Record process on the Continuum Random Tree

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Record process on the Continuum Random Tree

Romain Abraham, Jean-François Delmas

To cite this version:

Romain Abraham, Jean-François Delmas. Record process on the Continuum Random Tree. ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2013, 10, pp.251. �hal-00609467v3�

ROMAIN ABRAHAM AND JEAN-FRANC¸ OIS DELMAS

Abstract. By considering a continuous pruning procedure on Aldous’s Brownian tree, we construct a random variable Θ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical Galton-Watson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned bynleaves.

1. Introduction

The problem of randomly cutting a rooted tree arises first in Meir and Moon [28]. Given a rooted treeT_n with nedges, select an edge uniformly at random and delete the subtree not containing the root attached to this edge. On the remaining tree, iterate this procedure until only the edge attached to the root is left. We denote by X_n the number of edge-removals needed to isolate the root. The problem is then to study asymptotics of this random number X_n, depending on the law of the initial tree T_n.

In the original paper [28], Meir and Moon considered Cayley trees and obtained asymptotics for the first two moments ofX_n. Limits in distribution were then obtained by Panholzer [30]

for some simply generated trees, by Drmota, Iksanov, M¨ohle and Roesler [15] for random recursive trees, by Holmgren [24] for binary search trees, by Bertoin [11] for Cayley trees and by Janson [26] for conditioned Galton-Watson trees. The main result of [26] states that, if the offspring distribution of the Galton-Watson process is critical (that is with mean equal to 1) with finite variance, which we take equal to 1 for simplicity, then the following convergence in distribution of the conditional laws (specified by their moments) holds:

(1) L(X_n/√

n|T_n/√

n) −−−−−→_n^(d)

→+∞ L(Z_T | T)

whereT is the so-called continuum random tree (CRT) introduced by Aldous [8, 9] and can be seen as the limit in distribution ofT_n/√

n(see [9]). Furthermore, the random variable Z_T has (unconditional) Rayleigh distribution with densityxe^−x2/21_{x>0}. However, there is no constructive description ofZ_T conditionally onT.

The first goal of the paper is to give a continuous pruning procedure of the CRT that leads to a random variable that is indeed distributed, conditionally given the tree, as Z_T. In order to better understand the intuitive idea of the record process on the CRT, let us first consider the pruning of the simple tree consisting in the segment [0,1] divided into n segments of equal length, rooted at 0. Select an edge at random and discard what is located on the right of this edge. Then chose again an edge at random on the remaining segments and iterate the procedure until the segment attached to 0 is chosen. It is clear that the continuous

Date: February 1, 2013.

2010Mathematics Subject Classification. 60J80,60C05.

Key words and phrases. continuum random tree, records, cutting down a tree.

This work is partially supported by the “Agence Nationale de la Recherche”, ANR-08-BLAN-0190.

1

analogue of this procedure (when the number n of segments tends to +∞) is the so-called stick-breaking scheme: consider a uniform random variable U₁ on [0,1], then conditionally givenU1, consider a uniform random variableU2 on [0, U1] and so on. The sequence (Un)n≥0

corresponds to the successive cuts of the interval [0,1] in the continuous pruning. Moreover, this sequence can be obtained as the records of a Poisson point process. More precisely, if we consider a Poisson point measureP

i∈Iδ_(xi,ti)on [0,1]×[0,+∞) with intensity the Lebesgue measure, then the sequence (U_n) is distributed as the sequence of jumps of the record process

θ(x) = inf{t_i, x_i ∈[0, x]}.

In our case, the limiting object is Aldous’s CRT (instead of the segment [0,1]). More precisely, we consider a real tree T associated with the branching mechanism ψ(u) = αu² under the excursion measure N. This tree is coded by the height process p

2/αB_ex where B_ex is a positive Brownian excursion. This tree is endowed with two measures: the length measure ℓ(dx) which corresponds to the Lebesgue measure on the skeleton of the tree, and the mass measure m^T(dx) which is uniform on the leaves of the tree. Let σ = m^T(T) be the total mass ofT. Aldous’s CRT corresponds to the distribution of the treeT conditioned on the total massσ = 1, with α= 1/2. We then add cut points on T as above thanks to a Poisson point measure onT ×[0,+∞) with intensity

αℓ(dx)dθ

in the same spirit as in [10] (see also [6] for a direct construction, and [5] for the pruning of a general L´evy tree). We denote by (x_i, q_i) the atoms of this point measure, x_i represents the location of the cut point andq_i represents the time at which it appears. Forx ∈ T, we denote by

θ(x) = inf{q_i, x_i ∈[[∅, x]]}

where [[∅, x]] ⊂ T denotes the path between x and the root. When a mark appears, we cut the tree on this mark and discard the subtree not containing the root. Thenθ(x) represents the time at whichx is separated from the root. Then we define

Θ = Z

T

θ(x)m^T(dx) and Z = r2α

σ Θ.

We prove (see Theorem 3.2) that, conditionally on T, Z and Z_T have indeed the same law. The proof of this result relies on another representation of Θ in terms of the mass of the pruned tree (a similar result also appears in Addario-Berry, Broutin and Holmgren [7]).

More precisely, if we set

σ_q = Z

T

1_{θ(x)≥q}m^T(dx) the mass of the remaining tree at time q, then we have

Θ = Z +∞

0

σqdq.

Using this framework, we can extend in some sense Janson’s result by obtaining an a.s.

convergence in a special case. We consider, conditionally givenT,nleaves uniformly chosen (i.e. sampled according to the mass measure m^T) and we denote by T_n the sub-tree of T spanned by these n leaves and the root. The tree Tn is distributed under N[· | σ = 1] as a uniform ordered binary tree with n leaves (and hence 2n−1 edges) with random edge lengths. We denote byT_n^∗ the tree obtained by removing from T_n the edge attached to the root, and by X_n^∗ the number of discontinuities of the process (θ(x), x ∈ T_n^∗). The quantity X_n^∗+ 1 represents the number of cuts needed to reduce the binary tree T_n to a single branch

attached to the root. Notice that in our framework, several cuts may appear on the same branch, soX_n^∗ looks likeX_2n−1 for uniform ordered binary trees but is not exactly the same.

Then, we prove in Theorem 4.2 that N-a.e. or N[· |σ= 1]-a.s.:

n→+∞lim X_n^∗

√2n =Z.

This result can be extended by studying the fluctuations of the quantity X_n^∗/√

2n around its limit, this is the purpose of Hoscheit [25]. In this setting the fluctuations come from the approximation of the record process by its intensity, whereas there is no contribution from the approximation of T by Tn.

Using the second representation of Θ and results from Abraham, Delmas and Hoscheit [4]

on the pruning of general L´evy trees, we also derive a.s. asymptotics on the sizes (σi, i∈ I) of the removed sub-trees during the cutting procedure. According to Propositions 4.4 and 4.5, we have N-a.e.

n→lim+∞

√1 n

X

i∈I

1_{σi≥1/n}= lim

n→+∞

√nX

i∈I

σⁱ1_{σi≤1/n} = 2 rα

πΘ.

This result is extended to general L´evy trees in Abraham and Delmas [1].

The paper is organized as follows. In Section 2, we introduce the frameworks of discrete trees and real trees and define rigorously Aldous’s CRT, the mark process and the record process on the tree. Section 3 is devoted to the identification of the law of Θ conditionally given the tree. In Section 4, we prove the a.s. convergence ofX_n^∗ as well as the convergence results on the masses of the removed subtrees. Finally, we gathered in Section 5 several technical lemmas that are needed in the proofs but are not the heart of the paper.

2. The continuum random tree and the mark process

2.1. Real trees. We recall here the definition and basic properties of real trees. We refer to Evans’s Saint Flour lectures [20] for more details on the subject.

Definition 2.1. A real tree is a metric space (T, d) satisfying the following two properties for everyx, y∈ T:

• (Unique geodesic) There is a unique isometric map f_x,y from [0, d(x, y)] into T such that f_x,y(0) =x and f_x,y(d(x, y)) =y.

• (No loop) If ϕ is a continuous injective map from [0,1] into T such that ϕ(0) = x and ϕ(1) =y, then

ϕ([0,1]) =f_x,y([0, d(x, y)]).

A rooted real tree is a real tree with a distinguished vertex denoted∅ and called the root.

We denote by [[x, y]] = f_x,y([0,1]) the range of the mapping f_x,y, which is the unique continuous injective path betweenxandy in the tree, and [[x, y[[= [[x, y]]\{y}. A pointx∈ T is said to be a leaf if the setT \ {x} remains connected. We denote by Lf(T) the set of leaves of T. The skeleton of the tree is the set of non-leaves points T \Lf(T). As the trace of the Borelσ-field on the skeleton ofT is generated by the intervals [[x, y]], one can define a length measure denoted byℓ(dx) on a real tree by:

ℓ([[x, y]]) =d(x, y).

We will consider here only compact real trees and these trees can be coded by some continuous function (see [27] or [16]). We consider a continuous function ζ : [0,+∞) →

[0,+∞) with compact support [0, σ] and such that ζ(0) =ζ(σ) = 0. This function ζ will be called in the following the height function. For everys, t≥0, we set

m_ζ(s, t) = inf

r∈[s∧t,s∨t]ζ(r), and

d(s, t) =ζ(s) +ζ(t)−m_ζ(s, t).

We then define the equivalence relations∼tiff d(s, t) = 0. We setT the quotient space T = [0,+∞)/∼.

The pseudo-distance d induces a distance on T and we keep notation d for this distance.

We denote byp the canonical projection from [0,+∞) onto T. The metric space (T, d) is a compact real tree which can be viewed as a rooted real tree by setting∅=p(0).

On such a compact real tree, we define another measure : the mass measure m^T defined as the push-forward of the Lebesgue measure by the projection p. It is a finite measure supported by the leaves ofT and its total mass is

m^T(T) =σ.

This coding is very useful to define random real trees. For instance, Aldous’s CRT is the random real tree coded by 2Bex where Bex denotes a normalized Brownian excursion (i.e. a positive Brownian excursion with duration 1). Here, we will work under theσ-finite measure N which denotes the law of a real tree coded by an excursion away from 0 of

q2

α|B|where

|B| is a standard reflected Brownian motion. The tree T under N is then the genealogical tree of a continuous state branching process with branching mechanism ψ(u) = αu² under its canonical measure. In particular, under N,σ has density on (0,+∞):

(2) dr

2√

απ r^3/2·

We keep parameter α in order to stay in the framework of [2], and give the result in the setting of Aldous’s CRT (α= 1/2) or of Brownian excursion (α = 2).

Using the scaling property of the Brownian motion, there exists a regular version of the measureNconditioned on the length of the height processζ. We writeN^(r)for the probability measure N[· |σ = r]. In particular, we handle Aldous’s CRT if we work under N⁽¹⁾ with α= 1/2.

If x₁, . . . , x_n ∈ T, we denote by T(x₁, . . . , x_n) the subtree spanned by ∅, x₁, . . . , x_n, i.e.

the smallest connected subset ofT that containsx₁, . . . , x_nand the root. In other words, we have

T(x₁, . . . , x_n) =

n

[

i=1

[[∅, x_i]].

With an abuse of notation, we write for every t₁, . . . , t_n ≥ 0, T(t₁, . . . , t_n) for the subtree T(p(t₁), . . . , p(t_n)).

2.2. The mark process. We define now a mark process M on the tree T. Conditionally givenT, letM(dx, dq) be a Poisson point measure onT ×[0,+∞) with intensity 2αℓ(dx)dq.

An atom (x_i, q_i) of this random measure represents a mark on the tree T,x_i is the location of this mark whereas q_i denotes the time at which the mark appears.

Remark 2.2. The coefficient 2α in the intensity is added to have the same intensity as in the pruning procedures of [6, 5, 4] but, as we shall see, it does not appear in the law of the number of records.

In fact we will sometimes work with the restriction of M to T ×[0, a] for some a > 0.

To simplify the notations, we will always denote byM the mark process (even the restricted one) and will write M^T_a for the law ofM restricted toT ×[0, a], conditionally given T. We also write N_a[dT dM] =N[dT]M^T_a[dM], andN^(r)_a [dT dM] =N^(r)[dT]M^T_a[dM].

We set for every q≥0 andx∈ T:

(3) θ(x) = inf{q >0, M([[∅, x]]×[0, q])>0} and Tq={x∈ T;θ(x)≥q},

respectively the first time a mark appears between the root and x, and the tree obtained by pruning the original tree at the marks present at timeq. We also define the mass of the tree Tq:

σ_q=m^T(Tq).

According to [5],Tq is distributed under N_∞as a L´evy tree with branching mechanism ψ_q(u) =ψ(u+q)−ψ(q) =αu²+ 2αqu.

We will denote byN^ψq the distribution ofTqunderN. Moreover, thanks to Girsanov formula ([2], Lemma 6.2), we have, for every nonnegative Borel functionF

(4) N^ψq[F(T)] =N[F(Tq)] =N h

F(T) e^−αq2σi .

With the same abuse of notation as for the spanned subtree, we write for every t∈ R₊, θ(t) instead of θ(p(t)).

2.3. Discrete trees. We recall here the definition of a discrete ordered rooted tree according to Neveu’s formalism [29].

We consider U =

+∞

[

n=0

(N^∗)ⁿ the set of finite sequences of positive integers. The empty sequence ∅belongs to U. Ifu, v ∈ U, we denote byuv the sequence obtained by juxtaposing the sequencesu and v.

A discrete ordered rooted treeT is a subset of U satisfying the three following properties

• ∅ ∈T. ∅ is called the root of T.

• For every u∈ U and i∈N^∗, ifui∈T then u∈T.

• For every u∈T, there exists an integer k_u(T) such that ui∈T ⇐⇒ 1≤i≤ku(T).

The integer k_u(T) is the number of offsprings of the vertexu. The leaves of the tree are the u ∈T such that k_u(T) = 0. We will consider here only binary trees i.e, discrete trees such thatk_u(T) = 0 or 2.

We can add edge lengths to a discrete tree by considering weighted trees. A weighted tree is defined by a discrete ordered rooted tree T and a weight h_u ∈ [0,+∞) for every u ∈ T.

The elementsu∈T must be viewed as the edges of the tree andh_u is the length of the edge u. Obviously, such a weighted tree can be viewed as a real tree and we will always make the confusion between a discrete weighted tree and the associated real tree.

3. Janson’s random variable

LetT be a compact real tree and letM be a mark process on T. We set Θ =

Z

T

θ(x)m^T(dx).

Remark that this can be re-written using the coding by Θ =R_σ

0 θ(s)ds.

Using the tree-valued process (Tq, q ≥ 0), we can give another expression for Θ. Let (θ_i, i∈ I) be the set of jumping times of (σ_q, q≥0). We set:

(5) Tⁱ ={x∈ T;θ(x) =θ_i} and σⁱ =m^T(Tⁱ) =σ_θi−−σ_θi. According to [2], we have that M^T

∞-a.s. Tⁱ is a real tree for all i ∈ I. Then the following result is straightforward as by definition Θ =P

i∈Iθ_iσⁱ and σ_q=P

θi≥qσⁱ. Proposition 3.1. We have M^T

∞-a.s.:

Θ = Z _+∞

0

σ_qdq.

The main result of this section is the following theorem that identifies Θ as Janson’s random variable whose distribution is characterized by its moments.

Theorem 3.2. For every positive integerr, we have M^T

∞[Θ^r] = r!

(2α)^r Z

T^r

m^T(dx₁). . . m^T(dx_r) Q_r

i=1ℓ(T(x₁, . . . , x_i))· Proof. Using the expression of Proposition 3.1 for Θ, we have

M^T

∞[Θ^r] =r!M^T

∞

Z

0≤q1<q2<···<qr

dq₁. . . dq_rσ_q1. . . σ_qr

=r!M^T

∞

"

Z

0≤q1<q2<···<qr

dq₁. . . dq_r

r

Y

k=1

Z

T

m^T(dx_k)1_{xk∈Tqk}

#

=r!

Z

T^r

m^T(dx₁). . . m^T(dx_r) Z

0≤q1<q2<···<qr

dq₁. . . dq_rM^T

∞[x₁ ∈ Tq1, . . . , x_r ∈ Tqr].

To evaluate the probability that appears in the last equation, let us remark that, ify∈ Tq, theny∈ Tq^′ for everyq^′ < q. Therefore, we have

M^T

∞[x1∈ Tq1, . . . , xr∈ Tqr]

=M^T

∞[x₂ ∈ Tq2, . . . , x_r∈ Tqr

x₁ ∈ Tq1, . . . , x_r ∈ Tq1]M^T

∞[x₁ ∈ Tq1, . . . , x_r ∈ Tq1].

On one hand, we have M^T

∞[x₁∈ Tq1, . . . , x_r ∈ Tq1] =M^T

∞[M(T(x₁, . . . , x_r)×[0, q₁]) = 0]

= exp −2αq1ℓ(T(x1, . . . , xr)) .

On the other hand, by standard properties of Poisson point measures, we have M^T

∞[x₂∈ Tq2, . . . , x_r ∈ Tqr

x₁∈ Tq1, . . . , x_r∈ Tq1] =M^T

∞[x₂ ∈ Tq2−q¹, . . . , x_r∈ Tqr−q¹].

We finally obtain by induction, with the convention q₀ = 0:

M^T

∞[x1 ∈ Tq1, . . . , xr ∈ Tqr] =

r

Y

k=1

e^{−2α(qk−qk−1)ℓ(T(xk,...,xr))}.

Plugging this expression in the integral gives, after an obvious change of variables M^T

∞[Θ^r] =r!

Z

T^r

m^T(dx₁). . . m^T(dx_r) Z

0≤q1<···<qr

dq₁. . . dq_r

r

Y

k=1

e^{−2α(qk−qk−1)ℓ(T(xk,...,xr))}

=r!

Z

T^r

m^T(dx1). . . m^T(dxr)

r

Y

k=1

Z +∞ 0

da_ke^{−2αakℓ(T(xk,...,xr))}

= r!

(2α)^r Z

T^r

m^T(dx₁). . . m^T(dx_r) Q_r

k=1ℓ(T(x_k, . . . , x_r))·

We can then deduce from the results of [26], that forα= 2, underN⁽¹⁾_∞, Θ has Rayleigh dis- tribution. Using then scaling argument inr and αor directly Corollary 5.3 in the Appendix, we get the following result.

Corollary 3.3. For all r > 0, the random variable Z = q2α

r Θ is distributed under N^(r)_∞ according to a Rayleigh distribution with density xe^−x2/21_{x≥0}.

In particular, we have easily the first moments of Θ:

(6) N^(r)_∞ [Θ] = 1

2 rπr

α and N^(r)_∞ Θ²

= r α· 4. A.s. convergence

4.1. Statement of the main result. Let r ≥0 and let T be a tree distributed according to N^(r). Let (U₁, . . . , U_n) be n points uniformly chosen at random on [0, r], independent of T. We denote byT_n the random tree spanned by thesen points i.e.

T_n=T(U₁, . . . , U_n)

viewed as a discrete ordered weighted tree. Notice thatT_nhas 2n−1 edges. Let (h₁, . . . , h_2n−1) be the lengths of the edges given in lexicographic order. We consider the total length ofT_n:

L_n=ℓ(T_n) =

2n−1

X

k=1

h_k.

We definem_nas the first branching point of T_n, i.e.

(7)

n

\

k=1

[[∅, p(U_k)]] = [[∅, m_n]]

and we consider the length of the edge ofT_n attached to the root (8) h_∅,n:=d(∅, m_n) =ℓ([[∅, m_n]]) =h₁.

LetT_n^∗ be the sub-tree of Tn where we remove the edge [[∅, mn[[:

T_n^∗ =T_n\[[∅, m_n[[, and L^∗_n its total length i.e. L^∗_n=L_n−h_∅,n.

We set θ(x−) = inf{θ(y), y∈[[∅, x[[} and X_n^∗ the number of records on the tree T_n^∗: X_n^∗ = X

x∈T_n^∗

1_{{θ(x−)>θ(x)}},

Remark 4.1. The introduction of the tree T_n^∗ is motivated by the fact that the number X

x∈Tn

1_{{θ(x−)>θ(x)}}

of records on the whole tree is N_∞-a.e. infinite. Moreover, X_n^∗+ 1 represents the number of cuts that appears on the reduced tree Tn until a mark appears on the branch attached to the root which reduces the tree to a trivial one consisting of the root and a single branch attached to it. Hence it is the analogue of the discrete quantity X_n and is the right quantity to be studied.

We can then state the main result of this section which will be proven in Section 4.5.

Theorem 4.2. We have that, for all r >0, N^(r)_∞-a.s.:

n→lim+∞

X_n^∗

√2n = rα

2rΘ =Z.

Remark 4.3. Notice that the binary tree T_n has 2n−1 vertices; and it corresponds to a critical Galton-Watson tree with reproduction law taking values in{0,2}and with variance 1 conditionally on its number of edges being 2n−1. This and Theorem 1.6 in [26] for α= 1/2 and r= 1, imply that the number of edges with more than one cut is of order less that√

n.

We deduce from Theorem 4.2 and Corollary 4.9 that for all r >0,N^(r)_∞-a.s.:

n→lim+∞

X_n^∗ L_n =αΘ

σ·

In the left hand-side, we have the average of the number of records onT_n^∗ (asℓ(T_n^∗) is of the same order asL_n) and in the right hand-side, the ratio Θ/σ appears as the value ofθ(U) for a leaf chosen uniformly according to the normalized mass measurem^T/σ andα is a constant related to the branching mechanism. This result is then natural as intuitively the normalized mass measure is the weak limit of the normalized length measure on T_n.

4.2. Other a.s. convergence results. Recall the definition (3) of the pruned sub-treeTq. LetTbe the set of trees with their mass measure (see [3]). We define the backward filtration G= (Gq, q ≥0) with Gq =σ(Tr, r≥q). Following [4], we get that the random measure:

N(dT^′, dq) =X

i∈I

δ_Ti,θi(dT^′, dq) is underN_∞a point measure on T×R with intensity:

1_{q>0}2ασ_qN^q dT^′

dq.

This means that for every non-negative predictable process (Y(T^′, q), q ∈R₊,T^′ ∈T) with respect to the backward filtration G,

(9) N_∞

Z

Y(T^′, q)N(dT^′, dq)

=N_∞ Z

Yq1_{q>0}2ασ_q dq

, where (Yq=R

Y(T^′, q)N^q[dT^′], q∈R₊) is predictable with respect to the backward filtration G. We refer to [13, 14] for the general theory of random point measures.

Recallσⁱ =m^T(Tⁱ).

Proposition 4.4. We have N_∞-a.e.:

n→lim+∞

√1 n

X

i∈I

1_{σi≥1/n} = 2 rα

πΘ = r2σ

π Z.

Proof. Let K > 0 be large. We consider the G-stopping time τ_K = inf{q;σ_q < K/2α}. We define for everyθ >0 and every positive integer n,

Q_n(θ) =X

i∈I

1_{σi≥1/n}1_{θi>θ}.

We have Q_n(τ_K) =P

i∈I1_{σi≥1/n}1_{σ

θi+<K/2α} so that:

N_∞[Q_n(τ_K)] =N_∞

Z +∞ τ_K

dq σ_qNh

1_{σ≥1/n}e^−αq2σi

≤N_∞

Z +∞ 0

dq min

σ_q, K 2α

Nh

1_{σ≥1/n}e^−αq2σi

= Z _+∞

0

dqN

min

σ, K 2α

e^−αq2σ

Nh

1_{σ≥1/n}e^−αq2σi

= 1

4απ Z +∞

0

dq Z +∞

0

du u^3/2 min

u, K

2α

e^−αq2u Z +∞

1/n

dr

r^3/2e^−αq2r

= 1

8α^3/2√ π

Z

R²

+

du u^3/2

dr r^3/2 min

u, K

2α 1

√u+r1_{r>1/n},

where we used (9) for the first equality, Girsanov formula (4), and the density (2) of the distribution of σ under N. Elementary computations yields there exists a finite constant c which depends on K but not on nsuch that:

(10) N_∞[Q_n(τ_K)] =N_∞

Z +∞ τ_K

dq σ_qNh

1_{σ≥1/n}e^−αq2σi

≤c√

n(1 + log(n)).

Classical results on random point measures imply that the process (N_n(θ∨τ_K), θ≥0), with:

N_n(θ) =Q_n(θ)−2α Z +∞

θ

dq σ_qNh

1_{σ≥1/n}e^−αq2σi

is a backward martingale with respect toG. Moreover, since (Q_n(θ), θ ≥0) is a pure jump process with jumps of size 1, the process (M_n(θ∨τ_K), θ≥0), with:

M_n(θ) =N_n(θ)²−2α Z +∞

θ

dq σ_qNh

1_{σ≥1/n}e^−αq2σi is also a backward martingale with respect toG. Using (10), we get thatN_∞

h N_n⁴(τ_K)/n²2i is less than a constant timesn^−3/2; therefore

+∞

X

n=1

N_n⁴(τ_K) n²

2

is finite inL¹(N_∞) and thus isN_∞-a.e. finite. This implies that N_∞-a.e.:

n→+∞lim

N_n⁴(τ_K) n² = 0.

Moreover, we have by monotone convergence:

√2α n

Z +∞ τ_K

dq σ_qNh

1_{σ≥1/n}e^−αq2σi

= 2α Z +∞

τ_K

dq σ_q Z +∞

1

dr 2√

απr^3/2 e^−αq2nr

N_∞-a.e.

−−−−−→

n→∞ 2

rα π

Z +∞ τK

dqσ_q.

We get that the sequence (Q_n⁴(τ_K)/n², n ≥ 1) converges N_∞-a.e. toward 2p_α

π

R_+∞

τK dq σ_q. Since (Q_n(θ), n≥1) is non-decreasing, we deduce thatN_∞-a.e.:

n→lim+∞

Q_n(τ_K)

√n = 2 rα

π Z _+∞

τK

dq σ_q.

Since σ is finite N_∞-a.e., we get that N_∞-a.e. τ_K = 0 for K large enough. This gives the

result.

Proposition 4.5. We have N_∞-a.e.:

n→+∞lim

√nX

i∈I

σⁱ1_{σi≤1/n} = 2 rα

πΘ = r2σ

π Z.

Proof. The proof is very similar to the proof of Proposition 4.4. We set:

Q_n(θ) =X

i∈I

σⁱ1_{σi≤1/n}1_{θi≥θ}.

Mimicking the proof of Proposition 4.4, we have for some finite constantcwhich depends on K but not on n:

N_∞[Q_n(τ_K)] =N_∞

Z +∞ τK

dq σ_qNh

σ1_{σ≤1/n}e^−αq2σi

≤ 1 8α^3/2√

π Z

R²

+

du u^3/2

dr r^3/2 min

u, K

2α 1

√u+r r1_{r≤1/n}

≤cn^−1/2(1 + log(n))<+∞, as well as:

N_∞

Z +∞ τK

dq σ_qN

hσ²1_{σ≤1/n}e^−αq2σi

≤ 1 8α^3/2√

π Z

R²

+

du u^3/2

dr r^3/2 min

u, K

2α 1

√u+r r²1_{r≤1/n}

≤cn^−3/2(1 + log(n)).

Classical results on random point measures imply that the processes (N_n(θ∨τ_K), θ≥0) and (M_n(θ∨τ_K), θ≥0), with:

N_n(θ) =Q_n(θ)−2α Z +∞

θ

dq σ_qNh

σ1_{σ≤1/n}e^−αq2σi M_n(θ) =N_n(θ)²−2α

Z +∞ θ

dq σ_qN

hσ²1_{σ≤1/n}e^−αq2σi

are backward martingales with respect to G. We get that N_∞ h

n²N_n⁴(τK)2i

is less than a constant times n^−3/2. Following the proof of Proposition 4.4, we deduce that N_∞-a.e.

lim_n→+∞n²N_n⁴(τ_K) = 0. Furthermore, we have:

2α√ n

Z +∞ τK

dq σ_qN h

σ1_{σ≤1/n}e^−αq2σi

= 2α√ n

Z +∞ τK

dq σ_q Z ¹_n

0

dr 2√

απre^−αq2r

= 2α Z +∞

τ_K

dq σ_q Z 1

0

dr 2√

απre^−αq2nr

→2 rα

π Z _+∞

τK

dq σq.

We conclude the proof as in the proof of Proposition 4.4.

4.3. The record process on the real half-line. We consider here the half-line [0,+∞) instead of a real-tree T (the half-line is in fact a real tree that we supposed rooted at 0).

We define the mark processM under M_a (we omit theT = [0,+∞) in the notation), it is a Poisson point measure on [0,+∞)² with intensity 2α1_{{x≥0, 0≤q≤a}}dx dq and we set for every x≥0

θ(x) = min(a,inf{q_i;x_i≤x}) and X(x) =X(0) + X

0<y≤x

1_{{θ(y−)>θ(y)}}.

Remark 4.6. Let us denote by 1≥x₁ > x₂>· · · the jumping times of the process (θ(x),0≤ x≤1) underM_∞. By standard arguments on Poisson point measure, the random variablex₁ is uniformly distributed on [0,1]. Conditionally givenx1, the random variablex2is uniformly distributed on [0, x₁] and so on. We are thus considering the standard stick breaking scheme and the random vector (1−x₁, x₁−x₂, . . .) is distributed according to the Poisson-Dirichlet distribution with parameter (0,1).

For fixed x, θ(x) represents the first time a mark arrives between x and 0 (if it arrives before time a that is if θ(x) < a); and X(x)−X(0) denotes the number of (decreasing) records of the process (θ(u), u∈[0, x]). It is also the number of cuts that appear between x and 0 in the stick-breaking scheme before timea.

By construction θ and (θ, X) are Markov processes. Notice that θ is non-increasing and X is non-decreasing, and M_∞-a.s. X(x) = +∞ for everyx >0.

As most of our further proofs will be based on martingale arguments, let us first compute the infinitesimal generator of the former Markov processes. Notice first that inf{q_i;x_i ≤x} is distributed underM_∞ as an exponential random variable with parameter 2αx. Letg be a bounded measurable function defined on [0,+∞]. For every q∈[0,+∞] andx >0, we have

M_q[g(θ(x))] =M_∞[g(min(q, Y_x))] = e^−2αqxg(q) + Z q

0

g(u) 2αxe^−2αxu du,

whereY_xis exponentially distributed with parameter 2αx. Notice that ifgbelongs toC¹(R⁺) withg^′ bounded onR₊, we have by an obvious integration by parts that, forq ∈[0,+∞] and x >0,

M_q[g(θ(x))] =g(0) + Z q

0

g^′(u) e^−2αxu du.

We can then compute the infinitesimal generator of θ denoted by L. Let g be a bounded measurable function defined on [0,+∞] such that g−g(+∞) is integrable with respect to

the Lebesgue measure onR⁺. Forq∈[0,+∞], we have:

L(g)(q) = lim

x→0

M_q[g(θ(x))]−g(q) x

= lim

x→0−g(q)1−e^−2αqx

x +

Z q 0

2αg(u) e^−2αxu du

= 2α Z _q

0

(g(u)−g(q))du.

Therefore, we get that the processM^g = (Mx^g, x≥0) is a martingale underM_q, where M^g is defined by:

(11) M_x^g=g(θ(x)) + 2α Z _x

0

dy Z _θ(y)

0

g(θ(u))−g(y) du.

Remark 4.7. If furthermoreg belongs toC¹(R⁺) and ifx7→xg^′(x) is integrable with respect to the Lebesgue measure onR⁺, then we have forq∈[0,+∞]:

L(g)(q) =−2α Z q

0

xg^′(x)dx.

Similarly, we can also compute the infinitesimal generator of (θ, X), which we still denote byL. This quantity is of interest only forθ(0) finite. Letgbe a bounded measurable function defined on R⁺×N. For (q, k) ∈ R⁺×N, we denote byM_(q,k) the law of the process (θ, X) starting from (q, k). Standard computations on birth and death processes yield that for (q, k)∈R⁺×N:

L(g)(q, k) = lim

x→0

M_(q,k)[g(θ(x), X(x))]−g(q, k) x

= lim

x→0−g(q, k)1−e^−2αqx

x +

Z _q

0

2αg(u, k+ 1) e^−2αxu du+o(1)

= 2α Z _q

0

(g(u, k+ 1)−g(q, k))du.

In that case, we get that the processM^g= (Mx^g, x≥0) defined by:

(12) M_x^g =g(θ(x), X(x))−2α Z x

0

dy Z θ(y)

0

g(u, X(y) + 1)−g(θ(y), X(y))

du, is a bounded martingale underM_(q,k).

Finally, let us exhibit some martingales associated with the processXwhich show that this process can be viewed as a Poisson process with stochastic intensity 2αθ(u)du. Let n∈ N. Taking g(q, k) =k∧nin (12), we deduce that the process N⁽ⁿ⁾ = (Nx⁽ⁿ⁾, x≥0) defined for x≥0 by:

N_x⁽ⁿ⁾=X(x)∧n−2α Z x

0

θ(u)1_{X(u)<n} du

is a bounded martingale under M_(q,k) (for q < +∞). Notice that for (q, k) ∈ R⁺×N, we have:

M_(q,k)[|N_x⁽ⁿ⁾|]≤M_(q,k)[X(x)∧n] + 2α Z x

0

E_(q,k)[θ(u)]du

=k∧n+ 2α Z x

0

E_(q,k)[θ(u)1_{X(u)<n}]du+ 2α Z x

0

E_(q,k)[θ(u)]du

≤k+ 4αqx,

where we used that X is non-negative in the first equality, that N⁽ⁿ⁾ is a martingale in the second one, and that θ is non-increasing in the last one. As (N⁽ⁿ⁾, n ∈N) converges a.s. to the processN = (N_x, x≥0) defined forx∈R⁺ by:

(13) N_x=X(x)−2α

Z _x

0

θ(u)du,

we deduce thatN is a martingale under M_(q,k) for every (q, k)∈R⁺×N.

By takingg(q, k) =k² in (12) and using elementary stochastic calculus and similar argu- ments as above, we also get that the process M = (M_x, x≥0) defined forx≥0 by:

(14) Mx=N_x²−2α

Z x 0

θ(u)du is a martingale underM_(q,k) for every (q, k)∈R⁺×N.

4.4. Sub-tree spanned bynleaves. We recall here some properties of the sub-tree spanned by nleaves uniformly chosen.

We first recall the density of (h₁, . . . , h_2n−1) under N^(r), see [9] or [32] (Theorem 7.9), see also [18]. We denote by L_n the total length of T_n:

L_n=

2n−1

X

k=1

h_k.

Lemma 4.8. Under N^(r), (h1, . . . , h2n−1) has density:

f_n^(r)(h₁, . . . , h_2n−1) = 2(2n−2)!

(n−1)!

αⁿ

rⁿ L_ne^−αL2n/r1_{h1>0,...,h2n−1>0}.

The random variable L²_n, is distributed under N^(r) as rΓ_n/α where Γ_n is a γ(1, n) random variable with density 1_{x>0}xⁿ⁻¹e^−x/(n−1)!.

Corollary 4.9. We have thatN^(r)-a.s.

n→lim+∞L_n/√ n=p

r/α.

Proof. Using Lemma 4.8, we compute N^(r)

"_+∞

X

n=1

L²_n n − r

α 4#

= r α

+∞

X

n=1

E

"

Γn

n −1 4#

= r α

+∞

X

n=1

1 n²

3 + 1

n

<+∞. This implies thatN^(r)-a.s. P+∞

n=1

L²_n n −_α^r4

is finite which proves the corollary.

We end this section by studying the edge attached to the root defined in (7) whose length is denotedh_∅,n, see (8).

Proposition 4.10. The sequence (√

nh_∅,n, n ≥ 1) converges in distribution under N^(r) to pr/α E₁/2, where E₁ is an exponential random variable with mean 1.

Proof. Let k∈(−1,+∞). We set H_k= (α/r)^k/2N^(r)[h^k_∅,n]. We have using Lemma 4.8, H_k= 2(2n−2)!

(n−1)!

α^n+k/2 r^n+k/2

Z

R²ⁿ⁻¹

+

dh₁. . . dh_2n−1h^k₁ L_ne^−αL2n/r. Consider the change of variables:

u₁= rα

rh₁, · · · , u_2n−2 = rα

rh_2n−2, x= rα

rL_n, with Jacobian equal to ^α_rn−¹₂

.We get:

H_k= 2(2n−2)!

(n−1)!

α^n+k/2 r^n+k/2

Z

R²ⁿ⁻¹

+

r α

k/2

u^k₁r α

1/2

xe^−αx2/r 1_{{u1+···+u2n−2≤x}}r α

_n−¹₂

du₁· · ·du_2n−2dx

= 2(2n−2)!

(n−1)!

Z Z

R²

+

du₁dx1_{u1≤x}u^k₁xe^−x2 Z Z

R²ⁿ⁻³

+

du₂. . . du_2n−21_{{u2+···+u2n−2≤x−u1}}

= 2(2n−2)!

(n−1)!

1 (2n−3)!

Z +∞ 0

dx xe^−x2 Z x

0

dh h^k(x−h)²ⁿ⁻³. Sety=x², to get:

H_k= 2 (2n−2)!

(n−1)!

1

(2n−3)!β(k+ 1,2n−2) Z _+∞

0

dx x^2n+k−1e^−x2

= (2n−2)!

(n−1)!

1

(2n−3)!β(k+ 1,2n−2) Z +∞

0

dy y^n+k2−1e^−r

= (2n−2)!

(n−1)!

1 (2n−3)!

Γ(k+ 1)(2n−3)!

Γ(2n+k−1) Γ(n+k 2)

= Γ(k+ 1) 2^k

Γ(n−¹₂) Γ(n+^k₂ −¹₂),

where, for the last equality, we used twice the duplication formula:

(15) Γ(2n−1)

Γ(n) = 2²ⁿ⁻²Γ(n−1/2)

√π · We observe that lim_n→+∞N^(r)[n^k/2h^k_∅,n] = ₂^k!_{k α}^rk/2

= E[(√

rE₁/(2√

α))^k]. This gives the result, as the exponential distribution is characterized by its moments.

From the proof of Proposition 4.10, we also get the following result.

Lemma 4.11. For all k∈(−1,+∞), we have, whenn goes to infinity:

N^(r)[h^k_∅,n] =r α

k/2 Γ(k+ 1) 2^k

Γ(n−¹₂)

Γ(n+^k₂− ¹₂) ∼(r/α)^k/2n^−k/22^−kΓ(k+ 1).