Limits of multiplicative inhomogeneous random graphs and L\'evy trees: Limit theorems

We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-R\'enyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 1997]. In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic [EJP 1998] have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain L\'evy-type processes. We, instead, look at the metric structure of these components and prove their Gromov-Hausdorff-Prokhorov convergence to a class of random compact measured metric spaces that have been introduced in a companion paper. Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general"critical"regime, and relies upon two key ingredients: an encoding of the graph by some L\'evy process as well as an embedding of its connected components into Galton-Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of L\'evy trees, which allows us to give an explicit construction of the limit objects from the excursions of the L\'evy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via regime-specific proofs, for instance: the case of Erd\H os-R\'enyi random graphs obtained by Addario-Berry, Goldschmidt and B. [PTRF 2012], the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang [PTRF 2017], or the power-law case as considered by Bhamidi, Sen and van der Hofstad [PTRF 2018].


Introduction
Motivation and model. Random graphs have generated a large amount of literature. This is even the case for one single model: the Erdős-Rényi graph G(n, p) (graph with n vertices connected pairwise in an i.i.d. way with probability p ∈ [0, 1]). Since its introduction by Erdős and Rényi [23] more than fifty years ago, and the discovery of a phase transition where a "giant connected component" gets born, the pursuit of a deeper understanding of its structure has never stopped. Many landmark results by Bollobás [16], Łuczak [33], Janson, Knuth, Łuczak and Pittel [31] have shaped our grasp of this phase transition. From the point of view of precise asymptotics, one of the most important papers is certainly the contribution of Aldous [3], who introduced a stochastic process point of view and paved the way towards the study of scaling limits of critical random graphs. In that paper, he obtained the asymptotics for the sequence of sizes of the connected components of G(n, p) in the so-called critical window where the phase transition actually occurs. His work made possible the construction by Addario-Berry, Goldschmidt and B. [2] of the scaling limits of these connected components, seen as metric spaces, which also confirmed the limiting fractal (Brownian) nature.
Following [2], the question of identifying the scaling limits has been investigated for more general models of random graphs. Particular attention has been paid to the so-called inhomogeneous random graphs, which exhibit heterogeneity in the node degrees and whose behaviours are often quite different from the Erdős-Rényi graph. (See Fig. 1 for an illustration of this difference). Besides being a theoretical object with intriguing properties, these graphs are also commonly believed to offer more realistic modelling for the complex real-world networks [see, e.g. 34].
In the present work, we consider such an inhomogeneous random graph model that is defined as follows. Let w = (w 1 , w 2 , . . . , w n ) be a sequence of n positive real numbers sorted in nonincreasing order. Interpreting w i as the propensity of vertex i to form edges, we define a random graph G w as follows: the set of its vertices is {1, 2, . . . , n}, the events {i, j} is an edge of G w , 1 ≤ i < j ≤ n, are independent and (1) P {i, j} is an edge of G w = 1 − exp −w i w j /σ 1 (w) , where σ 1 (w) = w 1 + . . . + w n .
The graph G w extends the classical Erdős-Rényi random graph in allowing edges to be drawn with non uniform probabilities, while keeping the independence among edges.
The graph G w has come under different names in the literature, for instance, Poisson random graph in [8,35], the Norros-Reittu graph in [8] or rank-1 model in [11,12,17,38,39]. Here, we will refer to it as the multiplicative graph to emphasise its close connection with the multiplicative coalescent as pointed out by Aldous in [3]. This connection is the starting point of the work [4] of Aldous & Limic who identify the entrance boundary of multiplicative coalescent by looking at the asymptotic distributions of the sizes of the connected components found in G w . The asymptotic regime and the limiting processes found in Aldous & Limic [4] lie at the heart of this paper. Namely, we extend this result to the geometry of the connected components of G w by proving the weak convergence of these connected components as it has been done by Addario-Berry, Goldschmidt and B. [2] for the critical Erdős-Rényi graphs. Our approach relies on the results of a companion paper [19] where we provide a specific coding of G w and an embedding of G w into a Galton-Watson forest, and where we construct the continuous multiplicative graphs that are proven here to be the scaling limits of the discrete models.
More precisely, we equip G w with the graph distance d gr and we introduce the weight measure m w = 1≤i≤n w i δ i on G w . The goal of our article can be roughly rephrased as follows: we construct a class of (pointed and measured) compact random metric spaces (G, d, m) such that the graphs (G wn , ε n d gr , ε n m wn ) weakly converge to (G, d, m) along suitable subsequences (w n , ε n , ε n ). We also prove a similar result where m wn is replaced by the counting measure, the limit G being the same. Of course, here the scaling parameters, ε n and ε n go to 0, so that G is not discrete. The limits we consider hold in the sense of the weak convergence corresponding to Gromov-Hausdorff-Prokhorov topology on the space of (isometry classes of) compact metric spaces equipped with finite measures. To achieve the construction of the possible limiting graphs and to prove the convergence of rescaled multiplicative graphs, we rely on two main new ideas: (1) we code multiplicative graphs by processes derived from a LIFO-queue; (2) we embed multiplicative graphs into Galton-Watson trees whose scaling limits are well-understood. Before discussing further the connections with previous works and in order to explain the advantages of our approach, let us give a brief but precise overview of our results and of the two above mentioned ideas.
Overview of the results. Our approach relies first on a specific coding of w-multiplicative graphs G w via a LIFO-queue and a related stochastic process; the queue actually yields an exploration of G w and a spanning tree that encompasses almost all the metric structure of the graph. The LIFO-queue is defined as follows.
-A single server is visited by n clients labelled by 1, . . . , n; -Client j arrives at time E j and she/he requests an amount of time of service w j ; -The E j are independent exponentially distributed r.v. such that E[E j ] = σ 1 (w)/w j ; -A LIFO (last in first out) policy applies: whenever a new client arrives, the server interrupts the service of the current client (if any) and serves the newcomer; when the latter leaves the queue, the server resumes the previous service.
As mentioned above, the LIFO-queue yields a tree T w whose vertices are the clients: namely, the server is the root (Client 0) and Client j is a child of Client i in T w if and only if Client j interrupts the service of Client i (or arrives when the server is idle if i = 0). We introduce the following.
The quantity Y w t − J w t is the load of the server, i.e. the amount of service due at time t. We call sometimes Y w t the algebraic load of the server. Note that the LIFO-queue is coded by Y w . Then, observe that H w t is the number of clients waiting in the queue at time t. We easily see that H w is the contour (or the depth-first exploration) of T w ; this entails that the graph-metric of T w is entirely encoded by H w : namely, the distance between the vertices/clients served at times s and t in T w is H w t + H w s −2min r∈[s∧t,s∨t] H w r . To get to the graph from the tree T w , we need to include some surplus edges which are sampled from a Poisson point measure. More precisely, conditional on Y w , let (3) P w = 1≤p≤pw δ (tp,yp) be a Poisson pt. meas. on [0, ∞) 2 with intensity Note that a.s. p w < ∞, since Y w −J w is null eventually. We set: Next, we define the set of additional edges S w as the set of the edges connecting the clients served at times s p and t p , for all 1 ≤ p ≤ p w and we then define the graph G w by Namely, G w is the graph obtained by removing the root 0 from T w and adding the edges in S w . The following is proved in the companion paper [19].
Theorem 1.1 (Theorem 2.1 in [19]) G w is distributed as a w-multiplicative random graph as specified in (1).
From this representation of the discrete graphs, one expects that if Y w converges, then the graph should also converge, at least in a weak sense. However, since Y w is not Markovian, it is difficult to obtain a limit for the local-time functional H w , which is the function that encodes the metric. To circumvent this technical difficulty, we embed the non-Markovian LIFO-queue governed by Y w into a Markovian one that is defined as follows.
-A single server successively receives an infinite number of clients; -A LIFO policy applies; -Clients arrive at unit rate; -Each client has a type that is an integer ranging in {1, . . . , n}; the amount of service required by a client of type j is w j ; types are i.i.d. with law ν w = 1 1≤j≤n w j δ j . Namely, let τ k be the arrival-time of the k-th client and let J k be the type of the k-th client; then the Markovian LIFO queueing system is entirely characterised by k≥1 δ (τ k ,J k ) that is a Poisson point measure on [0, ∞) × {1, . . . , n} with intensity ⊗ ν w , where stands for the Lebesgue measure on [0, ∞). To simplify the explanation of the main ideas, we concentrate in this Overview only on the (sub)critical cases where the Markovian queue is recurrent, which amounts to assume that Here, for all r ∈ (0, ∞), we use the notation σ r (w) = 1≤j≤n w r j . The Markovian queue yields a tree T w that is defined as follows: the server is the root of T w and the k-th client to enter the queue is a child of the l-th one if the k-th client enters when the l-th client is being served. One easily checks that T w is a sequence of i.i.d. Galton-Watson trees glued at their root and that their common offspring distribution is (5) µ w (k) = 1≤j≤n w k+1 j σ 1 (w)k! e −w j , k ∈ N.
Observe that k∈N kµ w (k) = σ 2 (w)/σ 1 (w) ≤ 1, which implies that the GW-trees are finite. The tree T w is then coded by its contour process (H w t ) t∈[0,∞) : namely, H w t stands for the number of clients waiting in the Markovian queue at time t and it is given by (6) is the (algebraic) load of the Markovian server. These definitions make sense in the supercritical cases. Note that X w is a spectrally positive Lévy process with initial value 0; it is characterised by its Laplace exponent defined by E[e −λX w t ] = e tψw(λ) , for t, λ ∈ [0, ∞), that is explicitly given by: From this tractable model, we derive the LIFO-queue and the tree T w governed by Y w by a timechange that "skips" some time intervals and that is defined as follows. We colour in blue or red the clients of the Markovian queue in the following recursive way: (i) if the type J k of the k-th client already appeared among the types of the blue clients who previously entered the queue, then the k-th client is red; (ii) otherwise the k-th client inherits her/his colour from the colour of the client who is currently served when she/he arrives (and this colour is blue if there is no client served when she/he arrives: namely, we consider that the server is blue). Note that a client who is the first arriving of her/his type is not necessarily coloured in blue. We easily check that exactly n clients are coloured in blue and their types are necessarily distinct. Moreover, while a blue client is served, note that the other clients waiting in the line (if any) are blue too. Actually, the sub-queue of blue clients corresponds to the previous LIFO queue governed by Y w . More precisely, we set Blue = t ∈ [0, ∞) : a blue client is served at t and θ b,w t = inf s ∈ [0, ∞) : We refer to (98) in Section 3.3 for a precise definition of θ b,w . Then, We refer to Proposition 3.2 and Lemma 3.4 in Section 3.3 for a more precise statement of this equality. This explains how to code G w in terms of the two tractable processes X w and H w derived from the Markovian queue. Such Markovian queues and their coding processes (X w , H w ) have analogues in the continous time and space setting. In our context, the parameters governing such processes are those identified by Aldous & Limic [4] for the eternal multiplicative coalescent. Namely: (7) α ∈ R, β ∈ [0, ∞), κ ∈ (0, ∞) and c = (c j ) j≥1 decreasing and such that j≥1 c 3 j < ∞ .
To simplify, we restrict our explanations to the cases where X does not drift to ∞, which is equivalent to assuming that α ∈ [0, ∞). The tree corresponding to the clients of the continuous analogue of the Markovian queue that is driven by X, is actually the Lévy tree yielded by X, which is defined through its contour process as introduced by Le Gall & Le Jan [32]. To that end, we assume that ψ (as defined in (8)) satisfies the following: which implies that either j c 2 j = ∞ or β = 0; therefore X has infinite variation sample paths. Under Assumption (9), Le Gall & Le Jan [32] (see also Le Gall & D. [21]) prove that there exists a continuous process (H t ) t∈[0,∞) such that the following limit holds true for all t ∈ [0, ∞) in probability: (10) H t = lim We explain further how to make sense of this definition in the supercritical cases. The process H is called the height process associated with X and the processes (X, H) are the continuous analogues of (X w , H w ). We explain in Section 4.2 how to colour the Markovian queue driven by X: namely, we explain how to define a right-continuous increasing time-change (θ b t ) t∈[0,∞) that is the analogue of the discrete one θ b,w . We refer to (144) in Section 4.2 for a formal definition of θ b . Then we define the càdlàg process (11) Y t = X θ b t , t ∈ [0, ∞), that represents the load driving the analogue of the LIFO-queue (without repetitions). As we will see in (143) Section 4.2, Y can be written under the following form: (12) ∀t ∈ [0, ∞), Y t = −αt− where (B t ) t∈[0,∞) is a standard linear Brownian motion starting at 0 and where the E j are independent exponentially distributed r.v. that are independent from B and such that E[E j ] = (κc j ) −1 . The sum in (12), as it is, is informal: it has to be understood in the sense of L 2 semimartingales (see Section 4.2 for a precise explanation). The latter expression of Y can be found in Aldous & Limic [4] who proved that the lengths of the excursions of Y above its infimum (ranked in decreasing order) are distributed as the multiplicative coalescent. We refer to Theorem 4.2 in Section 4.2 for a precise statement of (11).
As it is proved in Theorem 2.6 in [19] (that is recalled in Theorem 4.7, Section 4.2) , there exists a continuous process (H t ) t∈[0,∞) that is an adapted functional of Y such that (13) ∀t ∈ [0, ∞), H t = H θ b t . Here, H is a.s. a continuous process that is called the height process associated with Y and we claim that (Y, H) is the continuous analogue of (Y w , H w ), as justified by limit theorems stated further.
As proved in [19] (and recalled in Lemma 4.8, Section 4.2), the excursion intervals of H above 0 and the excursion intervals of Y above its infimum are the same. Moreover, Proposition 14 in Aldous & Limic [4] (that is recalled in Proposition 4.5, Section 4.2) asserts that these excursions can be indexed in the decreasing order of their lengths. Namely, (14) t ∈ [0, ∞) : where the sequence ζ k = l k −r k , decreases. The continuous analogue of G w is derived from (Y, H) as follows: first, for all s, t ∈ [0, ∞), we define the usual tree pseudometric associated with H: Y u −J u > y p , p ≥ 1.
Here Π plays the role of Π w . Fix k ≥ 1. One can prove that if t p ∈ [l k , r k ], then s p ∈ [l k , r k ]. We define G k as the set [l k , r k ] where we have identified points s, t ∈ [l k , r k ] such that either d H (s, t) = 0 or (s, t) ∈ {(s p , t p ); p ≥ 1 : t p ∈ [l k , r k ]}. It actually yields a metric denoted by d k , on G k ; note that l k and r k are identified and we denote by k the corresponding point in G k ; we denote by m k the measure induced by the Lebesgue measure on [l k , r k ]. The continuous analogue of G w is then the sequence of pointed measured compact metric spaces that is called the (α, β, κ, c)-continuous multiplicative graph. We refer to Section 2.3 (and more specifically see (55)) for a more precise definition. As already mentioned, the main goal of the paper is to prove that G is the scaling limit of sequences of rescaled discrete graphs G wn for a suitable sequence of weights with finite support w n = (w (n) j ) j≥1 that are listed in the nonincreasing order: namely, w (n) j ≥ w (n) j+1 , and w (n) j = 0 for all sufficiently large j. Here, we first set (19) j n := sup j ≥ 1 : w (n) We don't require that j n is equal to n but we want lim n→∞ j n = ∞. Our main result (Theorem 2.4 in Section 2.2) asserts the following.
If the Markovian processes (X wn , H wn ), properly rescaled in time and space, weakly converge to (X, H), then (Y wn , H wn ) converges weakly to (Y, H) with the same scaling.
More precisely, the graphs G wn , or their coding functions, are rescaled by two factors a n and b n tending to ∞; a n is a weight factor and b n is an exploration-time factor. Namely, the rescaled processes to consider are 1 an X wn bn· (or 1 an Y wn bn· ) and it is natural to require a priori that b n = O(a 2 n ) by standard results on Lévy processes. Moreover, if the largest weight "persists" in the limit, then a n w (n) 1 and in general w (n) 1 = O(a n ). In the limit, if two large weights persist, they cannot fuse and they tend not to be connected by an edge. Namely, if the two largest weights persist, then 1−exp(−w (n) 1 w (n) 2 /σ 1 (w n )) → 0 and since w (n) 1 w (n) 2 a n , it entails lim n→∞ a 2 n /σ 1 (w n ) = 0. Next, since b n is an exploration-time factor, we require that b n E[C n ], where C n stands for the number of clients who are served before the arrival of Client 1 (i.e. the client corresponding to the largerst weight w (n) 1 ) in the w n -LIFO queue coding G wn . Let us denote by D n the sum of the weights of the vertices explored before visiting Client 1. It is easy to see that E[C n ] = j≥2 w (n) j /(w (n) j + w (n) 1 ) and that E[D n ] = j≥2 (w (n) j ) 2 /(w (n) j + w (n) 1 ). So, when w (n) 1 persists, we get σ 1 (w n ) a n E[C n ] and σ 2 (w n ) a n E[D n ]. Moreover, in the asymptotic regime that we consider, we require that the number of visited vertices has to be of the same order of magnitude as the sum of the corresponding weights: namely, E[C n ] E[D n ], which corresponds to the criticality assumption: σ 1 (w n ) σ 2 (w n ) that also implies a n b n σ 1 (w n ). These constraints amount to assuming the following a priori estimates: Note here that β 0 = 0 possibly. Then, a more precise statement of Theorem 2.4 is: If (a n , b n , w n ) satisfies (20), Necessary and sufficient conditions on the (a n , b n , w n ) for (21) to hold can be derived from previous results due to Le Gall & D. [21] (let us mention it is not direct: see Proposition 2.2). Namely, suppose that (a n , b n , w n ) satisfy (20); then (21) holds if and only if the following condition are satisfied where (Z wn k ) k∈N stands for a Galton-Watson Markov chain with offspring distribution µ wn given by (5) and with initial state Z wn 0 = a n . Let us mention that Proposition 2.3 shows that for all α ∈ R, β ∈ [0, ∞), β 0 ∈ [0, β], κ ∈ (0, ∞) and c such that j≥1 c 3 j < ∞ and such that Grey's condition (9) is satisfied, there exists a sequence (a n , b n , w n ) n∈N satisfying (20) and (23), so that (22) holds. Proposition 2.3 also shows that in (23), (A) does not necessarily imply (B). Moreover, Proposition 2.3 also provides a more tractable condition that implies (B) in (23) and that is satisfied in all the examples that have been considered previously.
By soft arguments (see Lemma 2.7), the convergence (22) of the coding functions implies that the rescaled sequence of graphs G wn converges, as random metric spaces. As already mentioned, the convergence holds weakly on the space G of (pointed and measure preserving) isometry classes of pointed measured compact metric spaces endowed with the Gromov-Hausdorff-Prokhorov distance (whose definition is recalled in (50) in Section 2.3). Actually, the convergence holds jointly for the connected components of G wn : namely, equip G wn with the weight-measure m wn = j≥1 w (n) j δ j ; let q wn be the number of connected components of G wn ; we index these connected component (G wn k ) 1≤k≤qw n in the decreasing order of their m wn -measure: namely, For the sake of convenience, we complete this finite sequence of connected components by point graphs with null measure to get an infinite sequence of G-valued r.v. (G wn k , d wn k , wn k , m wn k ) k≥1 , where d wn k stands for the graph-metric on G wn k , where wn k is the first vertex/client of G wn k who enters the queue and where m wn k is the restriction of m wn to G wn k . Then, Theorem 2.8 asserts the following: If (a n , b n , w n ) satisfy (20) and (21), then holds weakly on G N * equipped with the product topology. Moreover, Theorem 2.8 also asserts first that we can replace in (25) the weight-measure m wn by the counting measure # = 1≤j≤jn δ j , where j n := sup{j ≥ 1 : w (n) j > 0}, and it also asserts that under the additional assumption √ j n /b n → 0, the connected components can be listed in the decreasing order of their number of vertices: namely, Discussion We now briefly discuss connections to other works. We refer to Section 2.4 for more detailed comments on related papers.  [11,13]: the so-called asymptotic (Brownian) homogeneous case and the power-law case. In these papers the proof strategies greatly differ in these two cases. On the other hand, the remarkable work of Aldous and Limic [4] about the weights of large critical connected components deals with the inhomogeneity in a transparent way. We provide here such a unified approach for the geometry, which works not only for both cases but also for graphs which can be seen as a mixture of the two cases. Furthermore, an easy correspondence (see (58) below) allows us to link our parameters (α, β, κ, c) for the limit objects to the ones parametrising all the extremal eternal multiplicative coalescents, as identified by Aldous & Limic in [4]. We note that our limit theorems are valid in the Gromov-Hausdorff-Prokhorov topology, which controls the distances between all pairs of points, and not just in the Gromov-Prokhorov topology where only distances between finitely many typical points are controlled. (A general result has already been proved by Bhamidi, van der Hofstad & Sen [11] for the Gromov-Prokhorov topology in the special case when β = 0.) In light of this, we believe our work contains an exhaustive treatment of all the possible limits related to those multiplicative coalescents. In the mean time, we remove some technical conditions that had been imposed on the weight sequences in some of the previous works.
Avoiding to compute the law of connected components: The connected components of the random graphs may be described as the result of the addition of "shortcut edges" to a tree; this picture is useful both for the discrete models and the limit metric spaces. The work of Bhamidi, Sen & X. Wang and Bhamidi, van der Hofstad & Sen [10,11] yields an explicit description of the law of the random tree to which one should add shortcuts in order to obtain connected components with the correct distribution.
As in the case of classical random graphs treated in Addario-Berry, B. & Goldschmidt [2], this law involves a change of measure from one of the "classical" random trees, whose behaviour is in general difficult to control asymptotically. Our connected components are described as the metric induced on a subset of a Galton-Watson tree; the bias of the law of the underlying tree is somewhat transparently handled by the procedure that extracts the relevant subset.
More general models of random graphs. While we focus on the model of the multiplicative graphs, the theorems of Janson [30] on asymptotic equivalent models (see Section 2.4) and the expected universality of the limits confers on the results obtained here potential implications that go beyond the realm of this specific model: for instance, random graphs constructed by the celebrated configuration model where the sequence of degrees has asymptotic properties similar to the weight sequence of the present paper are believed to exhibit similar scaling limits; see Section 3.1 in [11] for a related discussion.
Upcoming work. The current version of the limit theorems consider the sequences of connected components in the product topology. The embedding of the graphs in a forest of Galton-Watson forest actually also yields a control on the tail of the sequence, which would allow to strengthen the convergence to p -like spaces as in [2] or [10]; this will be pursued somewhere else as well.
Organisation of the paper In Section 2, we state in precise forms the main results of the paper and Section 2.4 is devoted to the connection with previous results. Section 3 provides results on the discrete model: more specifically, in Section 3.3 a precise definition of the red and blue coding of the Markovian queue is recalled from [19] and new estimates are proved in Section 3.4. In Section 4, we recall from [19] the precise construction of the continuous state-space coding processes Y , H, θ b , etc. The proof of the main limit theorems is done in Section 5 and it proceeds through a sequence of lemmas. Section 5.3 is devoted to the proof of the limit theorem for the processes related to the Markovian queues. An appendix collects some general results (on Laplace transform, Skorokod's topology, limit theorems for random walks, Lévy processes and branching processes) that have been tailored in specific forms to adapt to our need here. We believe this facilitates the reading process.

Main results
Notation. Throughout the paper, N stands for the set of nonnegative integers and N * = N\{0}. A sequence of weights refers to an element of the set ↓ ∞ = (w j ) j≥1 ∈ [0, ∞) N * : w j ≥ w j+1 . For all r ∈ (0, ∞) and all w = (w j ) j≥1 ∈ ↓ ∞ , we set σ r (w) = j≥1 w r j ∈ [0, ∞]. The following subsets of ↓ ∞ will be of particular interest to us.

Convergence results for the Markovian queue.
We fix a sequence w n ∈ ↓ f , and two sequences a n , b n ∈ (0, ∞) that satisfy the a priori assumptions (20). As already mentioned the convergence of the graphs G wn is obtained thanks to the convergence of rescaled versions of Y wn and H wn and the convergence of these two processes is also obtained by the convergence of the Markovian processes into which they are embedded: namely, the asymptotic regimes of (Y wn , H wn ) and of (X wn , H wn ) should be the same. The purpose of this section is to state weak limit-theorems for X wn and H wn . Let us mention that many results of this section rely on standard limit-theorems on random walks, on results due to Grimvall in [26] on branching processes and on results due to Le Gall & D. in [21] on the height processes of Galton-Watson trees. However, the specific form of the jumps and of the offspring distribution of the trees actually requires a careful analysis done in the Proof-Section 5.3.
Recall from (6) the definition of X wn ; recall that the Markovian queueing system induced by X wn yields a tree that is an i.i.d. sequence of Galton-Watson trees with offspring distribution µ wn whose definition is given by (5). Denote by (Z wn k ) k∈N a Galton-Watson Markov chain with offspring distribution µ wn and with initial state Z wn 0 = a n . The following proposition is mainly based on Theorem 3.4 in Grimvall [26] p.1040, that proves weak convergence for Galton-Watson processes to Continuous States Branching Processes (CSBP for short). Recall that a (conservative) CSBP is a [0, ∞)-valued Markov process obtained from spectrally positive Lévy processes via Lamperti's timechange; the law of the CSBP is completely characterised by the Lévy process and thus by its Laplace exponent that is usually called the branching mechanism of the CSBP: we refer to Bingham [14] for more details on CSBP (and see Appendix Section B.2.2 for a very brief account). We denote by D([0, ∞), R) the space of càdlàg functions from [0, ∞) to R equipped with Skorokod's topology and we denote by C([0, ∞), R) the space of continuous functions from [0, ∞) to R, equipped with the topology of uniform convergence on all compact subsets. Proposition 2.1 Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20). Recall from above the definition of X wn and Z wn . Let (X t ) t∈[0,∞) and (Z t ) t∈[0,∞) be two càdlàg processes such that X 0 = 0 and Z 0 = 1. Then, the following holds true.
(i) The following convergences are equivalent.
(i-a) There exists t ∈ (0, ∞) such that 1 an X wn bnt → X t weakly on R.

Remark 2.1 The condition
√ j n /b n → 0 is explained by the second statement in Theorem 2.8 that provides a scaling limit for the connected components of multiplicative graphs listed in the decreasing order of their numbers of vertices.
Recall from (6) the definition of H wn , the height process associated with X wn . Note here that we also deal with supercritical cases. Proposition 2.2 Let α ∈ R, β ∈ [0, ∞), κ ∈ (0, ∞) and c = (c j ) j≥1 ∈ ↓ 3 and let ψ be given by (27). We assume that ψ satisfies (9): namely, ∞ dλ/ψ(λ) < ∞. Let X be a spectrally positive Lévy process with Laplace exponent ψ. Let H be its height process as defined in (10). Let a n , b n ∈ (0, ∞), w n ∈ ↓ f , n ∈ N, satisfy (20) with β 0 ∈ [0, β], (C1), (C2) and (C3). We also assume the following: Then, the joint convergence holds true , R) equipped with the product topology. We also get: Proof. See Section 5.3 (and more specifically Section 5.3.2). Proposition 2.2 strongly relies on Theorem 2.3.1 in Le Gall & D. [21]. However, its proof requires more care than expected at first glance because H wn is not exactly the height process as defined in [21] (it is actually a time-changed version of the so-called contour process as in Theorem 2.4.1 [21] p. 68).
The following proposition provides a practical criterion to check (C4): in particular, it shows that (C4) is always true when β 0 > 0; it also shows that Proposition 2.2 is never void.

Convergence of the processes coding the multiplicative graphs.
To deal with limits of sequences of pinching times, it is convenient to embed ([0, ∞) 2 ) p into (R 2 ) N * by extending any sequence ((s i , t i )) 1≤i≤p ∈ ([0, ∞) 2 ) p by setting (s i , t i ) = (−1, −1), for all i > p. Here, (−1, −1) plays the role of an unspecific cemetery point. We equip (R 2 ) N * with the product topology. Then, the main theorem of paper is the following.
Theorem 2.4 implies the convergence of the coding processes of the connected components of G wn , because each connected component of G wn is coded by an excursion above 0 of H wn and the corresponding pinching points. More precisely, denote by (l wn k , r wn k ), 1 ≤ k ≤ q wn , the excursion intervals of H wn above 0, that are exactly the excursion intervals of Y wn above its infimum process Here the indexation is such that ζ wn k ≥ ζ wn k+1 , where we have set ζ wn k = r wn k −l wn k (if ζ wn k = ζ wn k+1 , then we agree on the convention that l wn k < l wn k+1 ); the excursions processes are then defined as follows: (38) ∀k ∈ {1, . . . , q wn }, ∀t ∈ [0, ∞), H wn k (t) = H wn (l wn k +t)∧r wn k .
We next define the sequences of pinching points of the excursions: to that end, recall from (4) the definition of Π wn = (s p , t p ) 1≤p≤pw n that is the sequence of pinching times of G wn ; observe that if t p ∈ [l wn k , r wn k ], then s p ∈ [l wn k , r wn k ]; then, it allows to define the following for all k ∈ {1, . . . , q wn }: where (t k p ) 1≤p≤p wn k increases and where the (l wn k + s k p , l wn k + t k p )'s are exactly the terms (s p , t p ) of Π wn such that t p ∈ [l wn k , r wn k ]. As already specified, we trivially extend each finite sequence Π wn k as a random element of (R 2 ) N * . We pass to the limit for rescaled versions of ((H wn k , l wn k , r wn k , Π wn k )) 1≤k≤qw n . Since q wn tends to ∞, it is convenient to extend this sequence by taking for all k > q wn , H wn k as the null function, l wn k = r wn k = 0 and Π wn k as the sequence constant to (−1, −1). Similarly, recall from (14) the definition of the excursion intervals of H above 0: k≥1 (l k , r k ) = {t ∈ [0, ∞) : H t > 0}, where indexation is chosen in such a way that the sequence ζ k := r k −l k , k ≥ 1, decreases. We define the excursion processes as follows.
The pinching times are defined as follows: recall from (16) and (17) the definition of Π = (s p , t p ) p≥1 . If t p ∈ [l k , r k ], then note that s p ∈ [l k , r k ], by definition of s p . For all k ≥ 1, we then define: (41) Π k = (s k p , t k p ) 1≤p≤p k where (t k p ) 1≤p≤p k increases and where the (l k + s k p , l k + t k p )'s are exactly the terms (s p , t p ) of Π such that t p ∈ [l k , r k ]. Then the following theorem holds true. Theorem 2.5 Under the same assumptions as in Theorem 2.4, the following convergence Proof. See Section 5.2.1.

Convergence of the multiplicative graphs.
We recall here a generic procedure described in [19] which allows us to extract the w-graph G w from the coding processes (Y w , H w , Π w ) and the continuous multiplicative graph from (Y, H, Π). We begin with the coding of trees by real-valued functions.
Coding trees. Let h : [0, ∞) → [0, ∞) be a càdlàg function such that We further assume that one of the following conditions is satisfied: either (a) h takes finitely many values or (b) h is continuous.
Note that the (discrete) height process H w as defined in (2) verifies Condition (a), while in the continuous setting, the process H defined in (13) verifies Condition (b), as asserted by Theorem 4.7. For all We Then, d h induces a true metric on the quotient set T h that we keep denoting by d h and we denote by p h : [0, ζ h ) → T h the canonical projection. Note that if h is continuous, then p h is a continuous map. It follows that in that case the metric space T h is a compact real tree, namely a compact metric space where any pair of points is joined by a unique injective path that turns out to be a geodesic (see Evans [25] for more references on this topic). If, on the other hand, h satisfies Condition (a) in (44), then T h is compact but not connected. It is still tree-like, as d h satisfies the four-point inequality.
The metric space (T h , d h ) also inherits from h the following features: a distinguished point ρ h = p h (0), called the root of T h , and the mass measure m h , which satisfies that for any Borel measurable function f : Pinched metric spaces. Let (E, d) be a metric space and let Π = ((x i , y i )) 1≤i≤p where the elements (x i , y i ) ∈ E 2 , 1 ≤ i ≤ p, are referred to as pinching points. Let ε ∈ [0, ∞), that is interpreted as the length of the edges that are added to E (if ε = 0, then each x i is identified with y i ). Set A E = {(x, y) : x, y ∈ E} and for all e = (x, y) ∈ A E , set e = x and e = y. A path γ joining x to y is a sequence of e 1 , . . . , e q ∈ A E such that e 1 = x, e q = y and e i = e i+1 , for all 1 ≤ i < q. For all e = (x, y) ∈ A E , we then define its length by l e = ε ∧ d(x i , y i ) if (x, y) or (y, x) is equal to some (x i , y i ) ∈ Π; otherwise we set l e = d(x, y). The length of a path γ = (e 1 , . . . , e q ) is given by l(γ) = 1≤i≤q l e i , and we set: ∀x, y ∈ E, d Π,ε (x, y) = inf l(γ) : γ is a path joining x to y .
Limit theorems for multiplicative graphs. Recall from (1) the w n -multiplicative graph G wn . We equip its vertex set with a measure m wn = 1≤j≤jn w (n) j δ j . Recall from (37) the definition of the excursion intervals [l wn k , r wn k ), 1 ≤ k ≤ q wn , of H wn above 0, recall from (38) the definition of the corresponding excursions H wn k (·) of H wn above 0 and recall from (39) the corresponding sets of pinching times Π wn k . We recall that each excursion H wn k (·) corresponds to a connected component G wn k of G wn and we have m wn G wn k = ζ wn k = r wn k −l wn k . Thus, we get Then, G wn k is the pinched (measured pointed) metric space coded by (H wn k , Π wn k ). Namely, Thus, they define the same random element in the space G of the isometry classes of pointed compact measured metric spaces equipped with the Gromov-Hausdorff-Prokhorov distance δ GHP defined in (50). Here, we have denoted by d wn k the graph-distance, by wn k the first vertex explored via the LIFO coding and m wn k stands for the restriction to G wn k of the weight measure m wn . Since q wn tends to ∞, it is convenient to extend the sequence (G wn k ) 1≤k≤qw n by taking G wn k equal to the point space equipped with the null measure for all k > q wn .
Similarly, recall from (14) the definition of the excursion intervals (l k , r k ), k ≥ 1, of H above 0. Recall from (40) the definition of the excursion H k (·) of H above 0 and recall from (41) the definition of the set of pinching times Π k . We recall from (18) the definition of the continuous (α, β, c, κ)- Thus, they define the same random element in the space G. Then, Theorem 2.5 and Lemma 2.7 entail the following theorem.
Theorem 2.8 Under the same assumptions as in Theorem 2.4, the following convergence holds weakly on G N * equipped with the product topology. Denote by µ wn k = j∈G wn k δ j the counting measure on G wn k . Then, the following convergence holds weakly on G N * equipped with the product topology. Recall notation j n = max{j ≥ 1 : w (n) j > 0}. If we furthermore assume that √ j n /b n → 0, then (57) holds when the connected components are listed in the decreasing order of their numbers of vertices: namely, when µ wn 1 G wn 1 ≥ . . . ≥ µ wn qw n G wn qw n . Proof. See Section 5.2.2.

Remark 2.2
The assumption √ j n /b n → 0 may not be optimal for (57) to hold when the connected components are listed in the decreasing order of their numbers of vertices. However, for all α ∈ R, β ∈ [0, ∞), κ ∈ (0, ∞) and c = (c j ) j≥1 ∈ ↓ 3 satisfying (136), this statement is never void since the examples of (a n , b n , w n ) provided in Proposition 2.3 (iii) satisfy √ j n /b n → 0. Moreover, let us mention that all the cases that have been considered previously by other authors satisfy this assumption, as it is pointed out in the next Section 2.4.

Connections with previous results.
Entrance boundary of the multiplicative coalescent. The model of w-multiplicative random graphs appears in the work of Aldous [3] as an extension of Erdős-Rényi random graphs that have close connections with multiplicative coalescent processes. Relying upon this connection, Aldous and Limic determine in [4] the extremal eternal versions of the multiplicative coalescent in terms of the excursion lengths of Lévy-type processes Y (up to rescaling, as explained below); to that end, they consider in Proposition 7 [4] asymptotics of the masses of the connected components of sequences of multiplicative random graphs. The asymptotic regime in Proposition 7 [4] is very close to Assumptions (20) and (C1) -(C3) in our Theorem 2.8.
Let us briefly recall Proposition 7 in [4] since it is used in the proof of Theorem 2.8. Aldous & Limic fix a sequence of weights x n ∈ ↓ f , n ∈ N, and their notation for multiplicative graphs is the following: let (ξ i,j ) j>i≥1 be an array of independent and exponentially distributed r.v. with unit mean; let N (x n ) = max{j ≥ 1 : x (n) j > 0}; then for all q ∈ [0, ∞), Aldous & Limic consider the random graph G n (q) whose set of vertices is V (G n (q)) = {1, . . . , N (x n )} and whose set of edges . . stand for the (eventually null) sequence of the m n -masses of the connected components of G n (q). Then, it is easy to check that X n : q → (ζ k (x n , q)) k≥1 is a multiplicative coalescent process with finite support. Aldous & Limic describe the limit of the processes X n in terms of the excursion-lengths of a process (W κ AL ,−τ AL ,c AL s ) s∈[0,∞) whose law is characterized by three parameters: κ AL ∈ [0, ∞), τ AL ∈ R and c AL ∈ ↓ 3 ; this process is connected to the (α, β, κ, c)-process Y defined in (12) as follows: Proposition 7 [4] assumes the following: and it asserts that for all τ AL ∈ R, X n (σ 2 (x n ) −1 −τ AL ) → (ζ k ) k≥1 , weakly in ↓ 2 , where (ζ k ) k≥1 are the excursion-lengths of W κ AL ,−τ AL ,c AL above its infimum, listed in the decreasing order.
Assumptions (59) are close to (C2) and (C3). More precisely, let (α, β, κ, c) be connected with κ AL , τ AL and c AL as in (58); let a n , b n ∈ (0, ∞) and w n ∈ ↓ f satisfy (20) and (C1) -(C3); then, set: Recall from (1) the definition of G wn , the w n -multiplicative graph. Recall that m wn = j≥1 w (n) j δ j . Recall from Section 2.3 that the G wn k stand for the connected components of G wn listed in the nonincreasing order of their m wn -mass. Then, it is easy to check the following.
Multiplicative graphs in the same basin of attraction as Erdős-Rényi graphs. Bhamidi, van der Hofstad & van Leeuwaarden in [9] prove the scaling limit of the component sizes (number of vertices) for examples of multiplicative graphs which behave asymptotically like the Erdős-Rényi graphs. Bhamidi, Sen & X. Wang in [10] and Bhamidi, Sen, X. Wang & B. in [7] investigate instead the scaling limits of these graphs seen as measured metric spaces. The conditions under which these authors prove their limit theorems slightly differ. We give here a detailed account of these conditions so as to make a connection with our results. In all the cases covered by [7,9,10], the scalings appear to be a n = n 1/3 , b n = n 2/3 and w n is a sequence of length n having the following asymptotic behaviour: For all α ∈ R, set w n (α) = 1−αn − 1 3 w n = 1−αn − 1 3 w (n) j j≥1 . This is a situation covered by Theorem 2.8. Indeed, (62) easily implies that a n , b n , w n (α) satisfy (20), (C1), (C2), (C3), √ j n /b n = n −1/6 → 0, with the parameters α ∈ R, β 0 = 1, β = σ /σ, κ = 1/σ and c = 0. Thus, the branching mechanism is ψ(λ) = αλ + 1 2 σ σ λ 2 . Since β 0 = 1, Proposition 2.3 (i) implies (C4). Then, Theorem 2.8 applies in this setting, which allows us to extend − Theorem 1.1 in [9], which has been proved under the supplementary assumption that there exists a r.v. W : Ω → [0, ∞) such that (Assumption (b) in [9]. ) − Theorem 3.3 in Bhamidi, Sen & X. Wang in [10] that has been proved by quite different methods and under two additional technical assumptions (Assumptions 3.1 (c) and (d)). Turova in [36] also proved a result similar to Theorem 1.1 of [9] for i.i.d. random weight sequences. Let us mention that the convergence under the sole assumptions (62), that we proved, has been conjectured in [10], Section 5, part (c).
Gromov-Prokhorov convergence of multiplicative graphs without Brownian component. In light of the above mentioned result of Aldous & Limic [4] on the convergence of the component masses of the multiplicative graph in the asymptotic regime (59), it is natural to expect that the graph itself should also converge in some sense. The first affirmation in this direction is due to Bhamidi, van der Hofstad and Sen who prove the following in [11]: Denote by C i (q) the i-largest (in m n -mass) connected component of G n (q), that is, m n (C i (q)) = ζ i (x n , q). Equip each component C i (−τ AL + σ 2 (x n ) −1 ) with its graph distance rescaled by σ 2 (x n ) and with the mass measure m n , they prove that under (59) with κ AL = 0, the collection of rescaled metric spaces converge in the sense of Gromov-Prokhorov topology to a collection of measured metric spaces, which are not necessarily compact. They also give an explicit construction of the limiting spaces based upon a model of continuum random tree called ICRT. The Gromov-Prokhorov convergence is equivalent to the convergence of mutual distance of an i.i.d. sequence with law m n , which is weaker than the Gromov-Hausdorff-Prokhorov that we obtain in Theorem 2.8 under the compactness assumption ∞ dλ/ψ(λ). Let us mention that our approach via coding processes is quite distinct from that of Bhamidi, van der Hofstad & Sen in [11].
Lemma 2.10 implies that Theorem 2.8 applies to a n , b n and w n (α) as defined above. This extends Theorem 1.1 in Bhamidi, van der Hofstad & van Leeuwaarden [13] that proves the convergence of the component sizes under the more restrictive assumption that (1.6) in [13]) as well as Theorem 1.2 in Bhamidi, van der Hofstad & Sen [11] (Section 1.1.3) that asserts the convergence of the components as measured metric spaces under the supplementary assumptions that P(W ∈ dx) = f (x)dx, where f is a continuous function whose support is of the form [ε, ∞) with ε > 0, and such that x ∈ [ε, ∞) → xf (x) is nonincreasing (see Assumption 1.1 in [11], Section 1.1.3). Let us mention that both proofs in [13] and in [11] are quite different from ours.
Let us also mention that a solution to the Conjecture 1.3 on fractal dimensions of the components of G right after Theorem 1.2 in [11] is given in the companion paper [19], Proposition 2.7.
General inhomogeneous Erdős-Rényi graphs that are close to be multiplicative. In [30], Janson investigates strong asymptotic equivalence of general inhomogeneous Erdős-Rényi graphs that are defined as follows: denote by P the set of arrays p = (p i,j ) j>i≥1 of real numbers in [0, 1] such that N p = sup{j ≥ 2 : 1≤i<j p i,j > 0} < ∞; the p-inhomogeneous Erdős-Rényi graph G(p) is the random graph whose set of vertices is {1, . . . , N (p)} and whose random set of edges E (G(p)) is such that the r.v. (1 {{i,j}∈E (G(p))} ) 1≤i<j≤N (p) are independent and such that P({i, j} ∈ E (G(p))) = p i,j . The asymptotic equivalence is measured through the following function ρ that is defined for all More precisely, let p n , q n ∈ P , n ∈ N; then Theorem 2.2 in Janson [30] implies that there are couplings of G(p n ) and G(q n ) such that lim n→∞ P(G(p n ) = G(q n )) = 0 if and only if We then apply this result as follows: let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy the assumptions of Theorem 2.8; we set (20); next, as proved in Janson [30] In particular, let h : In this case, for all sufficiently large n, by (C2) and (20). Cases where h(x) = 1∧x have been studied by Chung & Lu [20] and van der Esker, van der Hofstad & Hooghiemstra [37]; the cases where h(x) = 1−e −x , was first studied by Aldous [3] and Aldous & Limic [4] and the previous cited papers [2,7,[9][10][11]13] [30]) Assume that a n , b n , w n satisfy the same assumptions as in Theorem 2.4 (and thus as in Theorem 2.8). We furthermore assume that √ j n /b n → 0. We define p n by (69). Let q n ∈ P . We define (u (n) i,j ) j>i≥1 by (69) and we suppose (70). Then, there exist couplings of G(q n ) and G wn such that (71) lim n→∞ P(G wn = G(q n )) = 0 and the weak limit (57) in Theorem 2.8 holds true in the same scaling for the connected components of G(q n ) that are listed in the decreasing order of their numbers of vertices and that are equipped with the graph distance and the the counting measure. In particular, its holds true when u (n) 3 Preliminary results on the discrete model.

Height and contour processes of Galton-Watson trees.
Let us briefly recall basic notation about the coding of trees. We first denote by U = n∈N (N * ) n the set of finite words written with positive integers; here, (N * ) 0 is taken as {∅}. The set U is totally ordered by the lexicographical order ≤ lex (the strict order is denoted by < lex ).
Let u = [i 1 , . . . , i n ] ∈ U be distinct from ∅. We set |u| = n that is the length or the height of u, with the convention that |∅| = 0. We next set ← − u = [i 1 , . . . , i n−1 ] that is interpreted as the parent of u (and if n = 1, then ← − u is taken as ∅). More generally, for all p ∈ {1, . . . , n}, we set u |p = [i 1 , . . . , i p ], with the convention: u |0 = ∅. Note that ← − u = u |n−1 . We will also use the following notation: A rooted ordered tree can be viewed as a subset t ⊂ U such that the following holds true.
is the i-th child of u; k u (t) + 1 is the degree of the vertex u in the graph t when u is distinct from the root. Implicitly, if k u (t) = 0, then there is no child stemming from u and assertion (c) is trivially satisfied. Note that the subtree stemming from u that is θ u t = {v ∈ U : u * v ∈ t} is also a rooted ordered tree.
Let T be the set of rooted ordered trees that is equipped with the sigma-field F (T) generated by the sets {t ∈ T : u ∈ t}, u ∈ U. Then, a Galton-Watson tree with offspring distribution µ (a GW(µ)-tree, pendent with the same law as τ under P. Assume that µ(1) < 1. Recall that τ is a.s. finite if and only if µ is critical or subcritical: namely, if and only if k≥1 kµ(k) ≤ 1.
A Galton-Watson forest with offspring distribution µ (a GW(µ)-forest, for short) is a random tree T such that k ∅ (T) = ∞ and such that the subtrees (θ [k] T) k≥1 stemming from ∅ are i.i.d. GW(µ)trees. We next recall how to encode a GW(µ)-forest T thanks to three processes: its Lukasiewicz path, its height process and its contour process. We denote by (u l ) l∈N the sequence of vertices of T such that u 0 = ∅ and such that for all l, u l+1 is the smallest vertex of T with respect to the lexicographical order that is larger than u l . If µ is critical or subcritical, then (u l ) l∈N exhausts all the vertices of T; however, if µ is supercritical (namely if k≥1 kµ(k) > 1), then (u l ) l∈N exhausts the vertices of T that are situated before (or on) the first infinite line of descent. We first set: l∈N is the Lukasiewicz path associated with T and (Hght T l ) l∈N is the height process associated with T. We recall from Le Gall & Le Jan [32] the following results.
(i) V T is distributed as a random walk starting from 0 and with jump-law T if and only if (u l ) |1 = p. Then, we get: The contour process of T is informally defined as follows: suppose that T is embedded in the oriented half plane in such a way that edges have length one and that orientation reflects lexicographical order of visit; we think of a particle starting at time 0 from ∅ and exploring the tree from the left to the right, backtracking as less as possible and moving continuously along the edges at unit speed. In (sub)critical cases, the particle crosses each edge twice (upwards first and then downwards). In supercritical cases, the particle only explores the edges that are situated before (or on) the first infinite line of descent in the lexicographical order: the edge on the infinite line of descent are visited once (upwards only) and the edge strictly before the infinite line of descent are visited twice (upwards first and then downwards). For all s ∈ [0, ∞), we define C T s as the distance at time s of the particle from the root ∅. The associated distance d C T as defined in (45) is the graph distance of T in the (sub)critical cases. We refer to Le Gall & D. [21] (Section 2.4, Chapter 2, pp. 61-62) for a formal definition and a formula relating the contour process to the height process.

Coding processes related to the Markovian queueing system.
We fix w = (w 1 , . . . , w n , 0, 0, . . .) ∈ ↓ f and we briefly recall the definition of the Markovian queue as in Introduction: a single server is visited by infinitely many clients; clients arrive according to a Poisson process with unit rate; each client has a type that is a positive integer ranging in {1, . . . , n}; the amount of time of service required by a client of type j is w j ; types are i.i.d. with law Let τ l stand for the time of arrival of the l-th client in the queue and let J l stand for her/his type; then, the queueing system is entirely characterised by the point measure that is distributed as a Poisson point measure on [0, ∞)×{1, . . . , n} whith intensity ⊗ν w , where stands for the Lebesgue measure on [0, ∞). We next introduce the following: Then, X w t − I w t is interpreted as the load of the Markovian queueing system at time t and X w t is the algebraic load of the queue. Note that X w is a spectrally positive Lévy process whose law is determined by its Laplace exponent ψ w : [0, ∞) → R in the following way: Here, recall that σ r (w) = w r 1 +. . .+w r n , r ≥ 0. We call the queueing system recurrent if a.s. lim inf t→∞ X w t = −∞, which means that all the clients will eventually depart. Let us observe that the system is recurrent if and only if σ 2 (w)/σ 1 (w) ≤ 1. If, on the other hand, σ 2 (w)/σ 1 (w) > 1, then α w < 0 and a.s. lim t→∞ X w t = ∞ (the queue will see an accumulation of infinitely many clients). In the sequel, we shall refer to the following cases: The LIFO queueing system governed by X w generates a tree that can be informally defined as follows: the clients are the vertices and the server is the root (or the ancestor); the j-th client to enter the queue is a child of the i-th one if the j-th client enters when the i-th client is served; among siblings, clients are ordered according to their time of arrival. In critical or subcritical cases, it fully defines a Galton-Watson forest; however in supercritical cases, it only defines the part of a Galton-Watson forest situated before the first infinite line of descent. To circumvent this problem, we actually define the tree first and then we couple it with the queueing system as follows.
In what follows, what we mean by a Poisson random subset Π on [0, ∞) with unit rate is the set of atoms of a unit rate Poisson random measure: namely, it is the random subset {e 1 + . . . + e n ; n ≥ 1}, where the e n are i.i.d. exponentially distributed r.v. with unit mean. For all u ∈ U\{∅}, let J(u) and Π u be independent r.v. whose laws are given as follows: J(u) has law ν w as defined in (75) and Π u is a Poisson random subset of [0, ∞) with unit rate. We next define Π ∅ as a Poisson random subset of [0, ∞) with unit rate that is assumed to be independent of (J(u), Π u ) u∈U\{∅} and by convenience, we set J(∅) = 0. For all u ∈ U, we index the points of Π u using the children of u. Formally, we define a map σ : {u * [p] ; p ≥ 1} → Π u as follows: Note that it defines a collection (σ(u)) u∈U\{∅} of r.v. It is easy to check that here is a unique random tree T w : Ω → T such that Clearly T w is distributed as a GW(µ w )-forest where µ w is given by Then we define the point process X w governing the Markovian queueing system as follows: denote by (u l ) l∈N the sequence of vertices of T w such that u 0 = ∅ and such that for all l, u l+1 is the smallest vertex of T w (with respect to the lexicographical order) that is larger than u l . Then we set We also set Recall from (72) that (V Tw l ) l≥0 stands for the Lukasiewicz path associated with T w ; we also recall the notation V Tw l for the quantity inf 0≤k≤l−1 V Tw k − 1.
Lemma 3.1 We keep the notation from above. Then X w as defined by (83) is a Poisson point measure on [0, ∞)×{1, . . . , n} with intensity ⊗ν w and therefore N w as defined by (84) is a Poisson process on [0, ∞) with unit rate. Let X w and I w be derived from X w by (77). For all t ∈ [0, ∞), then the following holds true: . Then, for all a, x ∈ (0, ∞), we get To that end, we need notation: fix u ∈ U\{∅}; then for all 0 ≤ p < |u|, we set: Note that J(u |0 ) = J(∅) = 0 and that w 0 = ∞ (by convention); thus, We next denote by G (u) the sigma-field generated by the r.v.
Elementary properties of Poisson point processes imply that conditionnally given G (u), the Q u p , 0 ≤ p < |u| are independent Poisson random subsets of [0, ∞) with unit rate: they are therefore independent of G (u); by construction they are also independent from the For all u ∈ U\{∅}, we next define s(u) ∈ U and e(u) ∈ [0, ∞) that satisfy s(u l ) = u l+1 and τ l+1 = e(u l ) + τ l . To that end, we first set q = sup p ∈ {0, . . . , |u|} : We also set: Elementary properties on Poisson point processes imply that e(u), J(s(u)) and G (u) are independent, that e(u) is exponentially distributed with unit mean and that J(u) has law ν w . Then, we easily derive from (81) that for all l ∈ N, u l+1 = s(u l ), as already mentioned. It is also easy to deduce from (83) that τ l+1 = e(u l ) + τ l . Thus, τ l+1 − τ l , J(u l+1 ) are independent and they are also independent of G (u l ) and therefore of the r.v. ((τ k , J(u k ))) 1≤k≤l . It implies that X w is a Poisson point measure on [0, ∞)×{1, . . . , n} with intensity ⊗ν w . We next prove inductively that for all l ≥ 1, Proof. Clearly, (87) holds for l = 1. Assume it holds true for l.
We first suppose that q ≥ 1. By comparing (86) and (87), we see that e(u l ) < Z l . Since τ l+1 −τ l = e(u l ) and since X w does not jump on [τ l , τ l+1 ) (by the definition (77)), we get: , the last equality being a consequence of (87) for l. It implies that −I w This proves that (87) holds for l + 1. It also completes the proof of (87) by induction.
Next, it is easy to check that By (87) and elementary properties of Poisson point processes, it shows that conditionally given X w τ l − I w τ l , V Tw l −V Tw l is distributed as a Poisson r.v. with mean X w τ l − I w τ l and elementary arguments entail (i). Next recall from (73) that −V Tw l = (u l ) |1 and recall from (87) that σ((u l ) |1 ) = −I w τ l . Namely, , which completes the proof of (ii). We next prove (85). We fix t ∈ [0, ∞) and to simplify we set It implies (85), which completes the proof of Lemma 3.1.
The contour of T w : estimates. Recall from (6) that H w t stands for the number of clients waiting in the line right after time t. More precisely, for all s, t ∈ [0, ∞) such that s ≤ t, we get The process H w is called the height process associated with X w by analogy with (74), but H w is actually closer to the contour process of T w . To see this, recall that (u l ) l∈N stands for the sequence of vertices of T w listed in the lexicographical order; we identify u l with the l-th client to enter the queueing system. For all t ∈ [0, ∞), we denote by u(t) the client currently served right after time t: namely Then, the length of the word u(t) is the number of clients waiting in the line right after time t: |u(t)| = H w t . We next denote by (ξ m ) m≥1 the sequence of jump-times of H w : namely, ξ m+1 = inf{s > ξ m : H w s = H w ξm }, for all m ∈ N, with the convention ξ 0 = 0. We then set: Note that (ξ m ) m≥1 is also the sequence of jump-times of u and that for all m ≥ 1, (u(ξ m−1 ), u(ξ m )) is necessarily an oriented edge of T w . We then set T w (t) = {u(s); s ∈ [0, t]}, that represents the set of the clients who entered the queue before time t (and the server ∅); T w (t) has N w (t) + 1 vertices (including the server represented by ∅); therefore, Recall from Section 3.1 the definition of the contour and the height processes of T w , denoted resp. by (C Tw t ) and (Hght Tw k ). Then, observe that Since N w is a homogeneous Poisson process with unit rate, Doob's L 2 -inequality combined with (91) and (92) imply the following inequality:

Red and blue processes.
This section contains no new result and we recall here more precisely the embedding of the LIFO queue without repetition coding the multiplicative graph G w into the Markovian queue considered in the previous Section 3.2. This embedding has been introduced in [19] (and it is informally recalled in Introduction). This embedding uses two auxiliary processes, the so-called blue and red processes, that are defined as follows. First, we introduce two independent random point measures on [0, ∞)× {1, . . . , n}: that are Poisson point measures with intensity ⊗ν w , where we recall that stands for the Lebesgue measure and that ν w = 1 1≤j≤n w j δ j . The blue process X b,w and the red process X r,w are defined respectively by Note that X b,w and X r,w are two independent spectrally positive Lévy processes with Laplace exponent ψ w given by (78). For all j ∈ {1, . . . , n} and all t ∈ [0, ∞), we next set: Then the N w j are independent homogeneous Poisson processes with jump-rate w j /σ 1 (w) and the r.v. ( Here Y w is the algebraic load of the following queue without repetition that codes the multiplicative graph G w (as explained in Introduction): a single server is visited by n clients labelled by 1, . . . , n; Client j arrives at time E w j and she/he requests an amount of time of service w j ; a LIFO (last in first out) policy applies: whenever a new client arrives, the server interrupts the service of the current client (if any) and serves the newcomer; when the latter leaves the queue, the server resumes the previous service.
We embed this queue without repetition into a Markovian one that is obtained from (Y w , A w ) and X r,w as follows. We first introduce the following time-change process that will play a prominent role: with the convention that inf ∅ = ∞. We next recall various properties of θ b,w that are used in the sequel.
To that end, let us first note that standard results on Lévy processes (see e.g. Bertoin's book [6] Chapter VII) assert that (γ r,w x ) x∈[0,∞) is a (possibly killed) subordinator whose Laplace exponent is given by: , the largest root of ψ w . Then, w = 0 in the subcritical or critical cases, while w > 0 in the supercritical case. Moreover, in the latter case, −I r,w ∞ := − inf t∈[0,∞) X r,w t is exponentially distributed with parameter w and γ r,w x < ∞ if and only if x < −I r,w ∞ . It follows that the explosion time for θ b,w is given by which is infinite in the critical and subcritical cases and which is a.s. finite in the supercritical cases. Note that θ b,w (T * w −) < ∞ in the supercritical cases. We shall also introduce the following processes: Both processes Λ b,w and Λ r,w are continuous and nondecreasing. Moreover, a.s. lim t→∞ Λ r,w t = ∞. In critical and subcritical cases, we also get a.s.
The following proposition was proved in [19].
Proposition 3.2 We keep the previous notation and we define the process X w by: Then, X w has the same law as X b,w and X r,w : namely, it is a spectrally positive Lévy process with Laplace exponent ψ w as defined in (78). Furthermore, we have Proof. See Proposition 2.2 in [19].
Recall that Blue and Red are the sets of times during which respectively blue and red clients are served (the server is considered as a blue client). Then formally these sets are given by: Note that the union defining Red is countably infinite in critical and subcritical cases and that it is a finite union in supercritical cases where . We next recall from (101) the definition of the time-changes Λ b,w and Λ r,w ; then, we easily check that We have the following properties of X w , θ b,w , etc. that are recalled from [19] (see Figure 2).
We next introduce the red time-change: Then, for all s, t ∈ [0, ∞), θ r,w s+t −θ r,w  Figure 2: Decomposition of X w into X b,w and X r,w . Above, the process X w : clients are in bijection with its jumps; their types are the numbers next to the jumps. The grey blocks correspond to the set Blue. Concatenating these blocks yields the blue process X b,w . The remaining pieces give rise to the red process X r,w . Concatenating the grey blocks but without the final jump of each block yields Y w . Alternatively, we can obtain Y w by removing the temporal gaps between the grey blocks in X w : this is the graphic representation of Observe also that each connected component of Red begins with the arrival of a client whose type is a repeat among the types of the previous blue ones, and ends with the departure of this red client, marked by × on the abscissa.
Embedding of the tree. The previous embedding of the LIFO queue without repetition governed by Y w into the Markovian queue governed by X w yields a related embedding of the trees associated with these queues. More precisely, consider first the queue governed by Y w : the LIFO rule implies that Client i arriving at time E i will leave the queue at the moment inf{t namely the first moment when the service load falls back to the level right before her/his arrival. It follows that the number of clients waiting in queue at time t is given by Recall that we denote by T w the tree formed by the clients in the queue governed by Y w . The process H w is actually the contour (or the depth-first exploration) of T w and the graph-metric d Tw of T w is encoded by H w in the following way: if we denote by V t ∈ {0, 1, . . . , n} the label of the client served at time t (with the understanding that V t = 0 if the server is idle), then Similarly for the Markovian queue governed by the process X w given in Proposition 3.2, we define its associated height process H w by setting H w t to be the number of the clients waiting at time t, namely, Then H w is the contour process of the i.i.d. Galton-Watson forest T w with offspring distribution µ w characterized by (81). Note that in (sub)critical cases, H w fully explores the whole tree T w . However in supercritical cases, the exploration of H w does not go beyond the first infinite line of descent. We shall use the following form of the previsouly mentioned embedding of T w into T w that is recalled from [19].
Lemma 3.4 Following the previous notation, we have 3.4 Estimates on the coloured processes.
We keep notation from the Section 3.3. In this section, we now provide estimates for where recall from (96) that E w j stands for the first jump-time of N w j ; E w j is therefore exponentially distributed with mean σ 1 (w)/w j . Elementary calculations combined with (113) immediately entail the following lemma.
Lemma 3.5 We keep the notation from above. For all (F t )-stopping time T and all a, t 0 , t ∈ (0, ∞), We next discuss the oscillations of X b,w Λ b,w and of X r,w Λ r,w . To that end, let us recall that D([0, ∞), R) stands for the space of R-valued càdlàg functions equipped with Skorokhod's topology. For all y ∈ D([0, ∞), R) and for all intervals I of [0, ∞), we set (116) osc(y, I) = sup |y(s)−y(t)|; s, t ∈ I , that is the oscillation of y on I. It is easy to check that for all a < b < c, where we recall that ∆y(b) = y(b) − y(b−). We also recall the definition of the càdlàg modulus of continuity of y: let z, η ∈ (0, ∞); then, we set Here the infimum is taken on the set of all subdivisions (t i ) 0≤i≤r , of [0, z], r being a positive integer; note that we do not require t r −t r−1 ≥ η. We refer to Jacod & Shiryaev [29] Chapter VI for a general introduction on Skorokod's topology. Recall from (100) the definition of T * w and from (101) the definition of Λ b,w and Λ r,w . Recall from (102) The following lemma is a key argument in the proof of Theorem 2.4. Lemma 3.6 We keep the notation from above. Let η ∈ (0, ∞). Then, the following holds true.
(i) Almost surely, for all (ii) Assume that we are in the supercritical cases (namely, First note that for all intervals I, we get: w is non-decreasing and continuous, and since θ b,w is strictly increasing, we get and by (117), we get There are two cases to consider: ; since it holds true for all subdivisions of [0, z + η] satisfying the conditions as above, we get We are now ready to prove (119). Let us fix z 0 , z ∈ (0, ∞) and let 0 = t 0 < . .
. By (123) (if S i has two points) and by (122) and (124) (if S i reduces to a single point), we get Since it holds true for all subdivisions (t i ) and since z → w z (y(·), η) is nondecreasing, it easily entails (119) if z 1 ≤ θ b,w z 0 ≤ z, which completes the proof of (i). Let us prove (ii). We assume that we are in the supercritical cases. The control of the càdlàg modulus of continuity of X b,w • Λ b,w is more complicated because this process becomes eventually constant after a last jump at time θ b,w (T * w −). To simplify notation we set τ = θ b,w (T * w −). We suppose that z > τ and z 0 > T * w > 2η. We fix z 1 ∈ (0, ∞). There are several cases to consider.
We then extend this to z 1 ≤ τ by using a basic property of the càdlàg modulus of continuity: which implies (120) when z 1 ≤ τ .
• We next assume that There are two subcases to consider.
The proof of (iii) is similar and simpler. Recall from (108) that θ r,w To complete the proof of (iii) we then argue as in the proof of (119).
4 Previous results on the continuous setting.
We next introduce the following process. Note that I ∞ is a.s. finite in supercritical cases and a.s. infinite in critical or subcritical cases. Observe that γ x < ∞ if and only if x < −I ∞ . Standard results on spectrally positive Lévy processes (see e.g. Bertoin's book [6] Ch. VII) assert that (γ x ) x∈[0,∞) is a subordinator (a killed subordinator in supercritical cases) whose Laplace exponent is given for all λ ∈ [0, ∞) by: We set = ψ −1 (0) that is the largest root of ψ. Note that > 0 if and only if α < 0. The following elementary lemma gather basic properties of X that are used further in the proofs.
Lemma 4.1 Let X be a spectrally positive Lévy process with Laplace exponent ψ given by (127) and with initial value X 0 = 0. We assume that there is λ ∈ (0, ∞) such that ψ(λ) > 0. Let ψ −1 be defined by (131) and recall that = ψ −1 (0) that is the largest root of ψ. Let X stand for a spectrally Lévy process with Laplace exponent ψ( + ·) and with initial value 0. Then, the following holds true.
Moreover, for all t ∈ [0, ∞) and for all nonnegative measurable Let E be an exponentially distributed r.v. with parameter that is independent from X (with the convention that a.s. E = ∞ if = 0). Then, (iv) Let (G x ) x∈[0,∞) be a right-continuous filtration such that for all x, y ∈ [0, ∞), γ x is G x -measurable and γ x+y −γ x is independent of G x . Let T be a (G x )-stopping time. Then, for all x, ε ∈ (0, ∞), Proof. The assertions in (i), (ii) and (iii) are (easy consequences of) standard results that can be found e.g. in Bertoin's book [6] Chapter VII. We only need to prove (iv). To that end, first note that the second inequality in (135) is a consequence of a standard inequality combined with (131). Then, note that in the critical or subcritical cases where γ = γ, the first inequality in (135) is a straightforward consequence of the fact that γ is a subordinator. Therefore, we now assume that > 0. Let γ * be a copy of γ that is independent of G ∞ . Then, we set γ = γ ·+T − γ T if T < ∞ and γ T < ∞, and γ = γ * otherwise. Then, γ is independent of G T and it is distributed as γ. We next set E = sup{x ∈ (0, ∞) : γ x < ∞}; we also define γ by setting γ Then observe that P(γ x > ε) ≤ P(γ x > ε) = P(γ x < ε), which completes the proof of (135).
Height process of X. We next define the analogue of H w . To that end, we assume that the function ψ (as defined in (127)) satisfies In particular, note that (136) implies that either β > 0 or σ 2 (c) = ∞: namely, (136) entails that X has infinite variation sample paths. Le Gall & Le Jan [32] (see also Le Gall & D. [21]) prove that there exists a continuous process H = (H t ) t∈[0,∞) such that the following limit holds true for all t ∈ [0, ∞) in probability : Note that (137) is a local time version of (111). We refer to H as to the height process of X.
Remark 4.1 Let us mention that in Le Gall & Le Jan [32] and Le Gall & D. [21], the height process H is introduced only for critical and subcritical spectrally positive processes. However, it easily extends to supercritical cases thanks to (132).
We next recall here that the excursions of X above its infimum process I are the same as the excursions of H above 0. More specifically, X − I and H have the same set of zeros: [21] Chapter 1). We also recall that since −I is a local time for X − I at 0, the topological support of the Stieltjes measure d(−I) is Z . Namely, We shall also recall here the following result: Here, γ x is given by (129)

The red and blue processes in the continuous setting.
In this section we recall from [19] the definition of the analogues in the continuous setting of the processes X b,w , X r,w , Y w , A w , θ b,w , etc. Let us start with some notation and some convention. Let (F t ) t∈[0,∞) be a filtration on (Ω, F ) that is specified further. A process (Z t ) t∈[0,∞) is said to be a (F t )-Lévy process with initial value 0 if a.s. Z is càdlàg, Z 0 = 0 and if for all a.s. finite (F t )-stopping time T , the process Z T + · −Z T is independent of F T and has the same law as Z.
Blue processes. We fix the parameters , j ≥ 1 be processes that satisfy the following. The blue Lévy process is then defined by Clearly X b is a (F t )-spectrally positive Lévy process with initial value 0 with Laplace exponent ψ as defined in (127). We next introduce the analogues of the processes A w and Y w defined in (97). To that end, note that E[c j (N j (t)−1) + ] = c j e −c j κt −1+c j κt ≤ 1 2 (κt) 2 c 3 j . So it makes sense to define the following: To view Y as in (12), Namely the jump-times of Y are the E j and ∆Y E j = c j .
Red and bi-coloured processes. We next introduce the red process X r that satisfies the following.
(r 1 ) X r is a (F t )-spectrally positive Lévy process starting at 0 and whose Laplace exponent is ψ as in (127).
(r 2 ) X r is independent of the processes B and (N j ) j≥1 .
We next introduce the following processes: with the convention: inf ∅ = ∞. For all t ∈ [0, ∞), we set I r t = inf s∈[0,t] X r s and I r ∞ = lim t→∞ I r t that is a.s. finite in supercritical cases and that is a.s. infinite in critical or subcritical cases. Note that γ r x < ∞ if and only if x <−I r ∞ . Recall that stands for the largest root of ψ: in supercritical cases, > 0 and −I r ∞ is exponentially distributed with parameter , as recalled in Lemma 4.1 (iii). We next set: In critical and subcritical cases, T * = ∞ and θ b only takes finite values. In supercritical cases, a.s. T * < ∞ and we check that θ b (T * −) < ∞. We next introduce the following.
Both processes Λ b and Λ r are continuous and nondecreasing. In critical and subcritical cases, we also get a.s. lim t→∞ Λ b t = ∞ and Λ b (θ b t ) = t for all t ∈ [0, ∞). However, in supercritical cases, a.s.
In the following theorem we recall from [19] the results about the previous processes that we need; in particular, it contains the analogue of Proposition 3.2.
(i) A.s. the process A is strictly increasing and the process Y has infinite variation sample paths.
X, X b and X r have the same law: namely, X is a spectrally positive Lévy process with initial value 0 and Laplace exponent ψ as in (127). Moreover, [19]; for (ii) and (iii), see Theorem 2.5 in [19].
The red and blue processes behave quite similarly as in the discrete setting as in Lemma 3.3. More precisely, we recall from [19] the various properties concerning the red and blue processes that are used in the proof. ( Proof. For (i) and (ii), see Lemma 5.4 in [19]; for (iii), (iv) and (v), see Lemma 5.5 in [19].
The excursions of Y above its infimum. Let X be derived from X b and X r by (147) and recall from (130) the notation I t = inf s∈[0,t] X s , for the infimum process of X. Recall from (139) that −I is a local-time for the set of zeros Z = {t ∈ [0, ∞) : X t = I t }. Let Y be defined by (142) and recall from (15) the notation J t = inf s∈[0,t] Y s . The following lemma (that is recalled from [19]) asserts that−J is a local-time for the set

Lemma 4.4
We keep the assumptions of Theorem 4.2. Then, the following holds true.
We next recall the following result due to Aldous & Limic [4] (Proposition 14, p. 20) that is used in our proofs.  (iii) Set M a = max{r−l ; r ≥ l ≥ a : (l, r) is an excursion interval of Y −J above 0}. Then, M a → 0 in probability as a → ∞.
Thanks to Proposition 4.5 (iii), the excursion intervals of Y −J above 0 can be listed as follows where ζ k = r k −l k , k ≥ 1, is decreasing. Then, as a consequence of Theorem 2 in Aldous & Limic [4], p. 4, we recall the following.
Proposition 4.6 (Theorem 2 [4]) We keep the assumptions of Theorem 4.2 and the previous notation. Then, (ζ k ) k≥1 , that is the ordered sequence of lengths of the excursions of Y −J above 0, is distributed as the (β/κ, α/κ, c)-multiplicative coalescent (as defined in [4]) taken at time 0. In particular, we get a.s. k≥1 ζ 2 k < ∞. Height process of Y . We define the analogue of H w in the continuous setting thanks to the following theorem that is recalled from various results in [19].
Theorem 4.7 Let (α, β, κ, c) be as in (7) and assume that (136) holds: namely, which implies the assumptions of Theorem 4.2. Let X be derived from X b and X r by (147). Let H be the height process associated with X as defined by (137) (and by Remark 4.1 in the supercritical cases). Then, there exists a continuous process (H t ) t∈[0,∞) such that for all t ∈ [0, ∞), H t is a.s. equal to a measurable functional of (Y ·∧t , A ·∧t ) and such that We refer to H as the height process associated with Y .
As for H and X−I, the following lemma (that is recalled from [19]) asserts that the excursion intervals of H and Y −J above 0 are the same.
Proof. The limit of b n ψ wn (λ/a n ) is a direct consequence of the equivalence (ii) ⇔ (iii) asserted in Proposition 2.1. Since the ψ wn are convex functions, the convergence is uniform in λ on all compact subsets of [0, ∞), which easily entails the convergence of the inverses.
We will use several times the following result from Ethier & Kurtz [24].
Proof. We repeatedly use the following estimates on Poisson r.v. N with mean r ∈ (0, ∞): By the definition (97), we get E[ Thus, by (C1)-(C3) and the Markov inequality, we get This shows that for any t ∈ [0, ∞), the laws of the 1 an A wn bnt are tight on R. We next prove (155) with R n t = 1 an A wn bnt , t ∈ [0, ∞). To that end, we fix z, ε ∈ (0, ∞) and k ∈ N, and we set T n := τ ε k (R n ). Then, (115) in Lemma 3.5 with a = a n ε, T = b n T n , t = b n η and t 0 = b n z implies the following: .
Recall from (95) the definition of X b,w and recall from (96) the definition of the Poisson processes N w j (·), j ≥ 1. Recall from (141) the definition of X b and that of the Poisson processes N j (·), j ≥ 1.
Proof. Let u ∈ R. Note that by (20) and (C3). Thus, for all t ∈ [0, ∞), N wn j (b n t) → N j (t) in law. Next, fix k ≥ 1 and set: Since we assume that Proposition 2.1 holds true, 1 an X b,wn bnt → X b t weakly on R. Since Q n t (resp. Q t ) is independent of (N wn j ) 1≤j≤k (resp. independent of (N j ) 1≤j≤k ), we easily check Thus, Q n t → Q t weakly on R. Since Lévy processes weakly converge in D([0, ∞), R) if an only if unidimensional marginals weakly converge on R (see Lemma B.8 in Appendix Section B, with precise references), we get Q n → Q and for all j ≥ 1, N wn j (b n ·) → N j , weakly on D([0, ∞), R). Since Q n , N wn 1 , . . . , N wn k are independent Lévy processes, they have a.s. no common jump-times and Lemma B.2 (in Appendix, Section B) asserts that Since X b,wn is a linear combination of Q n and the (N wn j ) 1≤j≤k , we get: which implies the weaker statement: ( 1 an X b,wn bn· , N wn 1 (b n ·), . . . , N wn k (b n ·)) −→ (X b , N 1 , . . . , N k ), weakly on (D([0, ∞), R)) k+1 equipped with the product topology. Since it holds true for all k, an elementary result (see Lemma B.7 in Appendix, Section B) entails (157).
. Then, to prove (158), we claim that it is sufficient to prove that for all t ∈ [0, ∞), Indeed, let t be such that ∆A t = ∆A t = 0 and let q, q be rational numbers such that q < t < q ; thus, A w n(p) b(n(p))q ≤ A w n(p) b(n(p))t ≤ A w n(p) b(n(p))q ; since ∆A t = 0, we get a.s. A w n(p) b n(p) t /a n(p) → A t ; the convergence in probability entails that A q ≤ A t ≤ A q ; since it holds true for all rational numbers q, q such that q < t < q , we get A t− ≤ A t ≤ A t which implies A t = A t since ∆A t = 0. Thus, a.s. A and A coincide on the dense subset {t ∈ [0, ∞) : ∆A t = ∆A t = 0}: it entails that a.s. A = A and the law of (A, X b , N j ; j ≥ 1) is the unique weak limit of the laws of ( 1 an A wn bn· , 1 an X b,wn bn· , N wn j (b n ·); j ≥ 1).
Next suppose that j * = sup{j ≥ 1 : → 0 it is possible to find a sequence (j n ) that tends to ∞ sufficiently slowly to get j * <j≤jn (v (n) j ) 3 → 0, which implies (162). Next, we use (162) to prove (160). To that end, we fix t ∈ [0, ∞) and we fix k ∈ N that is specified further; since j n → ∞, we can assume p is such that k < j n(p) . To simplify, we set ξ and we prove that each term in the right-hand side goes to 0 in probability. We is a Poisson r.v. with mean r p,j that is equal to v (n(p)) j b n(p) t/σ 1 (v n(p) ), by (156) we get E ξ p j = v (n(p)) j e −r p,j −1 + r p,j ). We next use the following elementary inequality: by (162). Next, note that j>j n(p) v (n(p)) j r 2 p,j = (b n(p) t/σ 1 (v n(p) ) 2 j>j n(p) (v (n(p)) j ) 3 −→ κβt 2 , which implies that d p (t) → 0 as p → ∞.
We next consider C p t : by (156), var(ξ p j ) ≤ (v (n(p)) j ) 2 r 2 p,j . Since the ξ p j are independent, we get by (162), which proves that C p t → 0 in probability when p → ∞. We next deal with R k,p t . By (156), (162) and (163), we first get: Similarly, observe that E[ξ j ] = c j e −κtc j −1 + κtc j ≤ 1 2 (κt) 2 c 3 j . This inequality combined with (164) entails: Finally, we consider D k,p . Since a.s. t is not a jump-time of N j , a.s. v n(p) Thus, for all k ∈ N, a.s. D k,p t → 0. These limits combined with (165) (and with the convergence to 0 in probability of C p t and d p (t)) easily imply (160), which completes the proof of the lemma.
Recall from (97) the definition of Y w and recall from (142) the definition of Y .
Proof. without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that the convergence in (158) holds true P-almost surely. We first prove that (( 1 an X b,wn bn· , 1 an A wn bn· )) → ((X b , A)) a.s. in D([0, ∞), R 2 ) thanks to Lemma B.1 (iii) (a standard result recalled in Appendix, Section B). To that end, first recall that by definition, the jumps of A (resp. of A wn ) are jumps of X b (resp. of X b,wn ): namely if ∆A t > 0, then ∆X b t = ∆A t . The same holds true for A wn and X b,wn .
Let t ∈ (0, ∞). First suppose that ∆A t > 0. Thus, ∆X b t = ∆A t . By Lemma B.1 (i), there exists a sequence of times t n → t such that 1 an ∆A wn bntn → ∆A t . Thus, for all sufficiently large n, 1 an ∆A wn bntn > 0, which entails 1 an ∆A wn bntn = 1 an ∆X b,wn bntn and we get 1 an ∆X b,wn bntn → ∆A t = ∆X b t . Suppose next that ∆A t = 0; by Lemma B.1 (i), there exists a sequence of times t n → t such that 1 an ∆X b,wn bntn → ∆X b t . Since ∆A t = 0, Lemma B.1 (ii) entails that 1 an ∆A wn bntn → ∆A t = 0. In both cases, we have proved that for all t ∈ (0, ∞), there exists a sequence of times t n → t such that 1 an ∆X b,wn bntn → ∆X b t and 1 an ∆A wn bntn → ∆A t : by Lemma B.1 (iii), it implies that (( 1 an X b,wn bn· , 1 an A wn bn· )) → ((X b , A)) a.s. in D([0, ∞), R 2 ). This entails (166), since the function (x, a) ∈ R 2 → (x, a, x−a) ∈ R 3 is Lipschitz and since X b,wn −A wn = Y wn and X b −A = Y .
Recall that X r,w (resp. X r ) is an independent copy of X b,w (resp. of X b ). Recall from (98) (resp. from (129)) the definition of γ r,w (resp. of γ r ). Recall that I r,w t = inf s∈[0,t] X r,w s and recall the notation I r,w ∞ = lim t→∞ I r,w t . Similarly, recall that I r t = inf s∈[0,t] X r s and recall the notation I r ∞ = lim t→∞ I r t . Recall from (133) in Lemma 4.1 the definition of γ r ; similarly we set Lemma 5.8 Let (α, β, κ, c) be as in (7). Recall from (127) the definition of ψ and assume that (136) holds: namely, ∞ dλ/ψ(λ) < ∞. Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20) and (C1)-(C3) as in (28) and in (29). Then, Proof. Let γ n (resp. γ) be a conservative subordinator with Laplace exponent a n ψ −1 wn (·/b n ) − a n wn (resp. ψ −1 (·)− ). By (153) in Lemma 5.1 , a n ψ −1 wn (λ/b n )−a n wn → ψ −1 (λ)− for all λ ∈ [0, ∞), which implies that for all x ∈ [0, ∞), γ n x → γ x weakly on [0, ∞). Since the γ n are Lévy processes, Theorem B.8 (in Appendix Section B) entails that γ n → γ weakly on D([0, ∞), R). Let E n (resp. E) be an exponentially distributed r.v. with parameter a n wn (resp. ) that is independent of γ n (resp. of γ), with the convention that a.s. E n = ∞ if wn = 0 (resp. a.s. E = ∞ if = 0). We then get . Under our assumptions, Proposition 2.1 implies that 1 an X r,wn bn· → X r · weakly on D([0, ∞), R). Then the laws of the processes on the left hand side of (168) are tight on D([0, ∞), R) 2 × [0, ∞]; we only need to prove that the joint law of the processes on the right hand side of (168) is the unique limiting law: to that end, let (n(p)) p∈N be an increasing sequence of integers such that where (γ , E ) has the same law as (γ r , −I r ∞ ). Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that the convergence in (169) holds P-a.s. and we only need to prove that (γ , E ) = (γ r , −I r ∞ ) a.s. We first prove that a.s. E = −I r ∞ . Since X r is a spectrally positive Lévy process, it has no fixed discontinuity.
Since E and −I r ∞ have the same law on [0, ∞], we get E = −I r ∞ a.s. We next prove that a.s. for all x ∈ [0, −I r ∞ ), γ x = γ x . Indeed, fix x < −I r ∞ such that ∆γ r x = 0. Then, by Lemma B.3 (iv), we get γ r,w n(p) a n(p) x /b n(p) → γ r x . Since x < −I r,w n(p) ∞ /a n(p) for all sufficiently large p, it shows that γ r,w n(p) a n(p) x /b n(p) → γ r x = γ r x . Thus, a.s. for all x ∈ [0, −I r ∞ ) such that ∆γ r x = 0, we get γ x = γ x , which implies the desired result. Note that it completes the proof of the lemma in the critical and subcritical cases.
To avoid trivialities, we now assume that we are in the supercritical cases. Namely, > 0 and −I r ∞ < ∞ a.s. To simplify notation, we set First note that the proof is complete as soon as we prove that t p * → t * . To prove this limit, we want to use Lemma B.3 (iii). To that end, we first fix x > −I r ∞ . Since (γ , E ) has the same law as (γ r , −I r ∞ ), γ is constant on [E , ∞) and since E = −I r ∞ , γ is constant on [−I r ∞ , ∞), which implies ∆γ x = 0 and thus γ r,w n(p) a n(p) x /b n(p) → γ x . We next fix t > γ x + t * . Thus, there is p 0 such that for all p ≥ p 0 , γ r,w n(p) a n(p) x /b n(p) < t and x > −I r,w n(p) ∞ /a n(p) , which implies that t p * = γ r,w n(p) a n(p) x /b n(p) . Since t > t p * ∨ t * , we get t p * = inf{s ∈ [0, t] : inf r∈[0,s] X r,w n(p) b n(p) r = inf r∈[0,t] X r,w n(p) b n(p) r } and t * = inf{s ∈ [0, t] : inf [0,s] X r = inf [0,t] X r }. Thus Lemma B.3 (iii) entails that t p * → t * , which completes the proof of the lemma. Recall from (98) the definition of θ b,w and recall from (167) the definition of γ r,w . We next set Lemma 5.9 Let (α, β, κ, c) be as in (7). Recall from (127) the definition of ψ and assume that (136) holds: namely, ∞ dλ/ψ(λ) < ∞. Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20) and (C1)-(C3) as in (28) and in (29). Then, the laws of the processes Proof. To simplify notation we set R n t = 1 bn θ b,wn bnt − t = 1 bn γ r,wn (A wn bnt ); we only need to prove that the R n are tight on D([0, ∞), R). To that end, we use Lemma 5.3. First, observe that for all K, z ∈ (0, ∞), P(R n t > K) = P 1 bn γ r,wn (A wn (b n t)) > K ≤ P 1 bn γ r,wn anz > K + P 1 an A wn bnt > z) .
This easily implies that for fixed t the laws of the R n t are tight on [0, ∞) since it is the case for the laws of γ r,wn anz /b n and A wn bnt /a n by resp. Lemma 5.8 and Lemma 5.4.
Next, denote by F t the σ-field generated by the r.v. N wn j (s) and γ r,wn (A wn s ) with s ∈ [0, t] and j ≥ 1; note that N wn j (t + ·)−N wn j (t) are independent of F t . Fix ε ∈ (0, ∞) and recall from (154) the definition of the times τ ε k (R n ): clearly b n τ ε k (R n ) is a (F t )-stopping time. Next, fix k ∈ N and set ∀x ∈ [0, ∞), g(x) = 1 bn γ r,wn a n (x + 1 an A wn (b n τ ε k (R n ))) − 1 bn γ r,wn (A wn (b n τ ε k (R n ))) .
x the sigma algebra generated by the processes (N wn j ) j≥1 and by γ r,wn y , y ∈ [0, x] and set G x = G o x+ . Then, it is easy to see that 1 This, combined with (171) and (153) in Lemma 5.1, implies the following: which completes the proof by Lemma 5.3.
Recall from (144) the definition of θ b and recall from (133) in Lemma 4.1 the definition of γ r . Then, we define Recall from (145) the definition of Lemma 5.10 Let (α, β, κ, c) be as in (7). Recall from (127) the definition of ψ and assume that (136) holds: namely, ∞ dλ/ψ(λ) < ∞. Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20) and (C1)-(C3) as in (28) and in (29). Then, Proof. Recall from (100) the definition of T * wn and recall from (145) the definition of T * . We first prove that 1 bn T * wn → T * in law on [0, ∞]. To that end, first observe that from the independence between the blue and red processes, we deduce that ( 1 an A wn bn· , − 1 an I r,wn In the (sub)critical cases α ∈ [0, ∞), −I r ∞ = ∞. Then, clearly 1 bn T * wn → T * in law on [0, ∞]. We next suppose α < 0; thus −I r ∞ is exponentially distributed with parameter > 0 (that is the largest root of ψ); namely −I r ∞ has a diffuse law which allows to apply Proposition 2.11 in Jacod & Shiryaev [29] (Chapter VI, Section 2a p. 341) that discusses continuity properties of specific hitting times; thus, we get that 1 bn T * wn → T * in law on [0, ∞]. By Lemmas 5.7, 5.8 and 5.9, the laws of the r.v. on the left hand side of (173) are tight on we only need to prove that the joint law of the processes on the right hand side of (173) is the unique limiting law. To this end, we note that by the aforementioned three lemmas, the independence between the red processes and blue ones, as well as the uniqueness of the limit law of ( 1 bn T * wn ) as implied by Jacod & Shiryaev's proposition, it suffices to consider the situation where (n(p)) p∈N is an increasing sequence of integers such that and then prove that θ = θ b . Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that (174) holds true P-almost surely. Since A has no fixed time of discontinuity, a.s. for all q ∈ Q ∩ [0, ∞), ∆A q = 0, and thus A w n(p) b n(p) q /a n(p) → A q . Since γ r has no fixed discontinuity and is independent of A, the same properties hold for γ r . Therefore, a.s. for all q ∈ Q ∩ [0, ∞), ∆γ r (A q ) = 0, which easily entails that γ r,w n(p) (A w n(p) (b n(p) q))/b n(p) → γ r (A q ); thus, θ b,w n(p) (b n(p) q)/b n(p) → θ b q for all q ∈ Q∩[0, ∞) a.s. Therefore, θ = θ b , which completes the proof.
Lemma 5.11 Let (α, β, κ, c) be as in (7). Recall from (127) the definition of ψ and assume that (136) holds: namely, ∞ dλ/ψ(λ) < ∞. Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20) and (C1)-(C3) as in (28) and in (29). Then, Proof. Without loss of generality (but with a slight abuse of notation), Skorokod's representation theorem allows to assume that (173) holds P-almost surely. To simplify notation, we next set R n = 1 an (X b,wn bn· , A wn bn· , Y wn bn· ) and R = (X b , A, Y ). Let us fix a ∈ (0, ∞). We consider several cases. We first suppose that ∆R a = 0. By Lemma B.1 (i), there is s n → a such that R n sn− → R a− , R n sn → R a and thus ∆R n sn → ∆R a . − Let us suppose more specifically that ∆Y a > 0. By definition of Y , we get ∆X b a = ∆Y a and ∆A a = 0. Suppose that a ∈ [0, T * ]; by Lemma 4.3 (ii), we get ∆θ b a = 0 and thus ∆θ b (a) = 0. − We finally suppose that ∆R a = 0; by Lemma B.1 (i), there exists a sequence s n → a such that Since, ∆R a = 0, Lemma B.1 (ii) entails that ∆R n s n → ∆R a . Thus, we have proved the following: for all a ∈ (0, ∞), there exists a sequence s n → a such that Recall next that for all t ∈ [0, ∞) and all n ∈ N, that Λ r,wn Lemma 5.12 Let (α, β, κ, c) be as in (7). Recall from (127) the definition of ψ and assume that (136) holds: namely, ∞ dλ/ψ(λ) < ∞. Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20) and (C1)-(C3) as in (28) and in (29). Recall the notation Q n (1) in (175). Then, the following convergence holds true Proof. Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that the convergence in (175) holds P-almost surely. Since Then, for all sufficiently large n, we get t < θ b,wn (T * wn )/b n and we can write Since θ b is strictly increasing on [0, T * ), standard arguments entail Λ b,wn (b n t)/b n → Λ b t . Suppose next that t > θ b (T * ), which is only meaningful in the supercritical cases. Then, for all sufficiently large n, we get t > θ b,wn (T * wn )/b n and we can write Λ b,wn bnt = T * wn and Λ b t = T * . Thus, we nondecreasing and continuous, a theorem due to Dini implies that 1 bn Λ b,wn bn· → Λ b uniformly on all compact subsets; it entails a similar convergence for Λ r , which completes the proof of (177).
Here is one of the key technical point of the proof that relies on the estimates of Lemma 3.6.
We derive a similar result for the red processes by a quite similar (but simpler) argument based on Lemma 3.6 (iii): we leave the details to the reader. (147)

Recall (178) and recall from
Lemma 5.14 Let (α, β, κ, c) be as in (7). Recall from (127) the definition of ψ and assume that (136) holds: namely, ∞ dλ/ψ(λ) < ∞. Let a n , b n ∈ (0, ∞) and w n ∈ ↓ f , n ∈ N, satisfy (20) and (C1)-(C3) as in (28) and in (29). Recall from (177) the notation Q n (2). Then  Proof. We first prove the following , R) 2 equipped with the product-topology. Note that the laws of Q n (3) are tight thanks to (177) and Lemma 5.13. We only need to prove that the joint law of the processes on the right hand side of (181) is the unique limiting law: to that end, let (n(p)) p∈N be an increasing sequence of integers such that , R) 2 equipped with the product topology. Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that the convergence in (182) holds P-a.s. and we only need to is, in general, not countable (it contains all the red intervals starting with a jump), so we have to proceed with care. To that end, we first set Then S is countable. We then consider several cases.
We first fix t ∈ (0, T * )\S and we assume that We then set a = Λ b t and we necessarily get a < T * , ∆θ b a > 0 and t ∈ [θ b a− , θ b a ]. Since t / ∈ S 2 , we then get t ∈ (θ b a− , θ b a ). To simplify the notation, we set By (182), R p → R a.s. on D([0, ∞), R 4 ). Since a < T * , ∆θ b a = ∆θ b a > 0 and a is a jump-time of R. By Lemma B.1 (i), there is a sequence s p → a such that (R p sp− , R p sp ) → (R a− , R a ): in particular, we get X b,w n(p) (b n(p) s p )/a n(p) → X b thus, for all sufficiently large p, we get . In (sub)critical cases, it simply means that Q b = X b Λ b . We now complete the proof that Q b = X b Λ b in the supercritical cases. To that end, we first observe the following. Let t 1 , t 2 ∈ (T * , ∞) be distinct times such that Then, by (182), we get t i > T * w n(p) /b n(p) for all sufficiently large p which implies X b,w n(p) Λ b,w n(p) (b n(p) t 1 ) = X b,w n(p) Λ b,w n(p) (b n(p) t 2 ) . Consequently, we get This argument easily implies that for all t ∈ [T * , ∞), Q b t = Q b T * . Thus, to complete the proof that Q b = X b Λ b in the supercritical cases, we only need to prove that X b,w n(p) (T * w n(p) )/a n(p) → X b (T * ). If ∆X b (T * ) = 0, then it is a consequence of (182) and of Lemma B.1 (ii). Therefore, it remains to address cases where ∆X b (T * ) > 0. In this case, we clearly get ∆θ b (T * ) = ∞; by Lemma 4.3 (ii) with a = T * , we get ∆Y (T * ) = 0 and therefore ∆X b (T * ) = ∆A(T * ) > 0 by definition of Y and A.
We first claim that it is sufficient to prove A w n(p) (T * w n(p) )/a n(p) → A T * . Indeed, suppose it holds true; since ∆Y (T * ) = 0, Lemma B.1 (ii) and (182) imply that Y w n(p) (T * w n(p) )/a n(p) → Y T * ; and it is sufficient to recall that X b,w n(p) = A w n(p) + Y w n(p) .
Thus, we assume that we are in the supercritical cases and that ∆X b (T * ) > 0, and we want to prove that A w n(p) (T * w n(p) )/a n(p) → A T * . By Lemma B.1 (i), there exists t p → T * such that A w n(p) (b n(p) t p −)/a n(p) → A T * − and A w n(p) (b n(p) t p )/a n(p) → A T * . Suppose that t p > T * w n(p) /b n(p) for infinitely many p; by the definition (100) of T * wn , it implies that A w n(p) (b n(p) t p −) ≥ −I r,w n(p) ∞ for infinitely many p and (182) implies This proves that a.s. t p ≤ T * w n(p) /b n(p) for all sufficiently large p. Then, Lemma B.1 (iv) in Appendix implies that A w n(p) (T * w n(p) )/a n(p) → A T * . As observed previously, it completes the proof of X b,w n(p) (T * w n(p) )/a n(p) → X b (T * ) and it completes the proof of in the supercritical cases. We next prove that Q r = X r Λ r : to that end, we set S 3 = {t ∈ [0, ∞) : (∆X r )(Λ r t ) > 0}. Lemma 4.3 (iv) entails that a.s. S 3 is countable and by Lemma B.1 (ii), a.s. for all t ∈ [0, ∞)\S 3 , we get X r,w n(p) Λ r,w n(p) (b n(p) t /a n(p) → X r (Λ r t ); this easily entails that a.s. Q r = X r • Λ r , which completes the proof of (181).
We now prove (180): without loss of generality (but with a slight abuse of notation), Skorokod's representation theorem allows to assume that (181) holds P-a.s. By Lemma 4.3 (v), a.s. for all t ∈ [0, ∞), ∆Q b t ∆Q r t = 0, and Lemma B.1 (iii) entails: Recall from (111) the definition of the height process H wn associated with X wn . Recall from (137) the definition of (H t ) t∈[0,∞) , the height process associated with X: H is a continuous process and note that (137) implies that H is an adapted measurable functional of X. Then, recall from (82) the definition of the offspring distribution µ wn and denote by (Z wn k ) k∈N a Galton-Watson Markov chain with initial state Z wn 0 = a n and offspring distribution µ wn ; recall from (31) Assumption (C4): there exists δ ∈ (0, ∞) such that lim inf n→∞ P(Z wn bnδ/an = 0) > 0.
Proof. We first prove that weakly on the appropriate product-space. By Proposition 2.2, the laws of the processes an bn H wn bn· are tight on C([0, ∞), R). Then, the laws of Q n (4) are tight thanks to (180). We only need to prove that the law of Q (4) is the unique limiting law, which is an easy consequence of (180), of the joint convergence (32) in Proposition 2.2 and of the fact that H is an adapted measurable deterministic functional of X.
To complete the proof of the lemma, we use a general (deterministic) result on Skorokhod's convergence for the composition of functions that is recalled in Theorem B.5 (see Appendix Section B.1). Without loss of generality (but with a slight abuse of notation), Skorokod's representation theorem allows to assume that (184) holds P-a.s.: − We set κ n = a n b n /σ 1 (w n ) and for all t ∈ [0, ∞) we set Z n where the sequence ζ k = l k − r k , k ≥ 1, decreases. Moreover, the sequence (ζ k ) k≥1 appears as the Lemma 5. 17 We make the same assumptions as in Theorem 2.4. We keep the previous notations. Then Proof. The laws of the Q n (6) are tight by (36) in Theorem 2.4 combined with the weak convergence We only need to prove that the law of Q(6) is the unique limiting law: to that end, let (n(p)) p∈N be an increasing sequence of integers such that Q n(p) (6) → (Y, H, Π, Z) weakly. It remains to prove that Z = k≥1 (ζ k ) 2 . Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that Q n(p) (6) → (Y, H, Π, Z) holds true P-a.s. Then, by Lemma 5.16, observe that for all l ≥ 1, Set Z = k≥1 (ζ k ) 2 ; by letting l go to ∞ in (196), we get Z ≥ Z , which implies Z = Z a.s. since Z and Z have the same law. This completes the proof of the lemma.
Proof of (57). We next prove the convergence of the connected components equipped with the counting measure. Recall from Introduction the definition of the discrete tree T wn coded by the w n -LIFO queue without repetition (namely, the tree coded by H wn ): the vertices of T wn are the clients; the server is the root (Client 0) and Client j is a child of Client i in T w if and only if Client j interrupts the service of Client i (or arrives when the server is idle if i = 0). We denote by C wn the contour process associated with T wn that is informally defined as follows: suppose that T wn is embedded in the oriented half plane in such a way that edges have length one and that orientation reflects lexicographical order of visit; we think of a particle starting at time 0 from the root of T wn and exploring the tree from the left to the right, backtracking as less as possible and moving continuously along the edges at unit speed. Since T wn is finite, the particle crosses each edge twice (upwards first and then downwards). For all s ∈ [0, ∞), we define C wn s as the distance at time s of the particle from the root of T wn . We refer to Le Gall & D. [21] (Section 2.4, Chapter 2, pp. 61-62) for a formal definition and the connection with the height process (see also the end of Section 3.2).
It is important to notice that the trees coded by C wn and by H wn are the same: the only difference is the measure induced by the two different coding functions. More precisely, C wn is derived from H wn by the following time-change: recall that j n = max{j ≥ 1 : w (n) j > 0} and let (ξ n k ) 1≤k≤2jn be the sequence of jump-times of H wn : namely, ξ n k+1 = inf{s > ξ n k : H wn s = H wn ξ n k }, for all 1 ≤ k < 2j n , with the convention ξ n 0 = 0. We then set that counts the number of clients who entered the w n -LIFO queue governed by Y wn . Recall here that E wn j is the first jump-time of N wn j : namely the E wn j are independent exponentially distributed r.v. with respective parameters w (n) j /σ 1 (w n ). In terms of the tree T wn , R n t is the number of distinct vertices that have been explored by H wn up to time t. By arguing as in the proof of (91), we easily check that We prove the following.
Lemma 5.21 We keep the previous notation. Then, the following holds.
Moreover, there exists a positive r.v. Q n that is a measurable function of (N wn j ) j≥1 , such that E[Q 2 n ] ≤ 4j n (recall that j n := max{j ≥ 1 : w (n) j > 0}) and such that and denote by (G t ) the natural filtration associated with the (N wn j ) j≥1 . It is easy to check that the M j are independent (G t )-martingales and that We easily check the following: which is nonnegative and nondecreasing in t so that sup s∈[0,t] |s−M (s)| = t−M (t). Moreover, for all j ≥ 1, we check that Since for all a ∈ [0, ∞), the function t → t ∧ a is 1-Lipschitz and since M is a convex combination of these functions, M is also 1-Lipschitz: namely, |M (t + s)−M (t)| ≤ s, which completes the proof of (208). By (206) and (207) we easily get for all t, ε ∈ (0, ∞) an bn H wn bns > a n ε .
Thus, by (20) and (36) in Theorem 2.4, we get lim n→∞ P sup s∈[0,t] | 1 bn Φ n (b n s)−2s| > 2 ε = 0. This prove that 1 bn Φ n (b n ·) converges to 2Id in probability on C([0, ∞), R), where Id stands for the identity map on [0, ∞). Then, standard arguments also imply that 1 bn φ n (b n ·) converges to 1 2 Id in probability on C([0, ∞), R). We also note that on any interval [k, k + 1] where k is an integer, C wn t is a linear interpolation between C wn k and C wn k+1 . These convergences combined with Theorem 2.4 imply 1 an A wn bn· , 1 an Y wn bn· , an bn H wn bn· , an bn C wn bn· , 1 weakly on the appropriate space. We now deal with the excursions of C wn above 0, that are the contour processes of the spanning trees T wn k , 1 ≤ k ≤ q wn , of the q wn connected components of G wn ; recall that the T wn k are also the connected components obtained from the tree T wn after removing its root. Recall from (37) that [l wn k , r wn k ) are the excursion intervals of H wn above 0: namely, 1≤k≤qw n [l wn k , r wn k ) = {t ∈ [0, ∞) : H wn t > 0}. Recall that the excursion intervals are listed in the decreasing order of their lengths; recall that H wn k (t) = H wn ((l wn k + t)∧ r wn k ), t ∈ [0, ∞), is the k-th longest excursion process of H wn above 0. Recall from (39) that Π wn k = ((s n,k p , t n,k p ); 1 ≤ k ≤ p n k ) is the sequence of pinching times falling into the k-th longest excursion. Then recall that m wn = j≥1 w (n) j δ j and recall that m wn k is the restriction to T wn k of m wn . Recall that T wn k , d gr , wn k , m wn k ) stands for the measured tree coded by H wn k and that G wn k , d wn k , wn k , m wn k is the measured graph coded by H wn k and the pinching setup (Π wn k , 1): namely, G wn k is isometric to the graph G(H wn k , Π wn k , 1) as defined in (49) and it is the k-th largest (with respect to the measure m wn ) connected component of G wn . We next set for all k ∈ {1, . . . , q wn }, l n k = Φ n (l wn k ), r n k = Φ n (r wn k ), Then, we easily check the following: is the contour process of T wn k . We denote by ν wn k the measure that the contour process induces on T wn k : namely, T wn k , d gr , wn k , ν wn k ) is the measured tree coded by C wn subset of vertices A. Since b −1 n ≤ a n /b n (for all sufficiently large n), we get d Pro 1 bn ν wn k , 2 bn µ wn k ≤ a n /b n . This combined with (211) entails weakly on G N * equipped with the product topology, which easily implies (57).
End of proof of Theorem 2.8. We next make the following additional assumption : √ j n /b n → 0 and we complete the proof of Theorem 2.8. To that end, it is sufficient to prove that for all fixed k ≥ 1, the probability that µ wn 1 2 Id) in probability on (C([0, ∞), R)) 2 . By Slutzky's theorem, we get a joint convergence of (b −1 n Q n , b −1 n Φ n (b n ·), b −1 n φ n (b n ·)) with (210). Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that the following convergence holds almost surely on the appropriate space.
Recall notation ζ k = r k − l k , ζ wn k = r wn k −l wn k = m wn k (G wn k ) and set ζ n k = r n k − l n k = ν wn k (G wn k ). First, we easily derive from the argument of the proof of (212) that ν wn k (G wn k ) = 2µ wn k (G wn k )+1. Let σ n be a permutation of {1, . . . , q wn k } such that (ζ n σn(k) ) 1≤k≤q wn k is nonincreasing. To complete the proof of Theorem 2.8, it is then sufficient to prove that for all k ≥ 1, there exists n k such that for all n ≥ n k , σ n (k) = k.
We say that a R-valued spectrally positive Lévy process (R t ) t∈[0,∞) with initial value R 0 = 0 is integrable if for at least one t ∈ (0, ∞) we have E[|R t |] < ∞. It implies that E[|R t |] < ∞ for all t ∈ (0, ∞). We recall from Section B.2.1 in Appendix that there is a one-to-one correspondence between the laws of R-valued spectrally positive Lévy processes (R t ) t∈[0,∞) with initial value R 0 = 0 that are integrable and the triplets (α, β, π) where α ∈ R, β ∈ [0, ∞) and π is a Borel-measure on (0, ∞) such that (0,∞) π(dr) (r∧r 2 ) < ∞. More precisely, the correspondence is given by the Laplace exponent of spectrally positive Lévy processes: namely, for all t, λ ∈ [0, ∞), The main result used to obtain the convergence of branching processes is a Theorem due to Grimvall [26], that is recalled in Theorem B.11: it states the convergence of rescaled Galton-Watson processes to Continuous State Branching Processes (CSBP for short). We say that a process (Z t ) t∈[0,∞) is an integrable CSBP if it is a [0, ∞)-valued Feller Markov process obtained from spectrally positive Lévy processes via Lamperti's time-change which further satisfies E[Z t ] < ∞ for all t ∈ [0, ∞).
The law of such a CSBP is completely characterised by the Laplace exponent of its associated Lévy process that is usually called the branching mechanism of the CSBP, which is necessarily of the form (215): see Section B.2.2 for a brief account on CSBP.
Let w n ∈ ↓ f , n ∈ N. Recall from (75) that ν wn = σ 1 (w n ) −1 j≥1 w (n) j δ j and recall from (82) that for all k ∈ N, Recall from Section 3.2 the definition of the Markovian LIFO-queueing system associated with the set of weights w n : clients arrive at unit rate; each client has a type that is a positive integer; the amount of service required by a client of type j is w (n) j ; the types are i.i.d. with law ν wn . If one denotes by τ n k the time of arrival of the k-th client in the queue and by J n k his type, then the queueing system is entirely characterised by X wn = k≥1 δ (τ n k ,J n k ) that is a Poisson point measure on [0, ∞)×N * with intensity ⊗ ν wn , where stands for the Lebesgue measure on [0, ∞). Next, for all j ∈ N * and all t ∈ [0, ∞), we introduce the following: Observe that (N wn j ) j≥1 are independent homogeneous Poisson processes with rates w (n) j /σ 1 (w n ) and X wn is a càdlàg spectrally positive Lévy process.
Let a n , b n ∈ (0, ∞), n ∈ N be two sequences that satisfy the following conditions.
(217) a n and b n a Remark 5.1 It is important to note that these assumptions are weaker than (20): namely, we temporarily do not assume that anbn σ 1 (wn) → κ ∈ (0, ∞), which explains why the possible limits in the theorem below are more general.
Theorem 5.23 Let w n ∈ ↓ f and a n , b n ∈ (0, ∞), n ∈ N, satisfy (217). Recall from (216) the definition of X wn t ; recall from (82) the definition of µ wn and let (Z (n) k ) k∈ N be a Galton-Watson process with offspring distribution µ wn and initial state Z (n) 0 = a n . Then, the following convergences are equivalent.
It remains to prove that β ≥ β 0 and that (I) ⇔ (IIIabc). Let (ζ n k ) k∈N be a sequence of i.i.d. random variables with law µ wn as defined in (82). By Theorem B.11, (I) is equivalent to the weak convergence on R of the r.v. R n := a −1 n 1≤k≤ bn ζ n k −1 . We next apply Lemma A.3 to ∆ n k := a −1 n (ζ n k − 1) q n = b n , which implies that (I) is equivalent to We next compute L n (λ) more precisely. To that end, let (W n k ) k∈N be an i.i.d. sequence of r.v. with the same law as w (n) , which implies: (220) L n (λ) = e λ bn /an E e −λζ n 1 /an bn = e λ bn /an E exp −W n 1 1 − e −λ/an bn .
We next recall from Section 3.2 that the Markovian w n -LIFO queueing system governed by X wn induces a Galton-Watson forest T wn with offspring distribution µ wn : informally, the clients are the vertices of T wn and the server is the root (or the ancestor); the j-th client to enter the queue is a child of the i-th one if the j-th client enters when the i-th client is served; among siblings, the clients are ordered according to their time of arrival. We denote by H wn t the number of clients waiting in the line right after time t; recall from (111) how H wn is derived from X wn : namely, for all s ≤ t, if one sets I wn,s We recall from Section 3.2 that X wn and H wn are close to the Lukasiewicz path and the contour process of T wn . Therefore, the convergence results for Lukasiewicz paths and contour processes of Galton-Watson trees in Le Gall & D. [21] (see Appendix Theorem B.12, Section B.2.3) allow us to prove the following theorem.
Theorem 5.24 Let X be an integrable (α, β, π)-spectrally positive Lévy process, as defined at the beginning of Section 5.3.1. Assume that (218) holds and that ∞ dz/ψ α,β,π (z) < ∞, where ψ α,β,π is given by (215). Let (H t ) t∈[0,∞) be the continuous height process derived from X as defined by (137). Let w n ∈ ↓ f and a n , b n ∈ (0, ∞), n ∈ N, satisfy (217). Let (Z (n) k ) k∈ N be a Galton-Watson process with offspring distribution µ wn (defined by (82)), and initial state Z (n) 0 = a n . Assume that the three conditions (IIIabc) in Theorem 5.23 hold true and assume the following: Then, the following joint convergence holds true: , equipped with the product topology. We also get: Proof. Recall from (72) (Section 3.1) the definition of the Lukasiewicz path V Tw n associated with the GW(µ wn )-forest T wn ; recall from (74) the definition its height process Hght Tw n and recall that C Tw n stands for the contour process of T wn . We first assume that (IIIabc) in Theorem 5.23 and that (222) hold true. Then, Theorem B.12 applies with µ n := µ wn : namely, the joint convergence (266) holds true and we get (224).
Recall that (τ n k ) k≥1 are the arrival-times of the clients in the queue governed by X wn and recall from (84) the notation N wn (t) = k≥1 1 [0,t] (τ n k ) that is a homogeneous Poisson process with unit rate. Then, by Lemma B.6 (see Section B.1 in Appendix) the joint convergence (266) entails the following. weakly on D([0, ∞), R) × (C([0, ∞), R)) 2 equipped with the product topology. Here X is an integrable (α, β, π)-spectrally positive Lévy process (as defined at the beginning of Section 5.3.1) and H is the height process derived from X by (137). By Theorem 5.23, the laws of the processes 1 an X wn bn· are tight in D([0, ∞), R). Thus, if one sets Q n (8) = ( 1 an X wn bn· , Q n (7)), then the laws of the Q n (8) are tight on D([0, ∞), R) 2 × (C([0, ∞), R)) 2 . Thus, to prove the weak convergence Q n (8) → (X, X, H, H ·/2 ) := Q(8), we only need to prove that the law of Q(8) is the unique limiting law: to that end, let (n(p)) p∈N be an increasing sequence of integers such that Actually, we only have to prove that X = X. Without loss of generality (but with a slight abuse of notation), by Skorokod's representation theorem we can assume that (225) holds P-almost surely. We next use (85) in Lemma 3.1: fix t, ε, y ∈ (0, ∞), set I wn t = inf s∈[0,t] X wn s ; by applying (85) at time b n t, with a = a n ε and x = a n y, we get the following.
Compared with (225), this implies that for all t ∈ [0, ∞) a.s. X t = X t and thus, a.s. X = X.
As explained right after Theorem 2.3.1 in Le Gall & D. [21] (see Chapter 2, pp. 54-55) Assumption (222) is actually a necessary condition for the height process to converge. However it is not always easy to check this condition in practice. The following proposition provides a handy way of doing it.
Proof. We first prove a lemma that compares the total height of Galton-Watson trees with i.i.d. exponentially distributed edge-lengths and the total height of their discrete skeleton. More precisely, let ρ ∈ (0, ∞) and let µ be an offspring distribution such that µ(0) > 0 and whose generating function is denoted by g µ (r) = l∈N µ(l)r l . Note that g µ ([0, 1]) ⊂ [0, 1]; let g •k µ be the k-th iterate of g µ , with the convention that g •0 µ (r) = r, r ∈ [0, 1]. Let τ : Ω → T be a random tree whose distribution is characterised as follows.
-The number of children of the ancestor (namely the r.v. k ∅ (τ )) is a Poisson r.v. with mean ρ; -For all l ≥ 1, under P( · | k ∅ (τ ) = l), the l subtrees θ [1] τ, . . . , θ [l] τ stemming from the ancestor ∅ are independent Galton-Watson trees with offspring distribution µ. We next denote by Z k the number of vertices of τ that are situated at height k + 1: namely, Z k = #{u ∈ τ : |u| = k + 1} (see Section 3.1 for the notation on trees). Then, (Z k ) k∈N is a Galton-Watson process whose initial value Z 0 is distributed as a Poisson r.v. with mean ρ. We denote by Γ(τ ) the total height of τ : namely, Γ(τ ) = max u∈τ |u| is the maximal graph-distance from the root ∅. Note that if µ is supercritical, then Γ(τ ) may be infinite). Observe that Γ(τ ) = min{k ∈ N : Z k = 0}. Thus, We next equip each individual u of the family tree τ with an independent lifetime e(u) that is distributed as follows.
We are now ready to prove Proposition 5.25. Recall from (82) the definition of the offspring distribution µ wn . We apply Lemma 5.26 with µ = µ wn , ρ = a n , q = b n /a n and we denote by r n (t) the solution of (229): the change of variable λ = a n r implies that r n (t) satisfies (232) an anrn(t) dλ b n g µw n 1− λ an −1 + λ an = t .

Proof of Propositions 2.1 and 2.2.
In this section we shall assume that the sequence (a n ) and (b n ) satisfy (217) and anbn σ 1 (wn) → κ where κ ∈ (0, ∞). This dramatically restricts the possible limiting triplets (α, β, π). To see this point, we first prove the following lemma.
We next set: We then see that a n b n /σ 1 (w n ) = κ, that sup n∈N w (n) 1 /a n < ∞. Moreover, we get b n a n 1 − which are the limits (C1), (C2) and (C3). It is easy to derive from (239) and from (240) that a n and b n /a n tend to ∞ and that b n /a 2 n tends to β 0 . Moreover, since j n ≤ n 8 + n 3 + n, it is also easy to check that √ j n /b n → 0. This completes the proof of Proposition 2.1 (iv).
which implies the desired result.

A Laplace exponents.
We state here a proposition on the Laplace transform of measures on R. To that end, we briefly recall standard results on the Laplace transform of finite measures on [0, ∞) and on [0, ∞]. Namely, let µ be a Borel-measure on the compact space [0, ∞]; its Laplace transform is given by L µ (λ) = We next easily deduce from (256) the following lemma.

B.1 General results.
In this section, we adapt and we recall from Jacod & Shiryaev's book [29] results on Skorokod's topology and weak convergence on D([0, ∞), R d ) that are used in the proofs.
Then, the following holds true.
(i) For all t ∈ [0, ∞), there exists a sequence of times t n → t such that x n (t n −) → x(t−), x n (t n ) → x(t) and thus, ∆x n (t n ) → ∆x(t). (ii) For all t ∈ [0, ∞) such that ∆x(t) = 0 and for all sequences of times s n → t, we get x n (s n −) → x(t) and x n (s n ) → x(t), and thus ∆x n (s n ) → 0. (iii) Assume that for all t ∈ (0, ∞) there is a sequence of times t n → t such that ∆x n (t n ) → ∆x(t) and ∆y n (t n ) → ∆y(t). Then ((x n (t), y n (t)) t∈[0,∞) −→ ((x(t), y(t)) t∈[0,∞) for the Skorokod topology on D [0, ∞), R d+d . In particular, this joint convergence holds true whenever x and y have no common jump-time. (iv) Let (t n ) be as in (i) and (s n ) be such that s n → t and s n ≥ t n , n ∈ N. Then, x n (s n ) → x(t).
As an immediate consequence of the Lemma B.1 (iii), we get the following lemma.
Then, the following holds true.
The point (ii) is an immediate consequence of a well-known theorem due to Dini.
We shall use the following elementary lemma whose proof is left to the reader.
We use Theorem B.5 (ii) several times under the following form.
Lemma B.6 Let (β n ) n∈N be a sequence of nonnegative real numbers such that β n → ∞. For all n ∈ N, let (σ n k ) k≥1 be an increasing sequence of random times such that lim k→∞ σ n k = ∞; then, for all t ∈ [0, ∞), we set M n t = k≥1 1 [0,t] (σ n k ). Let (R n ) n∈N be a sequence of R-valued càdlàg processes.
We shall say that Z satisfies the Grey condition if it has a positive probability to be absorbed in 0, namely if ∞ dz/ψ(z) < ∞; in that case, one can show that P(∃t : Z t = 0) = P(lim t→∞ Z t = 0) and if a.s. Z 0 = x, then we get: We refer to Bingham [14] for more details on CSBP. We next recall the following convergence result from Grimvall [26].
Theorem B.11 (Theorems 3.1 & 3.4 [26]) Let a n , b n ∈ (0, ∞), n ∈ N, such that both a n and b n /a n tend to ∞. For all n ∈ N, let µ n be a probability measure on N, let (Z (n) k ) k∈ N be a Galton-Watson process with offspring distribution µ n and initial state Z (n) 0 = a n , and let (ζ n k ) k∈N be an i.i.d. sequence of r.v. with law µ n . Then, the following assertions are equivalent.

B.2.3 Height and contour processes of Galton-Watson trees.
Let (µ n ) n∈N be a sequence of offspring distributions with finite mean and such that µ n (0) > 0. For all µ n , we denote by T n a Galton-Watson forest with offspring distribution µ n as defined in Section 3.1. Recall from this section the definition of the Lukasiewicz path, the height and the contour processes of T n that are denoted respectively by (V Tn k ) k∈N , (Hght Tn k ) k∈N and (C Tn t ) t∈[0,∞) . We shall use the following result from Le Gall & D. [21].
C Proof of Lemma 2.7.
Several key arguments of the proofs can be found in Le Gall & D. [ ; therefore our proof is brief. Recall the notation and the assumption of Lemma 2.7. We control the Gromov-Hausdorff distance by bounding the distorsion of an explicit correspondence between G and G . Namely, recall that a correspondence R between the two metric spaces (E, d) and (E , d ) is a subset R ⊂ E×E such that for all (x, x ) ∈ E×E , R ∩ ({x}×E ) and R ∩ (E×{x }) are not empty; the distorsion of R is then given by dis(R) = sup{|d(x, y)−d (x , y )|; (x, x ) ∈ R, (y, y ) ∈ R}. We first define a correspondence between T h and T h . Recall that p h : [0, ζ h ) → T h and p h : [0, ζ h ) → T h are the canonical projections and recall that the roots are defined by p h (0) = ρ h and p h (0) = ρ h . We first set R 0 = (p h (t), p h (t)); t ∈ [0, ∞) ∪ (p h (s i ), p h (s i )), (p h (t i ), p h (t i )); 1 ≤ i ≤ p , where we have adopted the convention that ρ h = p h (t) if t ≥ ζ h and ρ h = p h (t) if t ≥ ζ h : indeed, recall that for all t ≥ ζ h (resp. t ≥ ζ h ), h(t) = 0 (resp. h (t) = 0), which implies t ∼ h 0 (resp. t ∼ h 0). Then, R 0 is clearly a correspondence between (T h , d h ) and (T h , d h ) and we easily check that dis(R 0 ) ≤ 4 h−h ∞ + ω δ (h) . We next set Π = ((p h (s i ), p h (t i ))) 1≤i≤p and Π = ((p h (s i ), p h (t i ))) 1≤i≤p ; recall that (G, d) (resp. (G , d )) stands for the (Π, ε)-pinched metric space associated with (T h , d h ) (resp. the (Π , ε )pinched metric space associated with (T h , d h )); recall that d = d Π,ε (resp. d = d Π ,ε ) is given by (48); we denote by : T h → G and : T h → G the canonical projections and we set R = ( (x), (x )); (x, x ) ∈ R 0 .
We next construct an ambient space into which G and G are embedded: we first set E = G G and we define d E : E 2 → [0, ∞) as follows: first d E |G×G = d, d E |G ×G = d and for all x ∈ G and all x ∈ G , d E (x, x ) = inf d(x, z) + 1 2 dis(R) + d (z , x ) ; (z, z ) ∈ R .
Standard arguments easily imply that d E is a distance on E. Note that the inclusion maps of resp. G and G into E are isometries. Since G and G are compact, so is (E, d E ). Moreover, we easily check that d Haus E (G, G ) ≤ 1 2 dis(R). Recall that ρ = (ρ h ), that ρ = (ρ h ) and that (ρ, ρ ) ∈ R; thus, d E (ρ, ρ ) ≤ 1 2 dis(R).
Denote by M f (E) the space of finite Borel measures; recall that for all µ, ν ∈ M f (E), their Prokhorov distance is d Pro E (µ, ν) = inf{η ∈ (0, ∞) : ν(K) ≤ µ(K η ) + η and µ(K) ≤ ν(K η ) + η, for all K ⊂ E compact}; here, K η = {y ∈ E : d E (y, K) ≤ η}. Recall that m (resp. m ) is the pushforward measure of the Lebesgue measure Leb on [0, ζ h ) (resp. on [0, ζ h )) via the function •p h (resp. •p h ). Let K ⊂ G be compact; set C = ( •p h ) −1 (K) ∩ [0, ζ h ]: if h is a pure-jump function with finitely many jumps, C is a finite union of half-open half closed intervals; if h is continuous, so is •p h and C is also a compact of [0, ζ h ]. We next set C = [0, ζ h ] ∩ C and K = •p h (C ): if h is continuous, then K is a compact subset of G ; if h is pure-jump function with finitely many jumps, then K is a finite subset of G : it is also a compact subset. Note that C ⊂ ( •p h ) −1 (K ). Thus, we get Then, observe that for all x ∈ K , there is x ∈ K such that (x, x ) ∈ R, which implies d E (x, x ) ≤