruning at nodes.

Then, by definition of the pruning procedure, we have P

r_hG(α)=s, rhG(β)=t

=

ν∈rh−1t

p_k^(β)

νt

ν∈sⁱ

α β

k_νt−1

ν∈L(s)\L(t)

1−

α β

k_νt−1

=

ν∈sⁱ

p_k^(α)

νt

ν∈r_h−1t\sⁱ

p^(β)_k

νt

ν∈L(s),|ν|<h

1−

α β

kνt−1

1_{k_νt>0}

=

ν∈sⁱ

p_k^(α)

νt

ν∈rh−1t\s

p^(β)_k

νt

ν∈L(s),|ν|<h

p_k^(β)

νt

1−

α β

kνt−1

1_{k_νt>0}

=

ν∈rh−1s

p^(α)_k

νs

ν∈r_h−1t\s

p_k^(β)

νt

ν∈L(s),|ν|<h

p^(β)_k

νt

p^(α)₀

1− α

β k_νt−1

1_{kνt>0}

=

ν∈rh−1s

p^(α)_k

νs

ν∈r_h−1t\s

p_k^(β)

νt

ν∈L(s),|ν|<h

p_α,β(k_νt).

The definition ofGˆ(β)readily implies P

rhG(α)=s, rhGˆ(β)=t

=

ν∈r_h−1s

p_k^(α)

νs

ν∈rh−1t\s

p^(β)_k

νt

ν∈L(s),|ν|<h

pα,β(kνt)

which ends the proof.

Let #L(u)denote the number of leaves ofG(u). The latter proposition together with the description ofGˆ readily imply

G(α),#L(β) d

=

G(α),

ν∈L(α)

#L Gα,β^ν

. (3.5)

We can also describe the forward transition rates when the trees are finite. Ifsandtare two trees and ifν∈L(s), we define the tree obtained by graftingtonνby

r(s, ν;t):=s∪ {νw, w∈t}.

We also define, fors∈T,ν∈L(s)andk∈N^∗, the set of trees r

s, ν;tk(∞) :=

r(s, ν;t), k_∅t=k,#t= ∞ .

Corollary 3.3. Lets∈T,t∈T,t= {∅}and letν∈L(s).Then the transition rate at timeufromstor(s, ν;t)is given by

q_u

s→r(s, ν;t)

:=k_∅t−1 u

P(G(u)=t) p^(u)₀

. (3.6)

Lets∈T,ν∈L(s)andk≥1.Then the transition rate at timeufromsto the setr(s, ν;tk(∞))is q_u

s→r

s, ν;tk(∞)

:=k−1 u

1−F (u)^k p₀^(u)

p^(u)_k , (3.7)

and no other transitions are allowed.

Proof. Lets∈T,t∈T,t= {∅}and letν∈L(s). By Proposition3.2, we have P

G(u)=s,G(u+du)=r(s, ν;t)

=P

G(u)=s,Gˆ(u+du)=r(s, ν;t)

=P

G(u)=s P

G_u,u^ν _+du=t

˜ ν∈L(s)\{ν}

P

G_u,u^ν˜ _+du= {∅}

=P

G(u)=s P

G_u,u^ν _+du=t

p_u,u+du(0)^#L(s)−1. Using (3.3), we get

P

G(u+du)=r(s, ν;t)|G(u)=s

=P

G(u+du)=tpu,u+du(k_∅t) p_k^(u+du)

∅t

p_u,u+du(0)^#L(s)−1

=P

G(u+du)=t 1 p^(u)₀

1−

u u+du

k_∅t−1 p^(u₀^+du)

p₀^(u)

#L(s)−1

du∼→0P

G(u)=t 1 p^(u)₀

(k_∅t−1)du u · This gives Formula (3.6).

A similar computation replacingP(G(u)=t)byP(#G(u)= ∞|k_∅G(u)=k)=1−F (u)^kgives Formula (3.7).

Another similar computation gives that, iftis obtained by grafting two trees (or more) on the leaves ofs, P

G(u+du)=t|G(u)=s

=o(du)

and it is clear by construction that, in all the other cases, P

G(u+du)=t|G(u)=s

=0.

Let us define μ(u):=^∞

k=1

kp^(u)_k (3.8)

if it exists the mean ofp^(u). We setu₁=sup{u∈ [0,1], μ(u)≤1}. Recall the definition of functionMin (1.4).

Corollary 3.4. The process M

u,G(u)

,0≤u < u₁

(3.9) is a martingale with respect to the filtration generated by{G(u),0≤u < u₁}.

Proof. First, by the branching property of Galton–Watson process, for eachn≥1, 0≤u < u₁and ≥n, P

#L(u)= |k_∅G(u)=n

=P _n

i=1

Li=

,

whereL1, L2, . . .are i.i.d. copies of #L(u). This gives E

#L(u)

=p₀^(u)+E

k_∅G(u) E

#L(u)

which implies E

#L(u)

= p₀^(u)

1−μ(u)· (3.10)

A straightforward computation gives that the mean of the offspring distributionp_α,β is μα,β:=μ(β)−μ(α)

p^(α)₀ .

By the same reasoning, (3.3) and (3.5) imply, for 0≤α≤β < u1, E

#L(β)|G(α)

=#L(α)E

#L G^ν_α,β

=#L(α)

p_α,β(0)+μ_α,βE

#L(β)

=#L(α)

p_α,β(0)+μ_α,β p^(β)₀ 1−μ(β)

by (3.10). Then the martingale property of (3.9) follows from a simple calculation.

3.4. Pruning a Galton–Watson tree conditioned on non-extinction

Letp be a critical or sub-critical offspring distribution with meanμ such that p₀<1. We define the size-biased probability distributionp^∗ofpby

p^∗_k=kp_k

μ , k≥0.

LetG be a Galton–Watson tree with offspring distributionp. For a tree t, we denote by Z_nt=#gen(n,t)the number of individuals in thenth generation oft. We first recall a result in [6].

Proposition 3.5 (Kesten [6], Aldous and Pitman [4]).

(i) The conditional distribution ofGgiven{Z_nG>0}converges,asntends to+∞,toward the law of a random family treeG^∞specified by

P

rhG^∞=t

=μ^−h(Zht)P(rhG=t) ∀t∈T^(h), h≥0.

(ii) Almost surelyG^∞ contains a unique infinite path(∅=V₀, V₁, V₂, . . .)such that π(V_h+1)=V_h for every h=0,1,2, . . . .

(iii) The joint distribution of (V₀, V₁, V₂, . . .) and G^∞ is determined recursively as follows: for each h = 0,1,2, . . . ,given(V0, V1, V2, . . . , Vh)andrhG^∞,the numbers of children(kνG^∞, ν∈gen(h,G^∞))are independent with distributionpforν=Vh,and with the size-biased distributionp^∗forν=Vh;given also the numbers of children k_νG^∞forν∈gen(h,G^∞),the vertexV_h+1has uniform distribution on the set{(V_h, i),1≤i≤k_VhG^∞}.

We say thatG^∞is the Galton–Watson tree associated withp conditioned on non-extinction. We then define the process(G^∗(u),0≤u≤1)as a pruning process associated withG^∞. By Proposition3.1,G^∗(1−)=G^∗(1)=G^∞(1) almost surely. And since there exists a unique infinite path, we get thatG^∗(u)is finite almost surely for all 0≤u <1.

The distribution ofG^∗(u)for fixeduis given in the following proposition. Let us recall thatμ(u)is the mean of p^(u)defined in (3.8).

Proposition 3.6. For each0≤u <1, P

G^∗(u)=t

=

ν∈L(t)

1 μ(1)^|ν|+1

μ(1)−μ(u) p^(u)₀ P

G(u)=t

, t∈T. (3.11)

Proof. We prove (3.11) inductively. First, note that P

k_∅G^∞(1)=n

=p_n^∗=npn/μ(1).

Then P

G^∗(u)= {∅}

=

n≥1

1−uⁿ⁻¹ P

k_∅G^∞(1)=n

=

μ(1)−μ(u) /μ(1).

SinceP(G(u)= {∅})=p^(u)₀ , (3.11) holds fort= {∅}. On the other hand, by Proposition3.5, we have

P

TνG^∞(1)=t|ν=V_|ν|

=P

G^∞=t and

P

T_νG^∞(1)=t|ν=V_|ν|

=P(G=t) which gives

P

T_νG^∗(u)=t|ν∈G^∗(u), ν=V_|ν|

=P

G^∗(u)=t

(3.12) and

P

T_νG^∗(u)=t|ν∈G^∗(u), ν=V_|ν|

=P

G(u)=t

, (3.13)

respectively. Meanwhile, sinceG(u)is a Galton–Watson tree, P

TνG(u)=t|ν∈G(u)

=P

G(u)=t , which implies

P

G(u)=t

=p^(u)_k

∅t

1≤j≤k_∅t

P

G(u)=T(j )t

. (3.14)

For someh≥0, assume that (3.11) holds for all trees inT^h. By (3.12) and (3.13), we have fort∈T^h+1\T^h, P

G^∗(u)=t

=P

k_∅G^∗(u)=k_∅t^k∅t

i=1

P

∀1≤j≤k_∅t, T(j )G^∗(u)=T_{(j )}t|V₁=i

P(V₁=i)

=u^k∅t−1p^∗_k

∅t

1 k_∅t

k_∅t

i=1

P

G^∗(u)=T_(i)t

·

j=i,1≤j≤k_∅t

P

G(u)=T_{(j )}t

=p^(u)_k

∅t

μ(1)

k_∅t

i=1

ν∈L(T_(i)t)

1 μ(1)^|ν|

μ(1)−μ(u) p₀^(u) ·

1≤j≤k_∅t

P

G(u)=T_{(j )}t

=

ν∈L(t)

1 μ(1)^|ν|+1

μ(1)−μ(u) p^(u)₀ P

G(u)=t ,

where the last equality follows from (3.14) and the facts

1≤i≤k_∅t

iν: ν∈L(T(i)t)

=L(t)

and|iν| = |ν| +1. SinceT=_∞

h=1T^h, (3.11) follows inductively.

Remark 3.7. IfGis a critical Galton–Watson tree(μ(1)=1)withp₁<1,Formula(3.11)becomes P

G^∗(u)=t

=#L(t)(1−μ(u)) p^(u)₀ P

G(u)=t

=M(u,t)P

G(u)=t .

In other words,the law ofG^∗(u)is absolutely continuous with respect to the law ofG(u)with density the martingale of Corollary3.4.

4. The ascension process and its representation

In this section, we consider a critical offspring distributionpwithp₁<1 and set I=

u≥0,

+∞

k=1

u^k−1p_k≤1

.

Remark 4.1. We haveI= [0,u¯]withu¯≥1.Indeed,as the all the coefficients of the sum are non-negative,either the sum converges at the radius of convergenceRand is continuous on[0, R],or it tends to infinity whenu→R.In the latter case,there is a uniqueu < R¯ for which the generating function takes the value1 (by continuity).In the former case,either the value of the generating function atRis greater than1and the previous argument also applies,or the value of the generating function atRis less than1andu¯=R.

Let us give some examples:

Example 4.2 (The binary case). We consider the critical offspring distribution pdefined by p0=p2=1/2 (each individual dies out or gives birth to two children with equal probability).In that case,we have

+∞

k=1

u^k−1p_k=1 2u

and hence we haveu¯=2,I= [0,2]andp₀⁽²⁾=0.

Example 4.3 (The geometric case). We consider the critical offspring distributionpdefined by p_k=αβ^k−1 fork≥1,

p0=1−₁₋^α_β·

Then,for every u,p^(u) is still of that form.As the offspring distributionp is critical,we must have α=(1−β)², 0< β <1.In that case,we have

+∞

k=1

p_ku^k−1=(1−β)² 1−βu

and hence we haveu¯=2−β,I= [0,u¯],p₀^(u)¯ =0.

Example 4.4. We consider the critical offspring distributionpdefined by

p_k=_π⁶2

1

k³ fork≥1,

p0=1−_+∞

k=1p_k, thenu¯=1andp₀^(u)¯ =p0>0.

Foru∈I, let us define

p^(u)_k =u^k−1p_k, k≥1, p^(u)₀ =1−_+∞

k=1p^(u)_k . (4.1)

Then, foru∈I,p^(u)is still an offspring distribution, it is sub-critical foru <1, critical foru=1 and super-critical foru >1.

We consider a tree-valued process (G(u), u∈I ) such that the process (G(tu), t¯ ∈ [0,1])is a pruning process associated withG(u). Then this process¯ Gsatisfies the following properties:

• for everyu∈I,G(u)is a Galton–Watson tree with offspring distributionp^(u),

• for everyα, β∈I,α < β,G(α)is a pruning ofG(β).

We now consider{G(u), u∈I}as anascension processwith theascension time A:=inf

u∈I,#G(u)= ∞ with the convention inf∅= ¯u.

The state in the ascension process at time u is G(u) if 0≤u < A and t(∞) if A≤u where t(∞) is a state representing any infinite tree. Then the ascension process is still a Markov process with countable state-spaceT∪ {t(∞)}, wheret(∞)is an absorbing state.

Denote byF (u)the extinction probability of a Galton–Watson process with offspring distributionp^(u), which is the least non-negative root of the following equation with respect tos

s=g_u(s)=1−g₁(u)/u+g₁(us)/u, (4.2)

whereg₁ is the generating function associated with the offspring distributionp andg_u is the generating function associated withp^(u).

We set

F (u)¯ =1−F (u). (4.3)

Thus

uF (u)¯ +g1

u−uF (u)¯

=g1(u). (4.4)

The distribution of the ascension process is determined by the transition rates (3.6) and q_u

s→t(∞)

=#L(s) up^(u)₀

∞ k=2

(k−1)p^(u)_k

1−F (u)^k

. (4.5)

Define the conjugateuˆby ˆ

u=uF (u) foru∈I. (4.6)

In particular, foru≤1,F (u)=1 and consequentlyuˆ=u. On the contrary, foru >1, Proposition4.5shows that ˆ

u≤1.

We can restate Eq. (4.4) into

g₁(u)ˆ −g₁(u)= ˆu−u. (4.7)

Notice that this equation with the conditionuˆ≤1 characterizesu.ˆ

We first prove the following result which is already well known, see for instance [5], p. 52. We just restate this property in terms of our pruning parameter.

Proposition 4.5. For anyu∈I,u≥1 P

G(u)=t

=F (u)P

G(u)ˆ =t

, t∈T. (4.8)

In other words, the law ofG(u)conditioned to be finite isG(u), which explains the termˆ conjugateforu.ˆ Proof of Proposition4.5. By (2.1), we have

P

G(u)=t

=

ν∈t\L(t)

p^(u)_k

νt·

ν∈L(t)

p^(u)₀

and by (4.6), P

G(u)=t

=F (u)⁻⁽

ν∈t\L(t)(k_νt−1))

p₀^(u) p₀^(u)ˆ

#L(t)

P

G(u)ˆ =t .

We also have p⁽₀^u)ˆ =1−^∞

k=1

F (u)^k−1p_k^(u)=1+p^(u)₀ /F (u)−g_u F (u)

/F (u)=p₀^(u)/F (u). (4.9)

Then the desired result follows from the fact that given a treet∈T,

#L(t)=1+

ν∈t\L(t)

(k_νt−1).

In what follows, we will often suppose that

p⁽₀^u)¯ =0, (4.10)

which is equivalent to the condition

+∞

k=1

¯

u^k−1p_k=1

and which implies thatG(u)¯ is infinite.

We can however give the law ofA and of the treeG(A−)just before the ascension time in general, this is the purpose of the next proposition.

Proposition 4.6. Foru∈ [1,u)¯ andt∈T,

P(A≤u)= ¯F (u), (4.11)

P

G(A−)=t|A=u

=M(u,ˆ t)P

G(u)ˆ =t

, (4.12)

P

#G(A−) <+∞|A=u

=1. (4.13)

Furthermore,under assumption(4.10),

P(A <u)¯ =1 (4.14)

and

A,Aˆ A

=

A, F (A) d

=F¯⁻¹(1−γ ), γ

, (4.15)

whereF¯⁻¹:[0,1] → [1,u¯]is the inverse function ofF¯ andγis a r.v.uniformly distributed on(0,1).

Proof. We have P(A≤u)=P

#G(u)= ∞

= ¯F (u) which gives (4.11).

By the definition ofF (u)in Formula (4.2) and the implicit function theorem, functionF is differentiable on(1,u).¯ This gives foru∈(1,u)¯

P(A∈du)= −F(u)du and differentiating (4.2) gives

u

1−g₁(u)ˆ

F(u)=1−g₁(u)−F (u)

1−g₁(u)ˆ

. (4.16)

By (4.5), we have fort∈T P

G(A−)=t, A∈du

=#L(t)P(G(u)=t) up^(u)₀

∞ k=2

(k−1)p^(u)_k

1−F (u)^k

du. (4.17)

Now, using (4.8), we have P

G(A−)=t, A∈du

=#L(t)P(G(u)ˆ =t)F (u) up₀^(u)

∞ k=2

(k−1)p^(u)_k

1−F (u)^k du.

Easy computations give ∞

k=2

(k−1)p_k^(u)

1−F (u)^k

=g₁(u)−g₁(u)

u −F (u)g₁(u)ˆ +g(u)ˆ u

= −F(u)u

1−g₁(u)ˆ

+1−F (u)+g1(u)ˆ −g1(u) u

= −F(u)u

1−g₁(u)ˆ using first Eq. (4.16) and then Eq. (4.7).

This finally gives P

G(A−)=t, A∈du

= −#L(t)P(G(u)ˆ =t)F (u) p^(u)₀

F(u)

1−g₁(u)ˆ du

= −#L(t)P(G(u)ˆ =t) p₀^(u)ˆ

F(u)

1−μ(u)ˆ du by Eq. (4.9), which yields (4.12).

Summing (4.12) over all finite treestgives P

#G(A−) <+∞|A=u

=E M

ˆ u,G(u)ˆ

=1 by the martingale property, which is (4.13).

Finally, (4.11) gives

P(A= ¯u)=P(∀u <u, A > u)¯ =lim

u→ ¯u1− ¯F (u)=F (u¯−).

AsF is non-increasing,F (u¯−)indeed exists. Moreover, we have by taking the limit in (4.2) F (u¯−)=gu_¯

F (u¯−)