Zooming in at the root of the stable tree

We study the shape of the normalized stable Lévy tree T near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form as max( α, β ) → ∞ , where µ is the mass measure on T , H ( x ) is the height of x and σ r,x (resp. h r,x ) is the mass (resp. height) of the subtree of T above level r containing x . Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees.


Introduction
Stable trees are special instances of Lévy trees which were introduced by Le Gall and Le Jan [23] in order to generalize Aldous' Brownian tree [4]. More precisely, stable trees are compact weighted rooted real trees depending on a parameter γ ∈ (1, 2], with γ = 2 corresponding to the Brownian tree, which encode the genealogical structure of continuous-state branching processes with branching mechanism ψ(λ) = λ γ . As such, they are the possible scaling limits of Bienaymé-Galton-Watson trees with critical offspring distribution belonging to the domain of attraction of a stable distribution with index γ ∈ (1, 2], see Duquesne [10] and Kortchemski [22]. They also appear as scaling limits of various models of trees and graphs, see e.g. Haas and Miermont [20], and are intimately related to fragmentation and coalescence processes, see Miermont [25,26] and Berestycki, Berestycki and Schweinsberg [5]. Stable trees can be defined via the normalized excursion of the so-called height process which is a local time functional of a spectrally positive Lévy process. We refer to Duquesne and Le Gall [11] for a detailed account. See also Duquesne and Winkel [14], Goldschmidt and Haas [18], Marchal [24] for alternative constructions. In the present paper, we study the shape of the normalized stable tree T (i.e. the stable tree conditioned to have total mass 1) near its root. More precisely we show that, after zooming in at the root of T and rescaling, one gets the continuous analogue of the Kesten tree, that is a random real tree consisting of an infinite branch on which subtrees are grafted according to a Poisson point process. In particular, the (rescaled) subtrees near the root of T are independent and the conditioning for the total mass to be equal to 1 disappears when zooming in. This idea to zoom in at the root of the stable tree is closely related to the small time asymptotics -present in the works of Miermont [25] and Haas [19] -of the self-similar fragmentation process F − (t) obtained from the stable tree by removing vertices located under height t. See Remark 4.5 in this direction. As a consequence, we obtain the asymptotic behavior of additive functionals on T of the form with ∀x ∈ T , Z α,β (x) = H(x) 0 σ α r,x h β r,x dr, (1.1) where µ is the mass measure on T which is a uniform measure supported by the set of leaves, H(x) is the height of x ∈ T , that is its distance to the root, and σ r,x (resp. h r,x ) is the mass (resp. height) of the subtree of T above level r containing x. Before stating our results, we first introduce some notations. Let T be the space of weighted rooted compact real trees, that is the set of compact real trees (T, d) endowed with a distinguished vertex ∅ called the root and with a nonnegative finite measure µ. We equip the set T with the Gromov-Hausdorff-Prokhorov topology, see Section 2 for a precise definition.
Define a rescaling map R γ : T × (0, ∞) → T by R γ ((T, ∅, d, µ), a) = T, ∅, ad, a γ/(γ−1) µ . (1.2) In words, R γ ((T, ∅, d, µ), a) is the tree obtained from (T, ∅, d, µ) by multiplying all distances by a and all masses by a γ/(γ−1) . Moreover, define for every (T, ∅, d, µ) ∈ T norm γ (T ) = R γ (T, µ(T ) −1+1/γ ), (1.3) which is the tree T normalized to have total mass 1 and where distances are rescaled accordingly. Denote by N (1) the distribution of the normalized stable tree with total mass 1, see Section 3 for a precise definition. Under N (1) , let U be a uniformly chosen leaf, that is U is a T -valued random variable with distribution µ. Denote by T i , i ∈ I U the trees grafted on the branch ∅, U joining the root ∅ to the leaf U , each one at height h i and with total mass σ i = µ(T i ), see Figure 1. Fix f : (0, ∞) → (0, ∞) (this represents the speed at which we zoom in) and define for every ε > 0 a point measure on [0, ∞) 2 × T by δ (ε −1 hi,ε −γ/(γ−1) σi,normγ (Ti)) . (1.4) Finally, for any metric space X, we denote by M p (X) the space of point measures on X equipped with the topology of vague convergence.
Our first main result states that the measure N f ε (U ) converges to a Poisson point process which is independent of the underlying tree T and of H(U ). Theorem 1.1. Let T be the normalized stable tree with branching mechanism ψ(λ) = λ γ where γ ∈ (1,2]. Conditionally on T , let U be a T -valued random variable with distribution µ under N (1) . Let (T s , s 0) be a Poisson point process with intensity N B given by (4.1), independent of (T , H(U )). Let Φ : [0, ∞) 2 × T → [0, ∞) be a measurable function such that there exists C > 0 such that for every h 0 and T ∈ T, we have |Φ(h, b, T ) − Φ(h, a, T )| C|b − a|.  In other words, zooming in at the speed f(ε) = ε gives a finite branch on which subtrees are grafted in a Poissonian manner, whereas zooming in at a slower speed gives an infinite branch at the limit. Notice that the convergence (1.5) is stronger than convergence in distribution for the vague topology (1.6) as it holds for functions Φ with very few regularity assumptions: Φ(h, a, T ) is only Lipschitz-continuous with respect to a instead of (Lipschitz-)continuous with respect to (h, a, T ) with bounded support. In particular, this could allow to consider local time functionals of the tree.
As an application of this result, we study the asymptotic behavior as max(α, β) → ∞ of additive functionals Z α,β on the stable tree T . Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees, see Delmas, Dhersin and Sciauveau [9] or Abraham, Delmas and Nassif [1] where Zooming in at the root of the stable tree the finiteness of Z α,β . For example, let us mention the total path length and the Wiener index which when properly scaled converge respectively to Z 0,0 and Z 1,0 . Fill and Janson [16] considered the case γ = 2 and β = 0 (i.e. functionals of the mass on the Brownian tree) and proved that there is convergence in distribution as α → ∞ of Z α,0 properly normalized to Their proof relies on the connection between the normalized Brownian excursion which codes the Brownian tree and the three-dimensional Bessel bridge. Our aim is twofold: we extend their result to the non-Brownian stable case γ ∈ (1, 2) while also considering polynomial functionals depending on both the mass and the height. We use a different approach relying on the Bismut decomposition of the stable tree.
Going back to the connection with the self-similar fragmentation process Once this is established, one can argue that only the largest fragment F − 1 contributes to the limit, the others being negligible, then use [19,Corollary 17] which implies that 1 − F − 1 properly normalized converges in distribution to a (1 − 1/γ)-stable subordinator S, to get the convergence of Z α,0 to ∞ 0 e −St dt. In the present paper, we do not adopt this approach as it does not allow to consider functionals of the height (that is β = 0).
Let us briefly explain why we get a subordinator S at the limit. It is well known that µ is supported on the set of leaves of T . Let x ∈ T be a leaf and recall that σ r,x is the mass of the subtree above level r containing x. Since the total mass of the stable tree is 1, the main contribution to Z α,β (x) as α → ∞ comes from large subtrees T r,x with r close to 0. The height h r,x of such subtrees is approximately h − r. On the other hand, their mass is equal to 1 minus the mass we discarded from the subtrees grafted on the branch ∅, x at height less than r. By Theorem 1.1, subtrees are grafted on ∅, x according to a point process which is approximately Poissonian, at least close to the root ∅. Thus the mass σ r,x is approximately 1 − S r . Theorem 5.4 is slightly more general: we prove joint convergence in distribution of α 1−1/γ h −β Z α,β and α 1−1/γ h −β Z α,β (U ), where U ∈ T is a leaf chosen uniformly at random (i.e. according to the measure µ), to the same random variable. In other words, taking the average of Z α,β (x) over all leaves yields the same asymptotic behavior as taking a leaf uniformly at random. This is due to the following observations: a) a uniform leaf U is not too close to the root with high probability in the sense that its most recent common ancestor with x * has height greater than ε, where x * is the heighest leaf of T , b) when taking the average over all leaves, the contribution of those leaves whose most recent common ancestor with x * has height less than ε is negligible, and c) for those x ∈ T whose most recent common ancestor with x * has height greater than ε, the main contribution to Z α,β (x) comes from large subrees T r,x with r ε, these subtrees are common to all such leaves as T r,x = T r,x * . This is made rigorous in Lemma 5.3.
Let us make a connection with Theorem 1.18 of Fill and Janson [16]. Recall that the normalized Brownian tree with branching mechanism ψ(λ) = λ 2 is coded by √ 2B ex where B ex is the normalized Brownian excursion, see [11]. Thanks to the representation formula of [9,Lemma 8.6], we see that Fill and Janson's Y (α) = √ 2Z α−1,0 . Thus, we recover their result in the Brownian case γ = 2 when β = 0 (in which case c = 0).
Notice that as long as the exponent β of the height does not grow too quickly, viz. β/α 1−1/γ → 0, the additional dependence on the height makes no contribution at the limit. On the other hand, in the regime β/α 1−1/γ → ∞, the height h β r,x dominates the mass σ α r,x so we get the convergence in probability of Z α,β with a different scaling and there is no longer a subordinator at the limit. See Theorem 6.1 for a more general statement.
Now letting c → ∞, the right-hand side converges to h ∞ 0 e −t dt = h. Thus, one may view Theorem 1.3 as a special case of Theorem 1.2 by saying that, if β → ∞ and β/α 1−1/γ → c ∈ (0, ∞], then we have the convergence in distribution under N (1) We conclude the introduction by giving a decomposition of a general (compact) Lévy tree used in the proof of Theorem 1.2 which is of independent interest. Consider a Lévy tree T under its excursion measure N associated with a branching mechanism ψ(λ) = aλ + bλ 2 + ∞ 0 (e −λr − 1 + λr) π(dr) where a, b 0 and π is a σ-finite measure on (0, ∞) satisfying ∞ 0 (r ∧ r 2 ) π(dr) < ∞. We further assume that the Grey condition holds ∞ dλ/ψ(λ) < ∞ which is equivalent to the compactness of the Lévy tree. We refer to [11,Section 1] for a complete presentation of the subject. For every x ∈ T and every 0 r < r H(x), we let T [r,r ),x = (T r,x \ T r ,x ) ∪ {x r } where x r is the unique ancestor of x at height H(x r ) = r and T r,x is the subtree of T above level r containing x. The following result states that, when x ∈ T and 0 =: r 0 < r 1 < . . . < r n < r n+1 := H(x) are chosen "uniformly" at random under N, then the random trees T [ri−1,ri),x , 1 i n + 1 are independent and distributed as T under N[σ•], see Figure 2. In particular, this generalizes [1, Lemma 6.1] which corresponds to n = 1.
In particular, for every nonnegative measurable functions g i , 1 i n + 1 defined on T, A consequence of this decomposition is the following result giving the joint distribution of T y , the subtree of T above vertex y ∈ T , and H(y) when y is chosen according to the length measure (dy) on the stable tree T (which roughly speaking is the Lebesgue measure on the branches of T ). In particular, this generalizes [1, Proposition 1.6]. Corollary 1.6. Let T be the normalized stable tree with branching mechanism ψ(λ) = λ γ where γ ∈ (1, 2]. Let f and g be nonnegative measurable functions defined on T and [0, ∞) respectively. We have The paper is organized as follows. In Section 2 we define the space of real trees and the Gromov-Hausdorff-Prokhorov topology. In Section 3, we introduce the stable tree, recall some of its properties and prove Theorem 1.5 as well as some other useful results.
2 Real trees and the Gromov-Hausdorff-Prokhorov topology

Real trees
We recall the formalism of real trees, see [15]. A metric space (T, d) is a real tree if the following two properties hold for every x, y ∈ T .
(ii) (Loop-free). If ϕ is a continuous injective map from [0, 1] into T such that ϕ(0) = x and ϕ(1) = y, then we have A weighted rooted real tree (T, ∅, d, µ) is a real tree (T, d) with a distinguished vertex ∅ ∈ T called the root and equipped with a nonnegative finite measure µ. Let us consider a weighted rooted real tree (T, ∅, d, µ). The range of the mapping f x,y described above is denoted by x, y (this is the line segment between x and y in the tree). In particular, ∅, x is the path going from the root to x which we will interpret as the ancestral line of vertex x. We define a partial order on the tree by setting x y (x is an ancestor of y) if and only if x ∈ ∅, y . If x, y ∈ T , there is a unique z ∈ T such that ∅, x ∩ ∅, y = ∅, z . We write z = x ∧ y and call it the most recent common ancestor to x and y. For every vertex x ∈ T , we define its height by H(x) = d(∅, x). The height of the tree is defined by h(T ) = sup x∈T H(x). Note that if (T, d) is compact, then h(T ) < ∞. Let x ∈ T be a vertex. For every r ∈ [0, H(x)], we denote by x r ∈ T the unique ancestor of x at height r. Furthermore, we define the subtree T r,x of T above level r containing x as T r,x = {y ∈ T : H(x ∧ y) r} . (2.1) Equivalently, T r,x = {y ∈ T : x r y} is the subtree of T above x r . Then T r,x can be naturally viewed as a weighted rooted real tree, rooted at x r and endowed with the distance d and the measure µ |Tr,x (the restriction of µ to T r,x ). Note that T 0,x = T . We also define the subtree of T above x by T x := T H(x),x . Denote by σ r,x (T ) = µ(T r,x ) and h r, the total mass and the height of T r,x . For every α, β 0, we define We shall omit the dependence on T when there is no ambiguity, simply writing σ r,x , h r,x and Z α,β (x). For every 0 r < r H(x), we also introduce the notation which defines a weighted rooted real tree, equipped with the distance and the measure it inherits from T and naturally rooted at x r . The next lemma, whose proof is elementary, relates h r,x (T ), the height of the subtree of T above level r containing x, to the total height h(T ).
Zooming in at the root of the stable tree Lemma 2.1. Let T be a compact real tree. For every x ∈ T and r ∈ [0, H(x)], we have h(T ) h r,x (T ) + r.

The Gromov-Hausdorff-Prokhorov topology
We denote by T the set of (measure-preserving, root-preserving isometry classes of) compact real trees. We will often identify a class with an element of this class. So we shall write (T, ∅, d, µ) ∈ T for a weighted rooted compact real tree.
Let us define the Gromov-Hausdorff-Prokhorov (GHP) topology on T. Let (T, ∅, d, µ), (T , ∅ , d , µ ) ∈ T be two compact real trees. Recall that a correspondence between T and T is a subset R ⊂ T × T such that for every x ∈ T , there exists x ∈ T such that (x, x ) ∈ R, and conversely, for every x ∈ T , there exists x ∈ T such that (x, x ) ∈ R. In other words, if we denote by p : T × T → T (resp. p : T × T → T ) the canonical projection on T (resp. on T ), a correspondence is a subset R ⊂ T × T such that p(R) = T and p (R) = T . If R is a correspondence between T and T , its distortion is defined by Next, for any nonnegative finite measure m on T × T , we define its discrepancy with respect to µ and µ by where d TV denotes the total variation distance. Then the GHP distance between T and T is defined as where the infimum is taken over all correspondences R between T and T such that (∅, ∅ ) ∈ R and all nonnegative finite measures m on T × T . It can be verified that d GHP is indeed a distance on T and that the space (T, d GHP ) is complete and separable, see e.g. [3].
The next lemma gives an upper bound for the GHP distance between a tree (T, ∅, d, µ) ∈ T and the tree (T, ∅, ad, bµ) obtained from T by multiplying all distances by a > 0 and the measure µ by b > 0. The proof is elementary and is left to the reader.

Two decompositions of the general Lévy tree
Although in this paper we are only interested in the stable case ψ(λ) = λ γ , we state the results of this section in the general Lévy case. Let T denote a Lévy tree under its excursion measure N associated with a branching mechanism ψ(λ) = aλ + bλ 2 + ∞ 0 (e −λr − 1 + λr) π(dr) (3.1) Zooming in at the root of the stable tree where a, b 0 and π is a σ-finite measure on (0, ∞) satisfying ∞ 0 (r ∧ r 2 ) π(dr) < ∞. We further assume that ∞ dλ/ψ(λ) < ∞ so that the Lévy tree is compact.
Remark 3.1. The Brownian case ψ(λ) = λ 2 corresponds to a = 0, b = 1 and π = 0 while the non-Brownian stable case ψ(λ) = λ γ with γ ∈ (1, 2) corresponds to a = b = 0 and We shall need Bismut's decomposition of the stable tree on several occasions. This is a decomposition of the tree along the ancestral line of a uniformly chosen leaf. We refer the reader to [12,Theorem 4.5] and [2, Theorem 2.1] for more details. We will also need the probability measure P r on T which is the distribution of the Lévy tree starting from an initial mass r > 0. More precisely, take i∈I δ Ti a Poisson point measure on T with intensity r N and define P r as the distribution of the random tree T obtained by gluing together the trees T i at their root. See [2, Section 2.6] for further details.
Before stating the result, we first introduce some notations. Let (T, ∅, d, µ) be a (class representative of a)compact real tree and let x ∈ T . Denote by (x i , i ∈ I x ) the branching points of T which lie on the branch ∅, x , that is those points y ∈ ∅, x such that T \ {y} has at least three connected components. For every i ∈ I x , define the tree grafted on the branch ∅, We can now state Bismut's decomposition, see [12,Theorem 4.5] where we made the change of variables (s 1 , s 2 , . . . , s n+1 ) = (r 1 , r 2 − r 1 , . . . , r n+1 − r n ) for the second equality and used Bismut's decomposition (3.12) together with the fact that s. for the last.

The stable tree and its scaling property
Here, we define the stable tree and recall some of its properties. We refer to [12] for background. We shall work with the stable tree T with branching mechanism ψ(λ) = λ γ where γ ∈ (1, 2] under its excursion measure N: more explicitly, using the coding of compact real trees by height functions, one can define a σ-finite measure N on T with the following properties. (i) Mass measure. N-a.e. the mass measure µ is supported by the set of leaves Lf(T ) := {x ∈ T : T \ {x} is connected} and the distribution on (0, ∞) of the total mass σ := µ(T ) is given by We will make extensive use of the scaling property of the stable tree under N. Recall from (1.2) the definition of R γ and note that if T has total mass σ and height h then R γ (T, a) has total mass a γ/(γ−1) σ and height ah. Furthermore, it is straightforward to show that for all x ∈ T , r ∈ [0, H(x)] and a > 0: The scaling property of the stable tree can be written as follows: da a 1+1/γ · Informally, N (a) can be seen as the distribution of the stable tree T with total mass a.
The next result is a restatement of [17,Proposition 5.7] in terms of trees which gives a version of the scaling property for the stable tree conditioned on its total mass. Recall

Preliminary results on the stable tree
Let (T s , 0 s t) be a Poisson point process on T with intensity N B given by and denote by the random real tree obtained by grafting T s on a branch [t − r, t] at height s for every t − r s t and rooted at t − r, see Figure 3. We refer the reader to [2, Section 2.4] for a precise definition of the grafting procedure. Let  It is shown in the proof of [9, Lemma 4.6], see Section 8.6 and more precisely (8.20) therein, that in the stable case ψ(λ) = λ γ , both τ and S are subordinators defined on [0, t] with Laplace exponent ϕ(λ) = γλ 1−1/γ . In particular, thanks to [30,Section 4]   We now give the following form of Bismut's decomposition which we will use throughout the paper. Denote by D[0, ∞) the space of cadlag functions on [0, ∞) endowed with the Skorokhod J1 topology. By Theorem 3.2 we have, for every measurable function Notice that by definition τ t = S t and S r− = τ t − τ t−r for every r ∈ [0, t]. This will be used implicitly in the sequel. In particular, the following computation will be useful (3.13) where in the last equality we used Lemma 3.5-(i) with F ≡ 1.
Next, as an application of Theorem 3.4, we give the decomposition of the normalized stable tree into n + 1 subtrees. For functions f, g defined on (0, ∞), we denote by f * g their convolution defined by  (3.14) where R γ is defined in (1.2) and In particular, for every n 1 and all nonnegative measurable functions g i , Disintegrating with respect to σ and using the scaling property from Lemma 3.5-(ii), where L denotes the Laplace transform on [0, ∞). On the other hand, again disintegrating with respect to σ, we have (3.18) Zooming in at the root of the stable tree where we set Since this holds for every λ > 0, we deduce that da-a.e. on (0, ∞), Thanks to Lemma 2.2, the mapping a → R γ (T, a 1−1/γ ) is continuous on (0, ∞) for every T ∈ T. We deduce from the dominated convergence theorem that the F i are continuous on (0, ∞) and thus F 1 * . . . * F n+1 too. Similarly, the right-hand side of (3.19) is continuous with respect to a. Therefore the equality holds for every a ∈ (0, ∞). In particular, taking a = 1 proves (3.14) for continuous bounded functions f i : [0, ∞)×T → R. This extends to measurable functions f i : [0, ∞) × T → R thanks to the monotone class theorem. Finally, (3.15) is a direct consequence of (3.14).
In particular, the following corollary will be useful.

Zooming in at the root of the stable tree
In this section, we study the shape of the stable tree in a small neighborhood of its root. The main result, Theorem 4.2, states that after zooming in and rescaling, one sees a branch on which trees are grafted according to a Poisson point process on T with intensity N B given by where we recall from Section 3 that π is given by (3.2) and P r is the distribution of the random tree T obtained by gluing together at their roots a family of trees distributed according to a Poisson point measure with intensity r N.

(4.3)
We are now in a position to give the main result of this section.  Zooming in at the root of the stable tree (ii) If f(ε) = ε, then we have the following convergence in distribution Proof. We only prove (i), the proof of (ii) being similar. Let f : T → R and g : [0, ∞) → R be Lipschitz-continuous and bounded and assume that Φ : [0, ∞) 2 × T → [0, ∞) is measurable and satisfies (4.4). We shall consider the following modification of the Step 1. Set Using Lemma 3.5-(i) and Theorem 3.2, we have (4.7) Step 2. The proof of the following lemma is postponed to Section 7.1. To simplify notation, we introduce g(ε) = 1 − f(ε).
Since (T s , 0 s t) is a Poisson point process, it follows from the definition of T ↓ g(ε)t that (T s , 0 s f(ε)t) is independent of T ↓ g(ε)t . Thus, denoting by (T s , s 0) a Poisson point process with intensity N B which is independent of T ↓ g(ε)t , recalling that τ g(ε)t is a measurable function of T ↓ g(ε)t and making the change of variable u = g(ε)t, we have lim ε→0 Step 3. For fixed λ > 0, we have where we made the change of variable r = ε −1 g(ε)λ −1+1/γ s and used Lemma 4.1 with a = ελ 1−1/γ . (Notice that norm γ (T ) has the same distribution under a N B for every a > 0). Thus, we deduce that a.s. for every u > 0 Step 4. We deduce that a.s. for every u > 0 where the right-hand side is integrable with respect to 1 (0,∞) (u) du ⊗ P thanks to (3.13), it follows by dominated convergence that Step 5. Using Theorem 3.2 and Lemma 3.5-(i) again, we get that where, with a slight abuse of notation, we denote by (T s , s 0) a Poisson point process with intensity N B under N (1) , independent of (T , H(U )). Since H(U ) and (T s , s 0) are independent, this concludes the proof.
As a consequence of Theorem 4.2, the next result gives the asymptotic behavior of the total mass of the subtrees grafted near the root of the stable tree.
Proof. We adapt the arguments of [28, Chapter VII, Section 7.2], see also Theorem 3.1 and Corollary 3.4 in [29]. Since the process S has no fixed points of discontinuity, it is enough to show that the convergence (4.12) holds in T × R × D[0, r] for every r > 0. Fix r > 0 and let δ > 0. Define Recall that for a metric space X, we denote by M p (X) the space of point measures on X equipped with the topology of vague convergence. It is known (see [28, p. 215 Since uniform convergence on [0, T ] implies convergence for the Skorokhod J1 topology, 0 t r)) , (4.14) where S t = s t µ(T s ) is a stable subordinator with Laplace exponent ϕ, independent of (T , H(U )). Finally, we shall prove that for every η > 0 We have (4.16) where in the second inequality we used the Portmanteau theorem together with the following convergence in distribution   ∈ (1, 2). We briefly recall its definition. Consider the normalized stable tree T and denote by (T j , j ∈ J t ) the connected components of the set {x ∈ T : H(x) > t} obtained from T by removing vertices located . .) is defined as the decreasing sequence of masses (µ(T j ), j ∈ J t ). In [19, Section 5.1], Haas obtains the following functional convergence in distribution as a consequence of a more general result (S, F I), (4.17) where the convergence holds with respect to the Skorokhod J1 topology. Here F I is a fragmentation process with immigration and S is a stable subordinator with index 1 − 1/γ representing the total mass of immigrants. At least heuristically, this can be recovered from Theorem 4.2. Let U ∈ T be a leaf chosen uniformly at random. It is not difficult to see that for 0 t H(U ), with high probability as ε → 0, the biggest fragment at time εt is the one containing U . Thus we get 1 − F − 1 (εt) = hi εt σ i and is the decreasing rearrangement of the masses of T εt−hi i for the subtrees grafted at height h i εt. Here we denote by T r = T \ T <r = {x ∈ T : H(x) r} the set of vertices of T above height r. To recover (4.17), we may prove the joint convergence of 5 Asymptotic behavior of Z α,β in the case β/α 1−1/γ → c ∈ [0, ∞) We start by showing that if U ∈ T is a leaf chosen uniformly at random, Z α,β (U ) defined in (1.1) converges in disrtibution after proper rescaling.
Then we have the following convergence in distribution where (S t , t 0) is a stable subordinator with Laplace exponent ϕ given by (3.10), independent of (T , H(U )).
Now a simple application of the continuous mapping theorem gives where the last term converges to 0 as α → ∞. We deduce that lim sup α→∞ N (1) and, thanks to the dominated convergence theorem, The next lemma, whose proof is postponed to Section 7.3, states that taking a leaf uniformly at random or taking the average over all leaves yields the same limiting behavior for Z α,β (x). Recall from (1.1) the definition of Z α,β .  Theorem 5.4. Assume that α → ∞, β 0 and β/α 1−1/γ → c ∈ [0, ∞). Let T be the stable tree with branching mechanism ψ(λ) = λ γ where γ ∈ (1, 2]. Conditionally on T , let U be a T -valued random variable with distribution µ under N (1) . Then we have the following convergence in distribution where S is a stable subordinator with Laplace exponent ϕ given by (3.10), independent of (T , H(U )).
This, together with the estimates (6.7) and (6.8) yields the N (1) -a.s. convergence lim β→∞ E β = h which concludes the proof of the second part of the theorem. Recall that g(ε) = 1 − f(ε). Using the expression of F (ε) from (4.7), we write Therefore it follows that for every t > 0 P-a.s.
where the right-hand side is integrable with respect to 1 (0,∞) (t) dt ⊗ P thanks to (3.13), it follows from (7.4) and (7.5) that Using the inequality |e b − e a | 1 ∧ |b − a| for a b 0, we have Since τ is a stable subordinator with index 1 − 1/γ, we get that We deduce the following convergence in P-probability Thanks to (3.13), it follows from the dominated convergence theorem that Together with (7.7), this gives (7.9) Thanks to (3.13) and the dominated convergence theorem, it is clear that as the process τ is a.s. continuous at t. On the other hand, using the inequality where we used that τ t − τ g(ε)t is independent of τ g(ε)t and is distributed as τ f(ε)t for the first equality and that τ t  It follows from (7.1), (7.6), (7.8) and (7.13) that lim ε→0

Proof of Lemma 5.2
Recall from (5.3) the definition of I α .
Let η > 0. Using that h r,U h, we have εH(x) σ α r,x dr > η , where the last term vanishes thanks to (7.14) and the dominated convergence theorem.