A note on the scaling limits of contour functions of Galton-Watson trees

Recently, Abraham and Delmas constructed the distributions of super-critical L\'evy trees truncated at a fixed height by connecting super-critical L\'evy trees to (sub)critical L\'evy trees via a martingale transformation. A similar relationship also holds for discrete Galton-Watson trees. In this work, using the existing works on the convergence of contour functions of (sub)critical trees, we prove that the contour functions of truncated super-critical Galton-Watson trees converge weakly to the distributions constructed by Abraham and Delmas.


Introduction
In this note, we are interested in studying the scaling limits of contour functions of Galton-Watson trees. Since the (sub)critical case has been extensively studied, see e.g. [6] and [11], we mainly focus on the super-critical case.
In [6], it was shown that the scaling limits of contour functions of (sub)critical Galton-Watson trees are the height processes which encode the Lévy trees; see also [11] and references therein for some recent developments. In the super-critical case, however, it is not so convenient to use contour functions to characterize either Galton-Watson trees or Lévy trees which may have infinite mass. To this end, Abraham and Delmas in [1] showed that the distributions of super-critical continuous state branching processes ('CSBPs' in short) stopped at a fixed time are absolutely continuous w.r.t. (sub)critical CSBPs, via a martingale transformation. Since the Lévy trees code the genealogy of CSBPs, they further defined the distributions of the super-critical Lévy tree truncated at a fixed height by a similar change of probability.
In this work, we shall show that such distributions defined in [1] arise as the weak limits of scaled contour functions of super-critical Galton-Watson trees cut at a given level. Our main result is Theorem 3.1. A key to this result is the observation shown in Lemma 2.1 which could be regarded as a discrete counterpart of martingale transformation for Lévy trees constructed in [1]. Then by a collection of related results on convergence of Galton-Watson processes to CSBPs, we obtain our main result.
Let us mention some related works here. In [16], the authors studied the local time processes of contour functions of binary Galton-Watson trees in continuous time. Duquesne and Winkel in [7] and [8] also constructed super-critical Lévy trees as increasing limits of Galton-Watson trees. In their works, the trees are viewed as metric spaces and the convergence holds in the sense of the Gromov-Hausdorff distance; see Remark 3.3 below for a discussion of the relation between the present work and [8].
The paper is organized as follows. In Section 2, we recall some basic definitions and facts on trees and branching processes. In Section 3, we present our main result and its proof.

Trees and branching processes 2.1 Discrete trees and Galton-Watson trees
We present the framework developed in [17] for trees; see also [13] for more notation and terminology. Introduce the set of labels An element of U is thus a sequence w = (w 1 , . . . , w n ) of elements of N * , and we set |w| = n, so that |w| represents the generation of w or the height of w. If w = (w 1 , . . . , w m ) and v = (v 1 , . . . , v n ) belong to U, we write wv = (w 1 , . . . , w m , v 1 , . . . , v n ) for the concatenation of w and v. In particular w∅ = ∅w = w.
A (finite or infinite) rooted ordered tree t is a subset of U such that 3. For every w ∈ t, there exists a finite integer k w t ≥ 0 such that if k w t ≥ 1, then wj ∈ t for any 1 ≤ j ≤ k w t (k w t is the number of children of w ∈ t).
Then ∅ is called the root of tree t. Let T ∞ denote the set of all such trees t. For each u ∈ U, define T u = {t ∈ T ∞ : u ∈ t}. We endow T ∞ with the σ-algebra σ(T u , u ∈ U); see [17] for details. Given a tree t, we call an element in the set t ⊂ U a node of t. Denote the height of a tree t by |t| := max{|ν| : ν ∈ t}. For h = 0, 1, 2, . . . , define r h t = {ν ∈ t : |ν| ≤ h}, which is a finite tree. Denote by #t the number of nodes of t. Let T := {t ∈ T ∞ : #t < ∞} be the set of all finite trees. We say that w ∈ t is a leaf of t if k w t = 0 and set So L(t) denotes the set of leaves of t and #L(t) is the number of leaves of t.
To code the finite trees, we introduce the so-called contour functions; see [13] for details. Suppose that the tree t ∈ T is embedded in the half-plane in such a way that edges have length one. Imagine that a particle starts at time s = 0 from the root of the tree and then explores the tree from the left to the right, moving continuously along the edges at unit speed, until all edges have been explored and the particle has come back to the root. Then the total time needed to explore the tree is ζ(t) := 2(#t − 1). The contour function of t is the function (C(t, s), 0 ≤ s ≤ ζ(t)) whose value at time s ∈ [0, ζ(t)] is the distance (on the tree) between the position of the particle at time s and the root. We set C(t, s) = 0 for s ∈ [ζ(t), 2#t].
Given a probability distribution p = {p n : n = 0, 1, . . .} with p 1 < 1, following [17] and [4], call a T ∞ -valued random variable G p a Galton-Watson tree with offspring distribution p if i) the number of children of ∅ has distribution p: ii) for each h = 1, 2, . . . , conditionally given r h G p = t with |t| ≤ h, for ν ∈ t with |ν| = h, k ν G p are i.i.d. random variables distributed according to p.
The second property is called branching property. From the definition we see for t ∈ T where the product is over all nodes ν of t of height h. We have then where the product is over all nodes ν of t. Define m(p) = k≥0 kp k . We say G p is sub-critical (resp. critical, super-critical) if m(p) < 1(resp. m(p) = 1, m(p) > 1).
Given a tree t ∈ T ∞ , for a ∈ N, define r a t = {ν ∈ t : |ν| ≤ a}. Then r a t is a finite tree whose contour function is denoted by {C a (t, s) : s ≥ 0}. For k ≥ 0, we denote by Y k (t) the number of individuals in generation k: Given a probability measure p = {p k : k ≥ 0} with k≥0 kp k > 1. Let G p be a super-critical Galton-Watson tree with offspring distribution p. Then P(#G p < ∞) = f (p), where f (p) is the minimal solution of the following equation of s: Let q = {q k : k ≥ 0} be another probability distribution such that Then k≥0 kq k < 1. Let G q be a subcritical GW tree with offspring distribution q. Note that (Y k (G q ), k ≥ 0) is a Galton-Watson process starting from a single ancestor with offspring distribution q. We first present a simple lemma.
Lemma 2.1 Let F be any nonnegative measurable function on T. Then for t ∈ T, and for any a ∈ N, Proof. (2.2) is just (4.8) in [2]. The proof of (2.3) is straightforward. Set gen(a, t) = {ν ∈ t : |ν| = a}. By (2.1), for t ∈ T, we have We also have Then (2.3) follows from the fact that given a tree t ∈ T,

Remark 2.1
It is easy to see that (f (p) −Yn(G q ) , n ≥ 0) is a martingale with respect to F n = σ(r n G q ). In fact, by the branching property, we have for all 0 ≤ m ≤ n, Since contour functions code finite trees in T, we immediately get the following result.
Corollary 2.1 For any nonnegative measurable function F on C(R + , R + ) and a ∈ N, Lemma 2.1 could be regarded as a discrete counterpart of the martingale transformation for Lévy trees in Section 4 of [1]; see also (2.12) below in this paper. To see this, we need to introduce continuous state branching processes and Lévy trees.
In the sequel of this paper, we will assume the following assumptions on ψ are in force: (H1) The Grey condition holds: The Grey condition is equivalent to the a.s. finiteness of the extinction time of the corresponding CSBP. This assumption is used to ensure that the corresponding height process is continuous.
(H2) The branching mechanism ψ is conservative: for all ε > 0, The conservative assumption is equivalent to the finiteness of the corresponding CSBP at all time.
(ii) For any nonnegative random variable measurable w.

Height processes
To code the genealogy of the ψ-CSBP, Le Gall and Le Jan [15] introduced the so-called height process, which is a functional of the Lévy process with Laplace exponent ψ; see also Duquesne and Le Gall [6]. Assume that ψ is (sub)critical satisfying (H1). Let P ψ be a probability measure on D such that under P ψ , X = (X t , t ≥ 0) is a Lévy process with nonnegative jumps and with Laplace exponent ψ: The so-called continuous-time height process denoted by H with sample path in C(R + , R + ) is defined for every t ≥ 0 by: where the limit exists in P ψ -probability and I s t = inf s≤r≤t X r ; see [6]. Under P ψ , for a ≥ 0, the local time of the height process at level a is the continuous increasing process (L a s , s ≥ 0) which can be characterized via the approximation Furthermore, for any nonnegative measurable function g on R + , For any x > 0, define where I t = inf 0≤r≤t X r . By Theorem 1.4.1 of [6], the process (L a Tx , a ≥ 0) under P ψ is distributed as a ψ-CSBP started at x.
Let C := C(R + , R + ) be the space of nonnegative continuous functions on R + equipped with the supmum norm. Denote by (e t , t ≥ 0) the canonical process of C. Denote by P ψ x the law of (H t∧Tx , t ≥ 0) under P ψ . Then P ψ x is a probability distribution on C. Set Z a = L a Tx under P ψ x , i.e.,

Super-critical Lévy trees
Height processes code the genealogy of (sub)critical CSBPs. However, for super-critical CSBPs, it is not so convenient to introduce the height process since the super-critical CSBPs may have infinite mass. Abraham and Delmas in [1] studied the distributions of trees cut at a fixed level, where a super-critical Lévy trees was constructed via a Girsanov transformation. We recall their construction here. For any f ∈ C(R + , R + ) and a > 0, define where we make the convention that inf ∅ = +∞. Then we define Note that π a • π b = π a for 0 ≤ a ≤ b. Let ψ be a super-critical branching mechanism satisfying (H2). Denote by q * the unique (positive) root of ψ ′ (q) = 0. Then the branching mechanism ψ q (·) = ψ(· + q) − ψ(q) is critical for q = q * and sub-critical for q > q * . We also have γ > q * . Because super-critical branching processes may have infinite mass, in [1] it was cut at a given level to construct the corresponding genealogical continuum random tree. Define Define a filtration H a = σ(π a (e)) ∨ N , where N is the class of P ψq x negligible sets. By (2.9), we have M ψq,−q is H-adapted. Proof. See Theorem 2.2 and arguments in Section 4 in [1].
Define the distribution P ψ,a x of the ψ-CRT cut at level a with initial mass x, as the distribution of π a (e) under M ψq,−q a dP ψq x : for any non-negative measurable function F on C(R + , R + ), which do not depend on the choice of q ≥ q * ; see Lemma 4.1 of [1]. Taking q = γ in (2.11), we see and (e −γx+γZa , a ≥ 0) under P ψγ x is an H-martingale with mean 1.

Remark 2.2 P ψ,a
x gives the law of super-critical Lévy trees truncated at height a. Then the law of the whole tree could be defined as a projective limit. To be more precise, let W be the set of C(R + , R + )-valued functions endowed with the σ-field generated by the coordinate maps. Let (w a , a ≥ 0) be the canonical process on W. Proposition 4.2 in [1] proved that there exists a probability measureP ψ x on W such that for every a ≥ 0, the distribution of w a underP ψ x is P ψ,a x and for 0 ≤ a ≤ b π a (w b ) = π aP ψ x − a.s.

Remark 2.3
The above definitions of E ψ,a x andP ψ x are also valid for (sub)critical branching mechanisms.

From Galton-Watson forests to Lévy forests
Comparing (2.6) with (2.12), one can see that the l.h.s. are similar. The super-critical trees (discrete or continuum) truncated at height a are connected to sub-critical trees, via a martingale transformation. Motivated by Duquesne and Le Gall's work [6], which studied the scaling limit of (sub)critical trees, one may hope that the laws of suitably rescaled super-critical Galton-Watson trees truncated at height a could converge to the law defined in (2.12). Our main result, Theorem 3.1, will show it is true. For each integer n ≥ 1 and real number x > 0, • Let [x] denote the integer part of x and let ⌈x⌉ denote the minimal integer which is larger than x.
is a Galton-Watson process with offspring distribution p (n) starting from [nx].
• For a ∈ N, define the contour function of trees cut at level a, C p (n) ,x a = (C p (n) ,x a (t), t ≥ 0), by concatenating the contour functions (C(r a G p (n) • For a ∈ R + , define C p (n) ,x a = π a (C p (n) ,x ⌈a⌉ ).
• If k≥0 kp (n) k ≤ 1, then we define the contour function C p (n) ,x = (C p (n) ,x (t), t ≥ 0) by concatenating the contour functions (C(G p (n) Let (γ n , n = 1, 2, . . .) be a nondecreasing sequence of positive numbers converging to ∞. Define where g p (n) is the generating function of p (n) , and define a probability measure on [0, ∞) by We then present the following statements. By → we mean convergence in distribution.
Theorem 3.1 Let ψ be a branching mechanism satisfying (H1) and (H2). Assume that (A1) or (A2) holds. Suppose in addition that for every δ > 0, Then for x > 0, and for any bounded continuous function F on C(R + , R + ) and every a ≥ 0, Before proving the theorem, we would like to give some remarks.
which, together with (3.4), gives where the last inequality follows from our assumption (H1); see Chapter 10 in [12] for details.
Remark 3.3 Some related work on the convergence of discrete Galton-Watson trees has been done in [6] and [8]. In [6], only the (sub)critical case was considered; see Theorem 3.2 below. In Theorem 4.15 of [8], a similar work was done using a quite different formalism. The assumptions there are same as our assumptions in the Theorem 3.1. But the convergence holds for locally compact rooted real trees in the sense of the pointed Gromov-Hausdorff distance, which is a weaker convergence. Thus Theorem 3.1 implies that the super-critical Lévy trees constructed in [1] coincides with the one studied in [8]; see also [3] and [7].
We then present a variant of Theorem 2.3.1 and Corollary 2.5.1 in [6] which is essential to our proof of Theorem 3.1. [δγn] (0) n > 0. (3.6) and for any bounded continuous function F on C(R + , R + ) × D(R + , R + ), Proof. The comments following Theorem 2.3.1 in [6] give (3.7). And by Corollary 2.5.1 in [6], we have On the other hand, let C a be the set of discontinuities of π a . (2.10) yields Then by arguments on page 746 in [14], P ψ x (D a ) = 0. Then (3.8) follows readily from Theorem 2.7 in [5].
This contradicts the fact that f (p (n) ) = e −an/n is the minimal solution of g p (n) (s) = s. Thusγ = γ which implies that lim n→∞ a n = γ and lim n→∞ f (p (n) ) [nx] = e γx for any x > 0.
We are in the position to prove Theorem 3.1.
Proof of Theorem 3.1: With Theorem 3.2 in hand, we only need to prove the result when ψ is super-critical. The proof will be divided into three steps.
which is just the right hand side of (3.5). We have completed the proof. Remark 3.4 Write C n t = γ −1 n C p (n) ,x (2nγ n t) for simplicity and recall that (w a , a ≥ 0) denotes the canonical process on W. Suppose that the assumptions of Theorem 3.1 are satisfied. Then one can construct a sequence of probability measuresP p n x on W such that for every a ≥ 0, the distribution of w a underP p n x is the same as π a (C n ) and for 0 ≤ a ≤ b, π a (w b ) = π aP p (n) x − a.s.
We then haveP p n x →P ψ x as n → ∞. (3.21) Remark 3.5 In [1], an excursion measure ('distribution' of a single tree) was also defined. However, we could not find an easy proof of convergence of trees under such excursion measure.