Richard Mathieu

Keywords: 2000 Mathematics Subject Classification. Primary: 60G51, 60J80; Secondary: 60J85, 60G44, 60K25, 60G07, 60G57 Lévy process, height process, Doob harmonic transform, splitting tree, spine decomposition, size-biased distribution, queueing theory

Lévy process conditioned by its height process

Introduction

In this paper, we consider Lévy processes (X t , t ≥ 0) with no negative jumps (or spectrally positive), not drifting to +∞ and conditioned to reach arbitrarily large heights (in the sense of the height process H defined in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]) before hitting 0. Let P x be the law of X conditional on X 0 = x and (F t , t ≥ 0) be its natural filtration.

Many papers deal with conditioning Lévy processes in the literature. In seminal works by L. Chaumont [START_REF] Chaumont | Sur certains processus de Lévy conditionnés à rester positifs[END_REF][START_REF] Chaumont | Conditionings and path decompositions for Lévy processes[END_REF] and then in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF], for general Lévy processes, L. Chaumont and R. Doney construct a family of measures P ↑

x , x > 0 of Lévy processes starting from x and conditioned to stay positive defined via a h-transform and it can be obtained as the limit P ↑

x (Θ, t < ζ) = lim ε→0 P x (Θ, t < e/ε|X s > 0, 0 ≤ s ≤ e/ε) for t ≥ 0, Θ ∈ F t and for e an exponential r.v. with parameter 1 independent from the process X and where ζ is the killing time of X. In the spectrally positive case, when E[X 1 ] < 0, P ↑

x is a sub-probability while, if E[X 1 ] = 0, it is a probability. In [START_REF] Hirano | Lévy processes with negative drift conditioned to stay positive[END_REF], K. Hirano considers Lévy processes drifting to -∞ conditioned to stay positive. More precisely, under exponential moment assumption, he is interested in two types of conditioning events: either the process X is conditioned to reach (-∞, 0] after time s or to reach level s > 0 before (-∞, 0]. Then, at the limit s → ∞, in both cases, he defines two different conditioned Lévy processes which can be described via h-transforms. In [5, ch. VII], J. Bertoin considers spectrally negative Lévy processes, i.e. with no positive jumps, and also constructs a family of conditioned processes to stay positive via the scale function associated with X.

Here, we restrict ourselves to study spectrally positive Lévy processes and consider a new way to obtain a Lévy process conditioned to stay positive without additional assumptions and contrary to [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF], the law of the conditioned process is honest. The process X is conditioned to reach arbitrarily large heights before T 0 := inf{t ≥ 0; X t = 0}. The term height should not be confused with the level used in the previously mentioned conditioning of Hirano. It has to be understood in the sense of the height process H associated with X and defined below. More precisely, for t ≥ 0, Θ ∈ F t , we are interested in the limit

lim a→∞ P x Θ, t < T 0 sup 0≤s≤T 0 H s ≥ a . (1)
In the following, we will consider three different cases for the Lévy process X: a Lévy process with finite variation and infinite Lévy measure, a Lévy process with finite variation and finite Lévy measure and finally a Lévy process with infinite variation.

In the first case, as it is stated in Theorem 2.3, the conditioning in ( 1) is trivial because P x (sup 0≤t≤T 0 H t ≥ a) = 1 for all positive a.

In the second case, X is simply a compensated compound Poisson process whose Laplace exponent can be written as

ψ(λ) = λ - (0,∞) (1 -e -λr )Λ(dr)
where Λ is a finite measure on (0, ∞) such that m := (0,∞) rΛ(dr) ≤ 1 (without loss of generality, we suppose that the drift is -1). Thus, X is either recurrent or drifts to -∞ and its hitting time T 0 of 0 is finite a.s. In this finite variation case, the height H t at time t is the (finite) number of records of the future infimum, that is, the number of times s such that

X s-< inf [s,t] X.
The process X is then conditioned to reach height a before T 0 . In the limit a → ∞ in [START_REF] Abraham | Feller property and infinitesimal generator of the exploration process[END_REF], one obtains a h-transform defined via a martingale which depends on the jumps of the future infimum. In the particular case m = 1, this martingale is X •∧T 0 and we recover the same h-transform as the one obtained in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF]. The key result used in our proof is due to A. Lambert [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF]. Indeed, the process X can be seen as a contour process of a splitting tree [START_REF] Geiger | Depth-first search of random trees, and Poisson point processes[END_REF]. These random trees are genealogical trees where each individual lives independently of other individuals, gives birth at rate Λ(R + ) to individuals whose life-lengths are distributed as Λ(•)/Λ(R + ). Then, to consider X conditioned to reach height n before T 0 is equivalent to look at a splitting tree conditional on having alive descendance at generation n.

Notice that we only consider the case when the drift α of X equals 1. However, the case α = 1 can be treated in the same way because X is still the contour process of a splitting tree but visited at speed α.

We also obtain a more precise result about conditional subcritical and critical splitting trees. For n ∈ N, set P n the law of a splitting tree conditional on {Z n = 0} where Z n denotes the number of extant individuals in the splitting tree belonging to generation n. In fact, (Z n , n ≥ 0) is a Galton-Watson process. We are interested in the law of the tree under P n as n → ∞. We obtain that under a x log x-condition on the measure Λ, the limiting tree has a unique infinite spine where individuals have the size-biased lifelength distribution m -1 zΛ(dz) and typical finite subtrees are grafted on this spine.

The spine decomposition with a size-biased spine that we obtain is similar to the construction of size-biased splitting trees marked with a uniformly chosen point in [START_REF] Geiger | Size-biased and conditioned random splitting trees[END_REF] where all individuals on the line of descent between root and this marked individual have size-biased lifelengths. It is also analogous to the construction of size-biased Galton-Watson trees in Lyons et al. [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF]. These trees arise by conditioning subcritical or critical GW-trees on nonextinction. See also [START_REF] Aldous | Asymptotic fringe distributions for general families of random trees[END_REF][START_REF] Geiger | Elementary new proofs of classical limit theorems for Galton-Watson processes[END_REF][START_REF] Lambert | Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct[END_REF]. In [START_REF] Duquesne | Continuum random trees and branching processes with immigration[END_REF], T. Duquesne studied the so-called sin-trees that were introduced by D. Aldous in [START_REF] Aldous | Asymptotic fringe distributions for general families of random trees[END_REF]. These trees are infinite trees with a unique infinite line of descent. He also considers the analogous problem for continuous trees and continuous state branching processes as made by other authors in [START_REF] Lambert | The genealogy of continuous-state branching processes with immigration[END_REF][START_REF] Lambert | Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct[END_REF][START_REF] Li | Asymptotic behaviour of continuous time and state branching processes[END_REF].

We finally consider the case where X has paths with infinite variation. Its associated Laplace exponent is specified by the Lévy-Khintchine formula

ψ(λ) = αλ + βλ 2 + (0,∞) Λ(dr)(e -λr -1 + λr)
where α ≥ 0, (0,∞) Λ(dr)(r ∧ r 2 ) < ∞ and either β > 0 or (0,1) Λ(dr)r = ∞. In order to compute the limit [START_REF] Abraham | Feller property and infinitesimal generator of the exploration process[END_REF] in that case, we use the height process (H t , t ≥ 0) defined in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF][START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF] which is the analogue of the discrete-space height process in the finite variation case. We set S t := sup [0,t] X. Then, since 0 is regular for itself for S -X, H is defined through local time. Indeed, for t ≥ 0, H t is the value at time t of the local time at level 0 of S (t) -X (t) where X (t) is the time-reversed process of X at t

X (t) s := X t--X (t-s)-s ∈ [0, t]
(with the convention X 0-= X 0 ) and S (t)

s := sup 0≤r≤s X (t)
r is its past supremum. Under the additional hypothesis

[1,+∞) dλ ψ(λ) < ∞, (2)
which implies that X has paths with infinite variation and that the height process H is locally bounded, we obtain a similar result to the finite variation case: the limit in (1) allows us to define a family of (honest) probabilities (P ↑ x , x > 0) of Lévy processes conditioned to stay positive via a h-transform and the martingale Ht 0 ρ t (dz)e αz 1 {t≤T 0 } where ρ t (•) is a random positive measure on R + which is a slight modification of the exploration process defined in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF][START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF] and α = ψ (0) ≥ 0 (since X does not drift to +∞).

Again, in the recurrent case (i.e. if α = 0), we observe that the previous quantity equals X t∧T 0 and we recover the h-transform h(x) = x of [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] in the spectrally positive case. Indeed, for general Lévy processes, the authors consider the law of the Lévy process conditioned to stay positive which is defined via the h-transform

h(x) = E [0,∞) 1 {It≥-x} dL t
where I is the past infimum process and L is a local time at 0 for X -I. In the particular spectrally positive case, L = -I and h is the identity.

Under P ↑ x , the height process (H t , t ≥ 0) can be compared to the left height process ← -H studied in [START_REF] Duquesne | Continuum random trees and branching processes with immigration[END_REF]. In that paper, T. Duquesne gives a genealogical interpretation of a continuousstate branching process with immigration by defining two continuous contour processes ← -H and -→ H that code the left and right parts of the infinite line of descent. We construct two similar processes for conditioned splitting trees in Section 3.

We also obtain a path decomposition of X at its minimum: under P ↑

x , the pre-minimum and post-minimum are independent and the law of the latter is P ↑ which is, roughly speaking, the excursion measure of X -I conditioned to reach "infinite height". For similar results, see [START_REF] Chaumont | Sur certains processus de Lévy conditionnés à rester positifs[END_REF][START_REF] Chaumont | Conditionings and path decompositions for Lévy processes[END_REF][START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] and references therein. As in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF], the decomposition of X under P ↑

x implies the convergence of P ↑

x as x → ∞ to P ↑ . This probability can be viewed as the law of the Lévy process conditioned to stay positive and starting from 0.

The paper is organized as follows. In Section 2, we treat the finite variation case and investigate the limiting process after stating some properties about splitting trees. Section 3 is devoted to studying the conditioned splitting tree and Section 4 to considering Lévy processes with infinite variation and to giving properties of the conditioned process.

Finite variation case

2.1. Definitions and statement of result. Let Λ be a positive measure on (0, ∞) such that Λ = 0 and

(0,∞) (x ∧ 1)Λ(dx) < ∞
and let (X t , t ≥ 0) be a spectrally positive Lévy process with Lévy measure Λ and such that E 0 e -λXt = e tψ(λ) , λ > 0 where P x is the law of X conditioned to X 0 = x and

ψ(λ) = λ - (0,∞) (1 -e -λr )Λ(dr).
We denote by F t := σ(X s , 0 ≤ s ≤ t) the natural filtration of X. We will suppose that m := (0,∞) rΛ(dr) ≤ 1 that is, X is recurrent (m = 1) or drifts to -∞ (m < 1). Then the hitting time T 0 := inf{t ≥ 0; X t = 0} is finite almost surely. Observe that since X is spectrally positive, the first hitting time of (-∞, 0] is T 0 . Definition 2.1. The height process H associated with X is defined by

H t := # 0 ≤ s ≤ t; X s-< inf s≤r≤t X r .
We set

{s 1 t < • • • < s Ht t } := {0 ≤ s ≤ t < T 0 ; X s-< inf s≤r≤t X r }, I t s := inf s≤r≤t X r , 0 ≤ s ≤ t
and we denote the jumps of (I t s , s ≤ t) by ρ t i := inf s i t ≤r≤t X r -X s i t -for 1 ≤ i ≤ H t and ρ t 0 = inf 0≤r≤t X r (see Figure 1).

The assumption (0,∞) (x ∧ 1)Λ(dx) < ∞ implies that the paths of X have finite variation and then for all positive t, H t is finite a.s. (Lemma 3.1 in [START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF]).

Remark 2.2. The process X can be seen as a LIFO (last in-first out) queue [START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF][START_REF] Robert | Stochastic networks and queues[END_REF]. Indeed, a jump of X at time t corresponds to the entrance in the system of a new customer who requires a service ∆X t := X t -X t-. This customer is served in priority at rate 1 until a new customer enters the system. Then, the ρ t i 's are the remaining service times of the H t customers present in the system at time t.

The sequence (ρ t i , i ≤ H t ) can be seen as a random positive measure on non-negative integers which puts weight ρ t i on {i}. Its total mass is X t and its support is {0, ..., H t }. We denote by S the set of measures on N with compact support. For ν in S, set ν i := ν({i}), i ≥ 0 and

H(ν) := max{i ≥ 0; ν i = 0}.
Then, according to [26, p.200], the process (ρ t∧T 0 , t ≥ 0) is a S-valued Markov process. Its infinitesimal generator A is defined by

A(f )(ν) = (0,∞) f (ν 0 , ν 1 , . . . , ν H(ν) , r, 0, . . . ) -f (ν 0 , ν 1 , . . . , ν H(ν) , 0, . . . ) Λ(dr) - ∂f ∂x H(ν) (ν 0 , . . . , ν H(ν) , 0, . . . ) 1 {ν =0} . (3)
T 0 x t ρ t 0 ρ t 1 ρ t 2 H t = 2 s 1 t s 2 t Figure 1.
A trajectory of the process X started at x and killed when it reaches 0 and the remaining service times at time t ρ t i for i ∈ {0, . . . , H t }.

In the following proposition, we condition the Lévy process X to reach arbitrarily large heights before T 0 .

z 2 Λ(dz) < ∞ or m < 1 and [1,∞) z log(z)Λ(dz) < ∞ .
Then, for t ≥ 0 and Θ ∈ F t ,

lim a→∞ P x Θ, t < T 0 sup s≤T 0 H s ≥ a = 1 x E x [M t∧T 0 1 Θ ]
where

M t = Ht i=0 ρ t i m -i .
In particular, if m = 1, then M t = X t . Moreover, the process (M t∧T 0 , t ≥ 0) is a (F t )martingale under P x . (ii) If b = ∞, the conditioning with respect to {sup s∈(0,T 0 ) H s ≥ a} is trivial in the sense that for all a ≥ 0, P x (sup s∈(0,T 0 ) H s ≥ a) = 1.

Observe that if b < ∞, the process X is simply a compensated compound Poisson process whose jumps occur at rate b an are distributed as Λ(•)/b.

The proof of this result will be made in Section 2.3. It uses the fact that X can be viewed as the contour process of a splitting tree visited at speed 1. The integrability hypotheses about Λ are made in order to use classical properties of (sub)critical BGW processes that appear in splitting trees.

Notice that the case where X is a Lévy process with Laplace exponent ψ(λ) = αλ -(0,∞) (1 -e -λr )Λ(dr) and b < ∞ can be treated in a same way if m ≤ α. Indeed, in that case, X is still the contour process of a splitting tree but it is visited at speed α. Theorem 2.3 is still valid but the martingale becomes

M t∧T 0 = Ht i=0 ρ t i α m i 1 {t≤T 0 } .
Before the proof, we define the splitting trees and recall some of their properties.

Splitting Trees.

Most of what follows is taken from [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF]. We denote the set of finite sequences of positive integers by

U = ∞ n=0 (N * ) n where (N * ) 0 = {∅}. Definition 2.4. A discrete tree T is a subset of U such that (i) ∅ ∈ T (root of the tree) (ii) if uj ∈ T for j ∈ N, then u ∈ T (if an individual is in the tree, so is its mother) (iii) ∀u ∈ T , ∃K u ∈ N ∪ {∞}, ∀j ∈ {1, . . . , K u }, uj ∈ T (K u is the offspring number of u).
If u = (u 1 , . . . , u n ) ∈ T , then its generation is |u| := n, its ancestor at generation i is denoted by u|i and if v = (v 1 , . . . , v m ) we denote by uv the concatenation of u and v uv := (u 1 , . . . , u n , v 1 , . . . , v m ).

In chronological trees, each individual has a birth level α and a death level ω. Let p 1 , p 2 be the two canonical projections of U := U × [0, +∞) on U and [0, ∞). We will denote by T the projection of T ⊂ U on U T := p 1 (T) = {u : ∃σ ≥ 0, (u, σ) ∈ T}.

Definition 2.5. A subset T of U is a chronological tree if (i) ρ := (∅, 0) ∈ T (the root) (ii) T is a discrete tree (iii) ∀u ∈ T \{∅}, ∃0 ≤ α(u) < ω(u) ≤ ∞ such that (u, σ) ∈ T if and only if α(u) < σ ≤ ω(u). α(u) (resp. ω(u))
is the birth (resp. death) level of u (iv) if ui ∈ T , then α(u) < α(ui) < ω(u) (an individual has only children during its life) (v) if ui, uj ∈ T then i = j implies α(u i ) = α(u j ) (no simultaneous births).

For u ∈ T , we denote by ζ(u) := ω(u) -α(u) its lifetime duration. For two chronological trees T, T and x = (u, σ) ∈ T such that σ = ω(u) (not a death point) and σ = α(ui) for any i (not a birth point), we denote by G(T , T, x) the graft of T on T at x G(T , T, x)

:= T ∪ {(uv, σ + τ ) : (v, τ ) ∈ T }.
Recall that Λ is a σ-finite measure on (0, ∞] such that (0,∞) (r ∧ 1)Λ(dr) < ∞. A splitting tree [START_REF] Geiger | Size-biased and conditioned random splitting trees[END_REF][START_REF] Geiger | Depth-first search of random trees, and Poisson point processes[END_REF] is a random chronological tree defined as follows. For x ≥ 0, we denote by P x the law of a splitting tree starting from an ancestor individual ∅ with lifetime (0, x]. We define recursively the family of probabilities P = (P x ) x≥0 . Let (α i , ζ i ) i≥1 be the atoms of a Poisson measure on (0, x)×(0, +∞] with intensity measure Leb⊗Λ where Leb is the Lebesgue measure. Then P is the unique family of probabilities on chronological trees T such that

T = n≥1 G(T n , ∅ × (0, x), α n )
where, conditional on the Poisson measure, the (T n ) are independent splitting trees and for n ≥ 1, conditional on ζ n = ζ, T n has law P ζ . The measure Λ is called the lifespan measure of the splitting tree and when it has a finite mass b, there is an equivalent definition of a splitting tree:

• individuals behave independently from one another and have i.i.d. lifetime durations distributed as Λ(•) b , • conditional on her birthdate α and her lifespan ζ, each individual reproduces according to a Poisson point process on (α, α + ζ) with intensity b, • births arrive singly. We now display a branching process embedded in a splitting tree and which will be useful in the following. According to [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF], when b < ∞, if for n ∈ N, Z n is the number of alive individuals of generation n

Z n := #{v ∈ T : |v| = n}, (4)
then under P, (Z n , n ≥ 0) is a Bienaymé-Galton-Watson (BGW) process starting at 1 and with offspring distribution defined by

p k := (0,∞) Λ(dz) b (bz) k k! e -bz , k ≥ 0. ( 5
)
Notice that under P x , (Z n , n ≥ 1) is still a BGW process with the same offspring distribution but starting at Z 1 , distributed as a Poisson r.v. with parameter bx.

Because of the additional hypothesis m < 1 (resp. m = 1), the splitting trees that we consider are subcritical (resp. critical). Then, in both cases they have finite lengths a.s. and we can consider their associated JCCP (for jumping chronological contour process) (Y t , t ≥ 0) as it is done in [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF].

It is a càdlàg piecewise linear function with slope -1 which visits once each point of the tree. The visit of the tree begins at the death level of the ancestor. When the visit of an individual v of the tree begins, the value of the process is her death level ω(v). Then, it visits v backwards in time. If she has no child, her visit is interrupted after a time ζ(v); otherwise the visit stops when the birth level of her youngest child (call it w) is reached. Then, the contour process jumps from α(w) to ω(w) and starts the visit of w in the same way. When the visits of w and all her descendance will be completed, the visit of v can continue (at this point, the value of the JCCP is α(w)) until another birth event occurs. When the visit of v is finished, the visit of her mother can resume (at level α(v)). This procedure then goes on recursively until level 0 is encountered (0 = α(∅) = birth level of the root) and after that the value of the process is 0 (see Figure 2). For a more formal definition of this process, read Section 3 in [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF].

t x Y t T 0 t Figure 2.
On the left panel, a splitting tree whose ancestor has lifespan duration x (vertical axis is time and horizontal axis shows filiation) and its associated Jumping Chronological Contour Process (Y t , t ≥ 0) on the right panel.

Moreover, the splitting tree can be fully recovered from its JCCP and we will use this correspondence to prove Theorem 2.3. It enables us to link the genealogical height (or generation) in the chronological tree and the height process of the JCCP. Proposition 2.6 (Lambert, [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF]). The process (Y t , t ≥ 0) under P x has the law of the Lévy process (X t , 0 ≤ t ≤ T 0 ) under P x .

Moreover, if, as in Definition 2.1, for t ≥ 0, we consider

h(t) := #{0 ≤ s ≤ t; Y s-< inf s≤r≤t Y r },
then h(t) is exactly the genealogical height in T of the individual visited at time t by the contour process.

2.3.

Proof of Theorem 2.3. Thanks to Proposition 2.6, the process X is the JCCP of a splitting tree with lifespan measure Λ. For a ≥ 1, let τ a := inf{t ≥ 0; H t ≥ a}. Then, {sup s≤T 0 H s ≥ a} = {τ a < T 0 }. Furthermore, according to the second part of Proposition 2.6, the events {X reaches height a before T 0 } and {the splitting tree is alive at generation a} coincide. We first prove the simpler point (ii) of Theorem 2.3 where b = ∞. According to [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF], if for n ≥ 0, Z n denotes the sum of lifespans of individuals of generation n in the splitting tree

Z n := v∈T ,|v|=n ζ(v),
then under P x , the process (Z n , n ≥ 0) is a Jirina process starting at x and with branching mechanism

F (λ) := (0,∞) (1 -e -λr )Λ(dr) = λ -ψ(λ),
that is, Z := (Z n , n ≥ 0) is a time-homogeneous Markov chain with values in R + , satisfying the branching property with respect to initial condition, (i.e. if Z a and Z b are two independent copies of Z respectively starting from a and b, then Z a + Z b has the same distribution as Z starting from a + b) and such that

E x e -λZn = e -xFn(λ)
where

F n := F • • • • • F is the n-th iterate of F .
Hence, since H is the genealogical height in the splitting tree ,

P x (τ a < T 0 ) = P x (Z a = 0) = 1 -exp -x lim λ→∞ F a (λ) = 1
by monotone convergence and because the mass of Λ is infinite.

We now make the proof of Theorem 2.3(i) and suppose that b is finite.

P x Θ, t < T 0 sup s≤T 0 H s ≥ a = P x (Θ, τ a ≤ t < T 0 ) + P x (Θ, t < τ a < T 0 ) P x (τ a < T 0 ) (6)
We will investigate the asymptotic behaviors of the three probabilities in the last display as a → ∞.

As previously, H is the genealogical height in the splitting tree but in this case, we can use the process (Z a , a ≥ 1) defined by (4). As explained above, this process is a BGW process with offspring generating function defined by [START_REF] Bertoin | Lévy processes[END_REF] and such that under P x , Z 1 has a Poisson distribution with parameter bx. With an easy computation, one sees that the mean offspring number equals m = (0,∞) rΛ(dr) ≤ 1. Hence the BGW-process is critical or subcritical. We have

P x (τ a < T 0 ) = P x (Z a = 0) = k≥0 P x (Z a = 0|Z 1 = k)P x (Z 1 = k)
and by the branching property,

P x (Z a = 0|Z 1 = k) = 1 -(1 -P(Z a-1 = 0)) k ∼ a→∞ kP(Z a-1 = 0).
We first treat the subcritical case. According to Yaglom [START_REF] Yaglom | Certain limit theorems of the theory of branching random processes[END_REF]

, if (Z n , n ≥ 0) is subcritical (m < 1) and if k≥1 p k (k log k) < ∞, then there exists c > 0 such that lim n→∞ P(Z n = 0) m n = c. (7)
In the following lemma, we show that this log-condition holds with assumptions of Theorem 2.3.

Lemma 2.7. If [1,∞) z log(z)Λ(dz) < ∞, then k≥1 p k (k log k) < ∞. Proof. According to (5), k≥2 p k k log k = k≥2 k log kb -1 (0,∞) Λ(dz) (bz) k k! e -bz = b -1 (0,∞) Λ(dz)e -bz k≥2 k log k (bz) k k! by Fubini-Tonelli theorem. Since we have log k ≤ k -z z + log z, k ≥ 2, z > 0, k≥2 k log k (bz) k k! ≤ k≥2 k k z -1 + log z (bz) k k! ≤ k≥2 k z k-1 b k (k -1)! + z log z k≥2 z k-1 b k (k -1)! ≤ b(z + 1)e bz + bz log ze bz and k≥2 p k k log k ≤ (0,∞) Λ(dz)(z + 1 + z log z) < ∞.
Then if [1,∞) r log rΛ(dr) < ∞, the log-condition of ( 7) is fulfilled and there exists a constant c such that lim

a→∞ P(Z a-1 = 0|Z 1 = 1)/m a-1 = c.
Then,

lim a→∞ P(Z a = 0|Z 1 = k) m a-1 = kc.
Moreover,

P(Z a = 0|Z 1 = k) m a-1 ≤ kP(Z a = 0|Z 1 = 1) m a-1 ≤ Ck and k≥0 CkP x (Z 1 = k) = CE x [Z 1 ] = Cbx < ∞
where C is some positive constant. Hence, using the dominated convergence theorem,

lim a→∞ P x (τ a < T 0 ) m a-1 = cE x [Z 1 ] = cbx (8)
Similarly, if (Z n , n ≥ 0) is critical (m = 1), since the variance of its reproduction law

σ 2 = k≥1 k 2 p k -m 2 = b (0,∞) z 2 Λ(dz) -m + m 2 = b (0,∞) z 2 Λ(dz)
is finite, one also knows [START_REF] Athreya | Branching processes[END_REF] the asymptotic behavior of P(Z n = 0). Indeed, we have Kolmogorov's estimate

lim n→∞ nP(Z n = 0) = 2 σ 2 . ( 9
)
Then

lim a→∞ P x (τ a < T 0 ) a -1 = 2 σ 2 bx. ( 10
)
We are now interested in the behavior of P x (Θ, τ a ≤ t < T 0 ) as a → ∞. In fact, we will show that it goes to 0 faster than m a-1 (resp. 1/a) if m < 1 (resp. m = 1). Since b = Λ(R + ) < ∞, the total number N t of jumps of X before t has a Poisson distribution with parameter bt. Hence, since {τ a ≤ t} ⊂ {N t ≥ a},

P x (Θ, τ a ≤ t < T 0 ) ≤ P(N t ≥ a) = i≥a e -bt (bt) i i! ≤ (bt) a a! .
Thus, using the last equation and equation ( 8) or [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], the first term of the r.h.s. of ( 6) vanishes as a → ∞ for m ≤ 1. We finally study the term P x (Θ, t < τ a < T 0 ). For a word u, i ∈ N and x ∈ R, we denote by A(u, i, x) the event {u gives birth before age x to a daughter which has alive descendance at generation |u| + i}, that is, if |u| = j, A(u, i, x) := {∃v ∈ U; |v| = i, uv ∈ T and α(uv|j + 1) -α(u) ≤ x}.

Let v t be the individual visited at time t. Hence, using the Markov property at time t and recalling that Θ ∈ F t , we have

P x (Θ, t < τ a < T 0 ) = E x 1 {t<T 0 } 1 {t<τa} 1 Θ P x (τ a < T 0 |F t ) = E x 1 {t<T 0 } 1 {t<τa} 1 Θ P x Ht i=0 A(v t |i, a -i, ρ t i ) F t
and by the branching property,

P x Ht i=0 A(v t |i, a -i, ρ t i ) F t = 1 - Ht i=0 1 -P ρ t i (Z a-i = 0) a.s.
As previously, since computations for subcritical and critical cases are equivalent, we only detail the first one. We have, with another use of [START_REF] Chaumont | Conditionings and path decompositions for Lévy processes[END_REF],

P x Ht i=0 A(v t |i, a -i, ρ t i ) F t ∼ a→∞ Ht i=0 m a-i-1 ρ t i cb a.s.
We want to use the dominated convergence theorem to prove that

lim a→∞ m 1-a P x (Θ, t < τ a < T 0 ) = cbE x 1 {t<T 0 } 1 Θ Ht i=0 ρ t i m -i
and then, using [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF],

lim a→∞ P x Θ, t < T 0 sup s≤T 0 H s ≥ a = 1 x E x 1 {t<T 0 } 1 Θ Ht i=0 ρ t i m -i
so that the proof of the subcritical case would be finished. We have almost surely

m -a P x (τ a < T 0 |F t )1 {t<T 0 } ≤ m -a Ht i=0 P x (A(v t |i, a -i, ρ t i ))1 {t<T 0 } ≤ C Ht i=0 ρ t i m -i 1 {t<T 0 }
where C is a positive, deterministic constant. Hence, to obtain an integrable upper bound, since

E x [ρ t 0 1 {t<T 0 } ] ≤ x, it is sufficient to prove that E x Ht i=1 ρ t i m -i < ∞ (11)
in order to use the dominated convergence theorem. Recall that X (t) denotes the time-reversal of X at time t X (t) s := X t--X (t-s)-s ∈ [0, t], (X 0-= X 0 ) and S (t) is its associated past supremum process S (t) s := sup{X (t) r : 0 ≤ r ≤ s}. It is known that the process X (t) has the law of X under P 0 [5, ch.II]. We also have

H t = R t := # 0 ≤ s ≤ t; X (t) s = S (t)
s which is the number of records of the process S (t) during [0, t] and the ρ t i 's are the overshoots of the successive records. More precisely, if we denote by T1 < T2 < • • • the record times of X (t) , the overshoots are ρi :=

X (t) Ti -sup 0≤s< Ti X (t) s , i ≥ 1.
We come back to the proof of [START_REF] Geiger | Size-biased and conditioned random splitting trees[END_REF]. We have

Ht i=1 ρ t i m -i = Rt i=1 ρRt-i+1 m -i = Rt i=1 ρi m i-Rt-1 = i≥1 ρi m i-Rt-1 1 { Ti ≤t} .
We denote by ( F s , s ≥ 0) the natural filtration of X (t) . Thus, using Fubini-Tonelli theorem and the strong Markov property for X (t) applied at time Ti

E x Ht i=1 ρ t i m -i = i≥1 E ρi m i-Rt-1 1 { Ti ≤t} = m -1 i≥1 E ρi 1 { Ti ≤t} f (t -Ti )
where

f (s) = E 1 m Rs , s ≥ 0.
However, as previously, we have almost surely R t ≤ N t where N t is the number of jumps of X or X (t) before t and so has a Poisson distribution with parameter bt. Thus,

f (s) ≤ E[m -Ns ] = exp bs(m -1 -1) = e κs
where κ is some positive constant and

E x Ht i=1 ρ t i m -i ≤ m -1 e κt i≥1 E ρi ; Ti ≤ t .
Moreover,

E ρi ; Ti ≤ t = E ρi 1 { Ti -Ti-1 ≤t-Ti-1 } 1 { Ti-1 ≤t} = t 0 P( Ti-1 ∈ ds)E ρ1 ; T1 ≤ t -s ≤ E ρ1 ; T1 ≤ t P( Ti-1 ≤ t) since (ρ i , Ti -Ti-1 ), i ≥ 1 are i.i.d. random variables. Then E x Ht i=1 ρ t i m -i ≤ m -1 e κt E ρ1 ; T1 ≤ t i≥0 P( Ti ≤ t) = m -1 e κt E ρ1 ; T1 ≤ t i≥0 P(R t ≥ i).
The sum in the r.h.s equals E[R t ] which is finite since R t ≤ N t a.s. According to Theorem VII.17 in [START_REF] Bertoin | Lévy processes[END_REF], the joint law of

X (t) T1 , ∆X (t) T1
is given by

E F X (t) T1 , ∆X (t) T1 ; T1 < ∞ = (0,∞) Λ(dy) y 0 dxF (x, y). (12)
Thus, since

{ T1 ≤ t} ⊂ X (t) T1 -≥ -t , E ρ1 ; T1 ≤ t ≤ E ρ1 1 ∆X (t) T1 -X (t) T1 ≤t 1 { T1 <∞} = (0,∞) Λ(dy) y 0 x1 {y-x≤t} dx = (t,∞) Λ(dy)(yt -t 2 /2) + (0,t] Λ(dy)y 2 /2 < ∞
and the proof of ( 11) is completed. Finally, in the critical case m = 1, computations are similar. Thanks to [START_REF] Duquesne | Continuum random trees and branching processes with immigration[END_REF],

P x Ht i=0 A(v t |i, a -i, ρ t i ) F t ∼ a→∞ 1 a Ht i=0 ρ t i 2 σ 2 b = 1 a 2b σ 2 X t a.s.
by the definition of the ρ t i 's. Moreover, when m = 1, the process X is a (F t )-martingale and

E x [X t ] = E x [X 0 ] = x. Then, by the dominated convergence theorem, lim a→∞ aP x (Θ, t < τ a < T 0 ) = 2b σ 2 E x [X t∧T 0 1 Θ ]
and then, using [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF],

lim a→∞ P x Θ, t < T 0 sup s≤T 0 H s ≥ a = 1 x E x [X t∧T 0 1 Θ ] .
We conclude the proof by showing that (M t∧T 0 , t ≥ 0) is a (F t )-martingale. First, if m = 1, we have M t = X t and as it was stated just before, in this case, the process X is a martingale and so is X •∧T 0 . We now consider the case m < 1. Recall from (3) that the infinitesimal generator of (ρ t∧T 0 , t ≥ 0) is

A(f )(ν) = (0,∞) f (ν 0 , ν 1 , . . . , ν H(ν) , r, 0, . . . ) -f (ν 0 , ν 1 , . . . , ν H(ν) , 0, . . . ) Λ(dr) - ∂f ∂x H(ν) (ν 0 , . . . , ν H(ν) , 0, . . . ) 1 {ν =0} , ν ∈ S.
Let g be the application from S to R such that

g(ν) := H(ν) i=0 ν i m -i , ν ∈ S. (13)
Clearly, we have g(

ρ t∧T 0 ) = M t∧T 0 . According to [25, ch.VII], if G t := {σ(ρ r ), 0 ≤ r ≤ t}, to prove that M is a (G t )-martingale, it is sufficient to show that A(g) = 0. For ν ∈ S, we have A(g)(ν) = (0,∞) (g(ν) + rm -H(ν)-1 -g(ν))Λ(dr) -m -H(ν) 1 {ν =0} = (0,∞) rΛ(dr)m -H(ν)-1 -m -H(ν) 1 {ν =0} = 0. Then, M •∧T 0 is a (G t )-martingale. Moreover, since ρ t , 1 = X t , F t ⊂ G t and clearly G t ⊂ F t . Then, F t = G t and M •∧T 0 is a (F t )-martingale.

2.4.

A new probability.

Definition 2.8. For x > 0, we define a new probability measure P ↑ x on (Ω, F t ) by

P ↑ x (Θ) = 1 x E x [M t∧T 0 1 Θ ] , Θ ∈ F t .
First, notice that since M •∧T 0 is a martingale, this definition is consistent and that this new measure is honest, that is, it has mass 1. Second, (X t , t ≥ 0) is no more Markovian under P ↑ x while (ρ t , t ≥ 0) is because P ↑

x is obtained by a h-transform via the application g defined by (13).

Proposition 2.9. Under P ↑

x , in probability,

X t -→ t→∞ + ∞.
Proof. It is sufficient to prove that for all positive λ, E ↑ x e -λXt → 0 as t tends to ∞. First, in the critical case m = 1,

E ↑ x e -λXt = 1 x E x e -λXt X t∧T 0 -→ t→∞ 0 ( 14
)
using dominated convergence theorem, because (X t∧T 0 , t ≥ 0) is a non-negative martingale converging a.s. to 0 and the mapping x → e -λx x is bounded on R + .

We now suppose that m < 1. Then, for all positive a, distinguishing between H t > a and H t ≤ a, we have

E ↑ x e -λXt = 1 x E x e -λXt Ht i=0 m -i ρ t i 1 {t≤T 0 } ≤ 1 x E x e -λXt Ht i=1 m -i ρ t i 1 {t≤T 0 } 1 {Ht>a} + P x (t ≤ T 0 ) + m -a x E x e -λXt X t∧T 0 since i ρ t i = X t .
The second term of the r.h.s is an upper bound of the term i = 0 of the sum and it goes to 0 as t tends to ∞ since T 0 is finite a.s. So does the third term similarly to the critical case in [START_REF] Hirano | Lévy processes with negative drift conditioned to stay positive[END_REF]. Then, to finish the proof, it is sufficient to show that the first term tends to 0 uniformly in t as a goes to ∞. We have

A(t, a) := E x e -λXt Ht i=1 m -i ρ t i 1 {t≤T 0 } 1 {Ht>a} ≤ E x e -λ(Xt-It) Ht i=1 m -i ρ t i 1 {t≤T 0 } 1 {Ht>a}
and as in proof of Theorem 2.3, by time reversing at time t, we have

A(t, a) ≤ i≥1 E e -λSt m -(Rt-i+1) ρi 1 { Ti ≤t} 1 {Rt>a} = i≥1 m i-1 B i (t, a)
where the Ti 's are the times of successive records of the supremum process S, the ρi 's are the associated overshoots and R t is the number of records up to time t. In particular, S t = Rt j=1 ρj .

On the one hand, let us treat the case when i ≤ a. Since {R t > a} ⊂ { Ta ≤ t} and by applying the strong Markov property at time Ta ,

B i (t, a) ≤ E   Rt j=1 e -λρ j m -Rt ρi 1 { Ta≤t}   = E   a j=1 e -λρ j ρi 1 { Ta≤t} E   Rt j=a+1 e -λρ j m -Rt F Ta     = E   a j=1 e -λρ j ρi 1 { Ta≤t} m -a f (t -Ta )   (15)
where for s ≥ 0, f (s) := E Rs j=1 e -λρ j m -Rs . This function f is bounded. Indeed, for s ≥ 0,

f (s) = i≥1 m -i E   i j=1 e -λρ j ; Ti ≤ s < Ti+1   ≤ i≥1 m -i E   i j=1 e -λρ j 1 { Tj -Tj-1 ≤s}   ≤ i≥1 m -i E e -λρ 1 ; T1 < ∞ i
since the r.v. (ρ j , Tj -Tj-1 ) j≥1 are i.i.d. Applying equation [START_REF] Geiger | Elementary new proofs of classical limit theorems for Galton-Watson processes[END_REF] with the function F (x, y) = e -λx , we have

E e -λρ 1 1 { T1 <∞} = (0,∞) Λ(dy) 1 -e -λy λ = 1 - ψ(λ) λ .
Then,

f (s) ≤ i≥1 m -i 1 - ψ(λ) λ i < ∞
since 1 -ψ(λ)/λ < m for all positive λ because m = 1 -ψ (0) and ψ is strictly convex. We come back to [START_REF] Kennedy | Some martingales related to cumulative sum tests and single-server queues[END_REF]. If we denote by C an upper bound for f , for i ≤ a,

B i (t, a) ≤ Cm -a a j=1,j =i E e -λρ j 1 { Tj -Tj-1 ≤t} E e -λρ i ρi 1 { Ti -Ti-1 ≤t} ≤ Cm -a E e -λρ 1 1 { T1 <∞} a-1 E e -λρ 1 ρ1 1 { T1 <∞}
Applying again [START_REF] Geiger | Elementary new proofs of classical limit theorems for Galton-Watson processes[END_REF] with F (x, y) = xe -λx , we obtain

E ρ1 e -λρ 1 1 { T1 <∞} = ψ (λ) λ - ψ(λ) λ 2 < ∞. (16)
Hence,

a i=1 m i-1 B i (t, a) ≤ C(λ)m -a+1 1 - ψ(λ) λ a-1 a i=1 m i ≤ C(λ)am -a+1 1 - ψ(λ) λ a-1 . (17)
On the other hand, for i > a, as in previous computations, we apply the strong Markov property at time Ti to obtain

i≥a+1 m i-1 B i (t, a) ≤ i≥a+1 E   i j=1 e -λρ j ρi 1 { Ti ≤t} m -1 f (t -Ti )   ≤ Cm -1 i≥a+1 E e -λρ i ρi 1 { Ti -Ti-1 ≤t} 1 { Ti-1 ≤t} ≤ Cm -1 ψ (λ) λ - ψ(λ) λ 2 i≥a+1 P(R t ≥ i -1)
where we have first used the previous function f and its upper bound C and then that (ρ i , Ti -Ti-1 ) is independent from Ti-1 and equation [START_REF] Lambert | The genealogy of continuous-state branching processes with immigration[END_REF].

Then

i≥a+1 m i-1 B i (t, a) ≤ C(λ) i≥a P(R ∞ ≥ i) (18)
where C(λ) is a finite constant independent from t and a and R ∞ := lim t→+∞ R t ∈ N is the total number of records of S. Moreover, thanks to the strong Markov property, R ∞ follows a geometric distribution with success probability P( T1 = ∞) = 1 -m according to [START_REF] Geiger | Elementary new proofs of classical limit theorems for Galton-Watson processes[END_REF] with

F ≡ 1.
Finally, putting together ( 17) and ( 18), we obtain

A(t, a) ≤ i≥1 m i-1 B i (t, a) ≤ C(λ)   am -a+1 1 - ψ(λ) λ a-1 + i≥a P(R ∞ ≥ i)   (19)
where C(λ) is a finite constant independent from t and a. The first term of the r.h.s. of [START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF] goes to 0 as a goes to ∞ since 1 -ψ(λ)/λ < m. The second term also tends to 0 because E[R ∞ ] = (1 -m) -1 is finite. Hence, we have proved that A(t, a) tends to 0 as a → ∞ uniformly in t and E ↑

x [e -λXt ] → 0 which ends the proof.

Conditioned Subcritical and Critical Splitting Trees

In the previous section, we have defined the law of a conditioned Lévy process by using properties of (sub)critical splitting trees. Here, we study conditioned splitting trees by using the same properties about BGW-processes. More precisely, we are interested in the behavior as n → ∞ of a (sub)critical splitting tree conditioned to be alive at generation n and we want to give a spine decomposition of the limiting tree. We also give the distribution of its contour process which visits the left part of the spine.

We will use notation of Section 4.4 of [START_REF] Lambert | Population dynamics and random genealogies[END_REF]. For n ≥ 0 and u ∈ T , we denote by E n (u) the event {u has an extant descendance at generation |u| + n}. For simplicity of notation, we will denote by E n the event E n (∅). Let P n be the law of the splitting tree on this event

P n := P(•|E n ) = P(•|Z n > 0).
On the event E n , we define a distinguished lineage u n 0 u n 1 • • • u n n as the first lineage of the tree that reaches generation n as explained below. First, u n 0 = ∅ and one defines recursively:

for i ≥ 1, u n 0 u n 1 • • • u n i is the youngest daughter of u n 0 u n 1 • • • u n i-1
that has a descendance at generation n. We set U (n) := u n 0 u n 1 • • • u n n and we denote by

B n := {x ∈ T; p 1 (x) = U (n)
|k for 0 ≤ k ≤ n} this lineage of T defined on the event E n . For 0 ≤ k ≤ n -1, the age at which the individual U (n) |k gives birth to individual U (n) |k + 1 is

A n k := α(U (n) |k + 1) -α(U (n) |k), its residual lifetime is R n k := ω(U (n) |k) -α(U (n) |k + 1)
and its total lifelength is denoted by

T n k = A n k + R n k . Observe that modulo labelling, the sequence (A n k , R n k ), 0 ≤ k ≤ n -1 and A n n := ω(U (n) ) -α(U (n) ) characterizes the spine B n .
In the following, when we say that a tree (marked with a special lineage) converges for finite truncations, we mean that there is convergence for events that are measurable with respect to the σ-field generated by the truncations {x ∈ T; p 2 (x) ≤ σ}, σ > 0.

Theorem 3.1. We suppose that b < ∞ and that either m < 1 and [1,∞) z log(z)Λ(dz) < ∞ or m = 1 and (0,∞) z 2 Λ(dz) < ∞. Then, as n → ∞, the law of B n under P n converges for finite truncations to an infinite spine B whose distribution is characterized by an i.i.d sequence (A k , R k ) k≥0 such that

(A 0 , R 0 ) (d) = (U D, (1 -U )D)
where U is a uniform r.v. on (0, 1) independent from the size-biased r.v. D

P(D ∈ dz) = zΛ(dz) m z > 0.
Moreover, if (x i , i ∈ N) are the atoms of a Poisson measure on B with intensity b, then as n → ∞, the law of T under P n converges for finite truncations to an infinite tree with a unique infinite branch B on which are grafted at points (x i ) i.i.d. trees with common law P.

Proof. First, for t > 0 we denote by K n (t) the number of individuals at time t that have alive descendance at generation n. Then, as n → ∞, P(K n (t) = 1|Z n > 0) -→ 1. Hence, the limiting tree under P n has a unique infinite branch. We now investigate the law of this limiting spine. Let p be a natural integer. We denote by F p the σ-field generated by the lifespans of all individuals until generation p -1 and the numbers and birthdates of their daughters.

Under P n , we denote by N i the number of children of U (n) |i and by α ij , 1 ≤ j ≤ N i her age at their births. Then, for n ≥ p, for n 0 , . . . , n p-1 ∈ N * , for z 0 , . . . , z p-1 > 0, for 0 < x ij < z i where 0 ≤ i ≤ p -1, 1 ≤ j ≤ n i and for u 1 u 2 • • • u p any word of length p, we have

P n (E p ) := P n (T n i ∈ dz i , N i = n i , α ij ∈ dx ij , 0 ≤ i ≤ p -1, 1 ≤ j ≤ n i ; U (n) |p = u 1 • • • u p ) = 1 P(E n ) E   p-1 i=0   Λ(dz i ) b e -bz i (bz i ) n i n i ! n i j=1 dx ij z i u i+1 -1 l=1 1 (E n-i-1 (u 1 •••u i l)) c   1 E n-p (u 1 •••up)  
because by definition of B n , all the younger sisters of the marked individuals, that is, all individuals labeled by u 1 • • • u i l for any 0 ≤ i ≤ p -1 and any 1 ≤ l < u i+1 , have no alive descendance at generation n.

Conditioning with respect to the σ-field F p and thanks to the branching property, we have

P n (E p ) = p-1 i=0   Λ(dz i ) b e -bz i (bz i ) n i n i ! n i j=1 dx ij z i   p i=1 P(E c n-i ) u i -1 P(E n-p ) P(E n ) .
Since we consider subcritical or critical trees, P(E c n-i ) goes to 1 as n → ∞. Moreover, for m < 1, according to Lemma 2.7, if [1,∞) z log(z)Λ(dz) < ∞, then the log-condition of ( 7) is fulfilled and there exists c > 0 such that

P(E n ) m n = P(Z n = 0) m n -→ n→∞ c.
Furthermore, in critical case m = 1, if (0,∞) z 2 Λ(dz) is finite, according to (9), there is c > 0 such that lim

n→∞ nP(Z n = 0) = c .
In both cases, we obtain by the dominated convergence theorem

lim n→∞ P n (E p ) = p-1 i=0   Λ(dz i ) b e -bz i (bz i ) n i n i ! n i j=1 dx ij z i   m -p = p-1 i=0 z i Λ(dz i ) m • e -bz i (bz i ) n i -1 (n i -1)! • 1 n i • n i j=1 dx ij z j .
Thus, we see that in the limit n → ∞, the individuals of the spine have independent sizebiased lifelengths (first term of the r.h.s.), give birth to one plus a Poissonian number of children (second term) whose birth dates are independent uniform during the life of their mother (fourth term) and one of them is marked uniformly among them (third term). Hence, we obtain a limiting spine B characterized by a sequence (A k + R k , A k ) which are independent r.v. with the same joint distribution as (D, U D). Moreover, conditional on Intersecting the previous event E p with events involving trees descending from sisters of the marked individuals and applying the branching property leads by similar computations to the last statement of the theorem.

T k = A k + R k ,
We can now define two contour processes of the conditioned splitting tree. We denote by X ↑ (t), t ≥ 0 the contour of the limiting tree as defined previously for typical trees. Since the spine is infinite, only individuals on its left part are visited by this contour process. That is why we call it the left contour process. Notice that it does not characterize the conditioned tree because all information about the right part of the spine can not be recovered from X ↑ . That is why we need a second contour process X ↓ (t), t ≥ 0 .

For a non-conditioned (sub)critical tree, it is possible to define a process which visits it in the opposite direction by the time-reversal of the classical contour process X at T 0 : X := (X(T 0 -t), 0 ≤ t ≤ T 0 ) .

Then, for the conditioned tree, this reversed contour process is trapped on the right part of the infinite spine: it starts at level 0 and visits the right part of the infinite spine in the opposite direction of the typical contour process. We denote it by X ↓ . Roughly speaking, it is the contour process of the right part that we reverse at T 0 = +∞. In [START_REF] Duquesne | Continuum random trees and branching processes with immigration[END_REF], Duquesne obtains an equivalent result. He defines two processes ← -H and -→ H called left and right height processes and which describe the genealogy of a continuous state branching process with immigration. Notice that these height processes are linked to the height process (H t , t ≥ 0) that we use in Section 4 for Lévy processes with infinite variation.

A consequence of the previous theorem deals with the joint law of the processes X ↑ and X ↓ . Corollary 3.2. Let (U k , D k ) k be a sequence of i.i.d. pairs distributed as (U, D) in Theorem 3.1. We have

X ↑ , X ↓ (d) =   k≥0 X k (t -ς k )1 {t≥ς k } , k≥0 Xk (t -ςk )1 {t≥ς k } , t ≥ 0   where • (X k , Xk ) k is a sequence of i.i.d
. pairs of processes, where (X k , Xk ) is independent from (U j , D j ) for j = k. Moreover, conditionally on U k and D k , they are independent and X k is distributed as X started at level D k and stopped at the hitting time η k of level U k D k , and Xk is distributed as X started at level U k D k and reversed at the hitting time ηk of 0 by Y that is,

Xk (d) = X((η k -•) ∨ 0), • ς 0 = ς0 , ς k = η 1 + • • • + η k-1 and ςk = η1 + • • • + ηk-1 .
Remark 3.

3. An open question is to prove that the left contour process X ↑ of the conditioned tree has the same distribution as the conditioned process X under P ↑ defined in Section 2.

Infinite variation case

4.1. Definitions and main results. In this section, we consider the same setting as T. Duquesne and J.F. Le Gall in [10, Ch 1]. We assume that X is a Lévy process with no negative jumps, which does not drift to +∞ (so that X hits 0 a.s.) and has paths with infinite variation. Its law is characterized by its Laplace transform E 0 e -λXt = e tψ(λ) , λ > 0 specified by the Lévy-Khintchine formula

ψ(λ) = α 0 λ + βλ 2 + (0,∞) Λ(dr) e -λr -1 + λr1 {r<1}
where α 0 ∈ R, β ≥ 0 and Λ is a positive measure on (0, +∞) such that ∞ 0 Λ(dr)(1 ∧ r 2 ) is finite. Actually, since X does not drift to +∞, it has finite first moment and then (0,∞) Λ(dr)(r ∧ r 2 ) < ∞. We can rewrite ψ as

ψ(λ) = αλ + βλ 2 + (0,∞) Λ(dr)(e -λr -1 + λr).
Then, X does not drift to +∞ iff ψ (0) = α ≥ 0. More precisely, if α = 0, X is recurrent while if α < 0, it drifts to -∞.

According to Corollary VII.5 in [START_REF] Bertoin | Lévy processes[END_REF], X has infinite variation iff lim λ→∞ ψ(λ)/λ = +∞, which is satisfied iff

β > 0 or (0,1) rΛ(dr) = ∞. ( 20
)
In the following, we always suppose that (2) holds which implies [START_REF] Gall | Branching processes in Lévy processes: the exploration process[END_REF]. We denote by I the past infimum process of

X I t = inf 0≤s≤t X s .
The process X -I is a strong Markov process and because X has infinite variation, 0 is regular for itself for this process. Thus, we can consider its excursion measure N away from 0 and normalized so that -I is the associated local time.

The point 0 is also regular for itself for the Markov process X -S. We denote by L a local time at 0 for this process and by N * the associated excursion measure normalized so that for every Borel subset B of (-∞, 0),

N * σ 0 1 B (X s )ds = m(B) (21)
where m denotes the Lebesgue measure on R (see Lemma 1.1.1 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] for more details). This also fixes the normalization of the local time L.

We now define the height process H associated with X which is the counterpart of the height process in the finite variation case. As in Section 2, we denote by X (t) the timereversed process associated with X at time t and by S (t) its past supremum.

Definition 4.1. The height process H = (H t , t ≥ 0) associated with the Lévy process X is the local time at level 0 at time t of the process S (t) -X (t) with the normalization fixed in [START_REF] Li | Asymptotic behaviour of continuous time and state branching processes[END_REF] and H 0 := 0.

According to Theorem 1.4.3 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], the condition [START_REF] Aldous | Asymptotic fringe distributions for general families of random trees[END_REF] implies that H has a.s. continuous sample paths. There is an alternative definition of H t : there exists a positive sequence (ε k , k ≥ 0) such that a.s., for all t ≥ 0,

H t = lim k→∞ 1 ε k t 0 1 {Xt<I t s +ε k } ds < ∞ where for 0 ≤ s ≤ t, we set by I t s := inf s≤r≤t X r
the future infimum process of X at time t. Since this process is non-decreasing, one can define its right-continuous inverse

I -1 t I -1 t (u) := inf{s ≥ 0; I t s > u}, 0 ≤ u ≤ X t .
Observe that for 0 ≤ u ≤ I t , I -1 t (u) = 0. We now define the equivalent of the sequence (ρ t i , i ≤ H t ) defined in Section 2. Here, we consider the measure ρt defined in [10, p.25] and called exploration process to which we add a weight I t at 0. Definition 4.2. For t ≥ 0, the random positive measure ρ t is defined by

ρ t , f = ρt , f + I t f (0) = [0,t] d s I t s f (H s ) + I t f (0)
where d s I t s is the Stieltjes measure associated with the non-decreasing function s -→ I t s . Notice that this random measure has support [0, H t ], that ρ t ({0}) = I t and that its total mass is ρ t ,

1 = I t + (X t -I t ) = X t .
As in Section 2, we condition the Lévy process X to reach large heights before hitting 0 and the following theorem is the counterpart of Theorem 2.3. Theorem 4.3. Recall that α = ψ (0). Then, if (2) holds, for t ≥ 0 and Θ ∈ F t ,

lim a→∞ P x Θ, t < T 0 sup s≤T 0 H s ≥ a = 1 x E x Ht 0 ρ t (dz)e αz ; t ≤ T 0 , Θ .
Moreover, if

M t := Ht 0 ρ t (dz)e αz ,
(M t∧T 0 , t ≥ 0) is a martingale under P x . In particular, similarly to the finite variation case, if α = 0 (recurrent case), then M t = X t .

Remark 4.4. In the particular case Λ ≡ 0 (so we assume β > 0), we have ψ(λ) = αλ + βλ 2 and X is a Brownian motion with drift -α and variance 2β. Then, at it is noticed in [START_REF] Limic | A LIFO queue in heavy traffic[END_REF], the local time process at 0 for S -X is S/β. Also, for t ≥ 0, H t = (X t -I t )/β and ρt /β is the Lebesgue measure on [0, H t ]. Finally, in that particular case, if α > 0, the process

I t∧T 0 + β α e α β (Xt-It) -1 1 {t≤T 0 } , t ≥ 0
is a martingale. In fact, this process belongs to a larger class of martingales called Kennedy martingales [START_REF] Kennedy | Some martingales related to cumulative sum tests and single-server queues[END_REF] and studied in [START_REF] Azéma | Une solution simple au problème de Skorokhod[END_REF].

Before proving Theorem 4.3, we give two technical lemmas concerning integrability of the exploration process and the height process. Recall that ρ is the exploration measure process defined in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] ρt

, f = [0,t] d s I t s f (H s ), t ≥ 0.
We denote by e α the mapping

e α : x -→ e αx . ( 22
)
Lemma 4.5. For t > 0,

E [ ρt , e α ] = E Ht 0 ρt (dz)e αz < ∞.
Proof. First, for 0 ≤ s ≤ t, we have

H s ≤ H t d s I t s a.e. because on {∆I t s > 0} = {∆S (t) s > 0}, H t = L (t) t = L (t) s + L (s) s ≥ L (s) s = H s where for 0 ≤ u ≤ v, L (v)
u denotes the local time at level 0 at time u of the reverse process S (v) -X (v) . Then,

E Ht 0 ρt (dz)e αz = E t 0 d s I t s e αHs ≤ E t 0 d s I t s e αHt ≤ E (X t -I t )e αHt = E[S (t) t e αHt ].
By time-reversion at t, we have E S

t e αHt = E[S t e αLt ] where (L t , t ≥ 0) is a local time at 0 for S -X. First, if α = 0, E[S t e αLt ] = E[S t ] is finite according to [START_REF] Bertoin | Lévy processes[END_REF]Thm VII.4]. Moreover, if α > 0,

E[S t e αLt ] = ∞ 0 daE S t 1 {Lt≥ln a/α} = E[S t ] + ∞ 1 daE S t 1 {Lt≥ln a/α} . (23)
Since E[S t ] is finite, in order to prove the lemma, it is sufficient to show that

a -→ E S t 1 {Lt≥ln a/α} is integrable on [1, +∞). For b > 0, if L -1 (•) := inf{s ≥ 0; L(s) > •} denotes the right- continuous inverse of L, E S t 1 {Lt≥b} = E S L -1 (b) 1 {L -1 (b)≤t} + E S t -S L -1 (b) 1 {L -1 (b)≤t} (24)
On the one hand, if we denote by ϕ the bijective inverse mapping of ψ, the process

L -1 (b), S L -1 (b) , b < L(∞) (called
ascending ladder process) is a bivariate Lévy process killed at rate ψ (0) = α and whose bivariate Laplace exponent κ is

κ(µ, λ) := c µ -ψ(λ) ϕ(µ) -λ (25)
in the sense

E e -µL -1 (b)-λS L -1 (b) = e -bκ(µ,λ) .
Moreover, c is a constant depending on the normalization of the local time L (see Section VI.1 and Theorem VII.4 in [START_REF] Bertoin | Lévy processes[END_REF]). In fact, as it is proved in the proof of Lemma 1.1.2 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], with the normalization defined by ( 21), we have c = 1. Then, for µ > 0, using Markov inequality and ( 25), the first term of the r.h.s. of ( 24) satisfies

E S L -1 (b) 1 {L -1 (b)≤t} = E S L -1 (b) 1 {e -µL -1 (b) ≥e -µt } ≤ e µt E S L -1 (b) e -µL -1 (b) ≤ be µt ∂ ∂λ κ(µ, 0)e -bκ(µ,0) = be µt µ -αϕ(µ) ϕ(µ) 2 e -bµ/ϕ(µ) . (26)
On the other hand, by the strong Markov property applied at the stopping time L -1 (b), the second term of r.h.s. of ( 24) is

E S t -S L -1 (b) 1 {L -1 (b)≤t} = E S t-L -1 (b) 1 {L -1 (b)≤t}
where S has same distribution as process S and is independent from L -1 (b). Then,

E S t -S L -1 (b) 1 {L -1 (b)≤t} ≤ E[S t ] • P(L -1 (b) ≤ t).
According to [START_REF] Bertoin | Lévy processes[END_REF]Thm VII.4], E[S t ] is finite and by similar computations as before, for µ > 0,

P(L -1 (b) ≤ t) ≤ e µt e -bµ/ϕ(µ) . (27)
Then, using ( 26), [START_REF] Yaglom | Certain limit theorems of the theory of branching random processes[END_REF] and choosing b = ln a/α, we have

E S t 1 {Lt≥ln a/α} ≤ C(µ, t) ln a α + 1 a -µ ϕ(µ)α
where C(µ, t) is a constant only depending on µ and t. Since ϕ = ψ -1 is concave,

ϕ(µ) µ ≤ ϕ (0) = α -1
and since by hypothesis ψ(λ)/λ → ∞, for µ large enough, µ/(ϕ(µ)α) > 1. Hence, the application a -→ E S t 1 {Lt≥ln a/α} is integrable on [1, +∞) and the proof is complete.

Lemma 4.6. For any ξ > 0 and t ≥ 0, t 0 E e ξHs ds < ∞.

Proof. By time reversibility at time s, E e ξHs = E e ξLs . Using [START_REF] Yaglom | Certain limit theorems of the theory of branching random processes[END_REF] and similar techniques as in the previous proof, we have for all µ > 0

E e ξHs ≤ e µs ∞ 1 a -µ/(ϕ(µ)ξ) da + 1.
Then, t 0 E e ξHs ds is finite for µ large enough. Proof of Theorem 4.3.

Step 1: We begin the proof with showing that (M t∧T 0 , t ≥ 0) is a martingale. In the case α = 0, M = X and it is known that (X t , t ≥ 0) is a martingale.

We now suppose that α > 0. For t ≥ 0, we set

ν t := (I t + x)1 {It>-x} δ 0 + ρt and θ := inf{t ≥ 0; ν t = 0}.
In fact, ν is the exploration process starting at xδ 0 defined in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]. If we set Gt := σ(ρ s , 0 ≤ s ≤ t) and H t := σ(ν s , 0 ≤ s ≤ t), we have the inclusions Gt ⊂ H t ⊂ F t . The second one is trivial and the first one holds because ν t is the sum of the measure ρt (which satisfies ρt ({0}) = 0) and a measure whose support is {0}. Moreover, Gt = F t because ρt , 1 = X t -I t and because -I t is the local time at time t at level 0 of the process ρt . Hence, the three filtrations are equal and it is sufficient to prove that M •∧T 0 is a H-martingale.

According to Corollary 2.5 and Remark 2.7 in [START_REF] Abraham | Feller property and infinitesimal generator of the exploration process[END_REF], if f is a bounded non-negative function on [0, ∞) of class C 1 such that its first derivative f is bounded and such that f has a finite limit at +∞, then, under P 0 , the process M f t , t ≥ 0 defined by

M f t := e -ν t∧θ ,f + t∧θ 0 e -νs,f f (H s ) -ψ(f (H s )) ds (28)
is a martingale with respect to the filtration H. In fact, in [START_REF] Abraham | Feller property and infinitesimal generator of the exploration process[END_REF], the authors only proved this result in the case β = 0 and (0,1] Λ(dr)r = ∞. However, their proof can easily be adapted in the general setting using results of [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]Chap. 3].

Recall that e α : z → e αz and let (h K ) K≥1 be a sequence of functions of class

C 1 such that for all K ≥ 1, 0 ≤ h K ≤ K ∧ e α , 0 ≤ h K ≤ e α , h K is bounded and (h K ) K (resp. (h K ) K ) is
an increasing sequence which converges pointwise to e α (resp. e α ).

Then, for a > 0, K ≥ 1, applying (28

) with f = ah K , M a,K t := 1 -e -a ν t∧θ ,h K a + t∧θ 0 e -a νs,h K ψ(ah K (H s )) a -h K (H s ) ds
defines a martingale. We first have

M a,K t -→ a→0 M K t := ν t∧θ , h K + t∧θ 0 ψ (0)h K (H s ) -h K (H s ) ds a.s.
By Beppo Levi's theorem, since (h K ) K and (h K ) K are non-negative and non-decreasing, we have with probability 1

M ∞ t := lim K→∞ M K t = ν t∧θ , e α + t∧θ 0 αe αHs -αe αHs ds = ν t∧θ , e α .
We want to show that (M ∞ t ) t , that is the limit as a → 0 and then as K → ∞ of the martingales M a,K , is still a martingale. To do this, we use the dominated convergence theorem and we have to find an upper bound for M a,K t independent from a and K. We have

|M a,K t | ≤ ν t , e α + t 0 αe αHs ds + t 0 ψ(ah K (H s )) ah K (H s ) h K (H s )ds.
Moreover, there exists where T -x is the hitting time of (-∞, -x] by X under P 0 . Then, since (x + I t + ρt , e α ) 1 {t≤T -x } under P 0 has the same law as

C 1 , C 2 > 0 such that ψ(λ) λ ≤ C 1 + C 2 λ.
(I t + ρt , e α ) 1 {t≤T 0 } = M t∧T 0 under P x , M •∧T 0 is a martingale w.r.t. the filtration H = F.
Step 2:

We are now able to prove the main part of Theorem 4.3. For t > 0 and Θ ∈ F t , we have

P x Θ, t < T 0 sup s≤T 0 H s ≥ a = P x (Θ, τ a ≤ t < T 0 ) + P x (Θ, t < τ a < T 0 ) P x (sup s≤T 0 H s ≥ a) (30)
and we will investigate the asymptotic behaviors of the three probabilities in the last equation. First, assumption (2) enables us to define the bijective mapping φ : (0, ∞) → (0, ∞) such that

φ(t) := ∞ t dλ ψ(λ) , t > 0.
We denote by v its bijective inverse. Trivially, v(a) → 0 as a goes to ∞ and thanks to Lemma 2.1 in [START_REF] Lambert | Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct[END_REF], for u ≥ 0, we have

lim a→∞ v(a -u) v(a) = e αu . (31)
According to [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]Cor. 1.4.2], the mapping v can be linked to the excursion measure N :

v(a) = N (sup H > a), a > 0.
Then, by excursion theory and the exponential formula,

P x sup s≤T 0 H s ≥ a = 1 -exp(-xN (sup H > a)) = 1 -e -xv(a) . (32)
Hence,

P x sup s≤T 0 H s ≥ a ∼ a→∞ xv(a). (33)
Then, we consider the probability P x (Θ, τ a ≤ t < T 0 ) ≤ P x (τ a ≤ t) and we want to prove that it goes to 0 faster than v(a) as a → ∞. We define

g a := sup{t < τ a , X t = I t } (34)
the left-end point of the first excursion of X -I which reaches height a and set ε a := τ a -g a .

Let (e(t), t ≥ 0) be the excursion point process of X -I at level 0, that is, for t ≥ 0

e(t) = (X -I) s+I -1 (t-) , 0 ≤ s ≤ I -1 (t) -I -1 (t-) if I -1 (t-) < I -1 (t) ∂ otherwise
where ∂ is a cemetery point.

For a generic excursion ε with duration σ = σ(ε), we denote by

h * := sup [0,σ] H(ε)
its maximum height. For t ≥ 0, we denote by ∆ t the length of the excursion e(t) of X -I and we set h * t := h * (e(t)). Then, since (∆ t , h * t ) t≥0 is the image by a measurable application of the Poisson point process (e(t), t ≥ 0), it is a Poisson point process on (0, ∞) × (0, ∞). Distinguishing its atoms (δ, h) between h > a and h < a, we obtain that

Y a t := s≤t ∆ s 1 {h * s <a} and Ỹ a t := s≤t ∆ s 1 {h * s >a}
are independent. Moreover, (Y a t , t ≥ 0) is a subordinator with Lévy measure N (∆ ∈ dr; sup H < a). If ϕ a denotes its Laplace exponent, as a tends to ∞, ϕ a → ϕ where ϕ = ψ -1 is the Laplace exponent of the subordinator s≤t ∆ s (see Theorem VII.1 in [START_REF] Bertoin | Lévy processes[END_REF]).

Furthermore, γ a := inf{s ≥ 0; Ỹ a s = 0} is independent from Y a and follows an exponential distribution with parameter N (sup H > a) = v(a). Hence,

E e -λga = E e -λY a (γa) = v(a) ∞ 0 dte -v(a)t E e -λY a t = v(a) ∞ 0 dte -v(a)t e -ϕ a (λ)t = v(a) v(a) + ϕ a (λ) . (35)
Hence, since g a and ε a are independent, P x (τ a ≤ t) = P x (g a + ε a ≤ t) = E x P x (g a ≤ tε a |ε a )1 {εa≤t} and by Markov inequality, for all λ > 0, P x (τ a ≤ t) ≤ E x e λ(t-εa) v(a) v(a) + ϕ a (λ)

1 {εa≤t} = v(a)e λt v(a) + ϕ a (λ)
E x e -λεa 1 {εa≤t} .

Then,

P x (τ a ≤ t) v(a) ≤ e λt P x (ε a ≤ t) v(a) + ϕ a (λ) -→ a→∞ 0 (36)
since ϕ a → ϕ and the r.v. ε a is the time to reach height a for an excursion of X -I conditioned on sup H > a, so that P x (ε a ≤ t) vanishes in the limit a → ∞.

We finally study the asymptotic behavior of the last term of (30). We have

P x (Θ, t < τ a < T 0 ) = E x 1 Θ 1 {t≤T 0 ∧τa} P x (τ a < T 0 |F t ) and 1 -P x (τ a < T 0 |F t ) = E x   exp   - 0≤u≤Xt χ(u, e(I -1 t (u))   F t  
where for u > 0 and for a generic excursion ε of X -I of duration σ = σ(ε),

χ(u, ε) = ∞ if sup [0,σ] H(ε) > a -H I -1 t (u)
where we remind the reader that I -1 t (•) is the right-continuous inverse of (I t s , s ≤ t). Then, by the exponential formula for Poisson point processes,

1 -P x (τ a < T 0 |F t ) = exp - Xt 0 du N (dε) 1 -e -χ(u,ε) = exp - Xt 0 du N h * > a -H I -1 t (u) = exp - t 0 (d s I t s + I t δ 0 (s)) N (h * > a -H s ) = exp (-ρ t , v(a -•) ) .
Hence, since ρ t , v(a -•) ≤ X t v(a -H t ) goes to 0 as a goes to ∞ and according to (31),

P x (τ a < T 0 |F t ) v(a) = 1 -exp (-ρ t , v(a -•) ) v(a) -→ a→∞ ρ t , e α .
Then, using Fatou's Lemma, lim inf

a→∞ P x (Θ, t < τ a < T 0 ) v(a) ≥ E x 1 Θ 1 {t≤T 0 } ρ t , e α = E x [1 Θ M t∧T 0 ] .
Replacing Θ by Θ c in the latter display, we have lim inf

a→∞ P x (t < τ a < T 0 ) v(a) -lim sup a→∞ P x (Θ, t < τ a < T 0 ) v(a) ≥ E x [M t∧T 0 ] -E x [1 Θ M t∧T 0 ] . Since M •∧T 0 is a martingale, E x [M t∧T 0 ] = x. Moreover, P x (t < τ a < T 0 ) v(a) = P x (τ a < T 0 ) v(a) - P x (τ a < T 0 , τ a ≤ t) v(a)
and in the limit a → ∞, the first term of the r.h.s. tends to x according to (33) and the second vanishes thanks to (36). Finally, we have lim sup

a→∞ P x (Θ, t < τ a < T 0 ) v(a) ≤ E x [1 Θ M t∧T 0 ] ≤ lim inf a→∞ P x (Θ, t < τ a < T 0 ) v(a) (37)
and putting together (33), ( 36) and (37), the proof is completed.

4.2.

Properties of the conditioned process. As in the finite variation case, we define a new probability P ↑ x as a h-transform via the martingale M •∧T 0

P ↑ x (Θ) = 1 x E x [M t∧T 0 1 Θ ] , Θ ∈ F t . (38)
If α = 0, M t = X t and, as previously mentioned in the introduction, our definition of P ↑ x is the same as that of [START_REF] Chaumont | Sur certains processus de Lévy conditionnés à rester positifs[END_REF][START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF] in spectrally positive case. Then, to condition X to reach arbitrarily high heights is equivalent to condition X to hit 0 after arbitrarily long times. In this particular case, X is a Markov process under P ↑

x . The following proposition is the counterpart of Proposition 2.9 in the finite variation case. Proposition 4.7. Under P ↑

x , in probability X t -→ t→∞ +∞.

Proof. We want to prove that under P ↑ x , X t goes to infinity in probability as t → ∞. We make a similar proof as that of Proposition 2.9, that is, we want to prove that for λ > 0,

E ↑

x e -λXt → 0 as t tends to ∞. First, in the case α = 0,

E ↑ x e -λXt = 1 x E x e -λXt X t∧T 0 -→ t→∞ 0
using dominated convergence theorem. We now suppose that α > 0. Then, for all positive a,

E ↑ x e -λXt = 1 x E x e -λXt I t + t 0 d s I t s e αHs ; t ≤ T 0 ≤ P x (t ≤ T 0 ) + 1 x E x e -λXt t 0 d s I t s e αHs 1 {Ht≤a} + 1 {Ht>a} 1 {t≤T 0 } ≤ P x (t ≤ T 0 ) + e aα x E x e -λXt (X t -I t ); t ≤ T 0 + 1 x E x e -λ(Xt-It) t 0 d s I t s e αHs ; H t > a . (39)
The first and the terms of the r.h.s. vanish as t goes to ∞ thanks to the dominated convergence theorem and because T 0 is finite a.s. We now want to show that the third term of (39) vanishes as a goes to infinity uniformly in t. By time-reversing at time t, we have

B(t, a) := E x e -λ(Xt-It) t 0 d s I t s e αHs ; H t > a = E e -λSt t 0 d r S r e α(Lt-Lr) ; L t > a .
Then, observing that {L t > a} = {L -1 (a) < t} and applying the strong Markov property at the stopping time L = E e -λS L -1 (a) e αa 1 {Lt>a} f 1 (t -L -1 (a))

L -1 (a) 0 d r S r e -αLr +e -αa f 2 (t -L -1 (a)) (40)
where for u ≥ 0, We first show that these two functions are bounded on (0, ∞). As in [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF], we have

f 1 (u) = E e -λSu
f 1 (u) = E[e -λSu e αLu ] = ∞ 1 daE e -λSu 1 {Lu>ln a/α} + E e -λSu .
For b > 0, using [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF],

E e -λSu 1 {Lu>b} ≤ E e -λS L -1 (b) 1 {b<L(∞)} = exp -b ψ(λ) λ .
Hence, choosing b = ln a/α,

f 1 (s) ≤ ∞ 1 a -ψ(λ)/(λα) da + 1 < ∞
because ψ is convex and ψ(λ)/λ > ψ (0) = α as soon as λ > 0.

We do the same with f 2 . For b > 0, if M denotes an upper bound of the mapping

x → xe -λx , E e -λSs S s 1 {Ls>b} = E e -λS L -1 (b) -λ(Ss-S L -1 (b) ) S s -S L -1 (b) + S L -1 (b) 1 {Ls>b} ≤ M E e -λS L -1 (b) 1 {Ls>b} + E e -λS L -1 (b) S L -1 (b) 1 {Ls>b} ≤ exp -b ψ(λ) λ M + b ψ (λ)λ -ψ(λ) λ 2 using that (S L -1 (b) , b < L(∞)
) is a subordinator with Laplace exponent ψ(λ)/λ according to [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF]. Then, taking b = ln a/α,

f 2 (s) ≤ C(λ) ∞ 1 (1 + ln a) a -α -1 ψ(λ)/λ da + M < ∞.
We come back to (40). If C 1 (resp. C 2 ) denotes an upper bound of f 1 (resp. f 2 ) and observing that

L -1 (a) 0 d r S r e -αLr ≤ S L -1 (a) a.s., we have B(t, a) ≤ C 1 e αa E e -λS L -1 (a) S L -1 (a) 1 {Lt>a} + C 2 E e -λS L -1 (a) 1 {Lt>a} ≤ C 1 ψ (λ)λ -ψ(λ) λ 2
ae αa e -a ψ(λ) λ + C 2 e -a ψ(λ) λ .

Thus, B(t, a) goes to 0 uniformly in t as a tends to +∞ and the proof is completed.

The definition of P ↑ x in (38) does not make sense for x = 0. However, since X has infinite variation, it is possible to define the law P ↑ of the conditioned process starting at 0. Roughly speaking, it is the law of an excursion of X -I conditioned to reach an infinite height. We recall that N denotes the excursion measure of X -I away from 0. By doing similar computations as those in the proof of Theorem 4.3, one can prove that for t ≥ 0 and for Θ

∈ F t , lim a→∞ N (Θ| sup H > a) = N Ht 0 ρt (dz)e αz 1 Θ =: P ↑ (Θ) ( 41
)
where ρ is defined by [START_REF] Limic | A LIFO queue in heavy traffic[END_REF]. In comparison with the previous proof, P a) is replaced by N (sup H > a) = v(a) and the weight I t at 0 in the definition of the measure ρ t is no more present. We then justify that, under this probability, X starts at 0 by proving that P ↑ is the limit (in Skorokhod space) of the family of measures P ↑

x (sup [0,T 0 ] H > a) = 1 -e -xv(
x as x → 0. This kind of problem has already been treated by Bertoin in [5, Prop VII.14] and by Chaumont and Doney in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF]Thm 2] for different hypotheses about X as mentioned previously.

To obtain the convergence of P ↑ x , as in [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF], we first get a path decomposition of the process (X, P ↑ x ) at its minimum. Let η := sup{t ≥ 0, X t = I t } be the time at which the ultimate infimum of X is attained. Then, we defined the pre-minimum and post-minimum processes as ← -X := (X(t), 0 ≤ t < η) and -→ X := (X(t + η) -X η-, t ≥ 0).

We get the following path decomposition of the conditioned process.

Proposition 4.8. Under P ↑ x , we have that (i) X η-is uniformly distributed on [0, x] and P ↑

x (X η-= X η ) = 1, (ii) the two processes ← -X and -→ X are independent, (iii) conditional on X η = u, ← -X has the law of X under P x and killed at time

T u := inf{t ≥ 0; X t = u}, (iv) - → X is distributed as P ↑ .
Proof. We begin by proving (i). For 0 < x ≤ y, by applying (38) at the (F t ) t≥0 -stopping time T y ,

P ↑ x (X η-≤ y) = P ↑ x (T y < ∞) = 1 x E x M Ty∧T 0 1 {Ty<∞} . (42)
Under P x , since X has no negative jumps, we have a.s. T y < T 0 < ∞ and M Ty = I Ty = y by the definition of M and since H Ty = 0. Hence, P ↑ x (X η-≤ y) = x/y and under P ↑ x , X η-is uniformly distributed on [0, x].

We now prove that P ↑ x (X η-= X η ) = 1. We set ϑ := inf t < T 0 ; H t = sup s∈[0,T 0 ] H s the hitting time of the maximal height reached by X before T 0 and let g := sup {t < σ; X t = I t } be the left-end point of the last excursion of X -I that reaches this maximal height. Notice that under P x , ϑ is finite a.s. because H is continuous and T 0 < ∞ a.s. and that under P ↑ x , g = η a.s. according to Proposition 4.7.

For a > 0, we recall from (34) that g a is the left-end point of the first excursion that reaches height a and we set d a := inf{t > τ a , X t = I t } its right-end point. The main idea that we use in the following is that under P x ( • |τ a < T 0 ), with high probability, τ a > g = g a . Indeed, applying the Markov property at time τ a , P x (τ a ≤ g) ≤ E x 1 {τa<T 0 } P Xτ a (τ a < T 0 ) ≤ P x (τ a < T 0 ) 2 and P x (τ a ≤ g|τ a < T 0 ) ≤ P x (τ a ≤ T 0 ) -→ a→∞ 0.

(43)

Using that and Theorem 4.3, we have P ↑

x (X η-= X η ) = lim a→∞ P x ( X g-= X g | τ a ≤ T 0 ) = lim a→∞ P x ( X g-= X g , τ a > g| τ a ≤ T 0 ) .

Moreover, since g a = g on {T 0 > τ a > g}, P ↑ x (X η-= X η ) = lim a→∞ P x ( X ga-= X ga , τ a > g| τ a ≤ T 0 ) = lim a→∞ P x ( X ga-= X ga | τ a ≤ T 0 )

and since X has infinite variation paths, 0 is regular for (0, ∞) and according to [START_REF] Millar | Zero-one laws and the minimum of a Markov process[END_REF], X can not jump just after attaining a local minimum. Then, conditional on {g a < ∞}, X ga-= X ga a.s. and P ↑ x (X η-= X η ) = 1. We now prove points (ii), (iii) and (iv). For l > 0, let D (resp. D l ) be the set of càdlàg functions from [0, +∞) (resp. [0, l]) to R ∪ where we remind that δ is a cemetery point. We endow these spaces with the Skorokhod's topology.

It is sufficient to show that for all bounded and continuous function F on D, for all l > 0 and for all bounded and continuous function G on D l ,

E ↑ x F ( ← - X )G( - → X • k l ) = x 0 dy x E x f (X • k Ty ) • P ↑ [G(X • k l )]
where k l is the killing operator at time l. Let X a 1 := (X(t), 0 ≤ t < g a ) and X a 2 = (X(t + g a ) -I ga , 0 ≤ t ≤ (d a -g a ) ∧ l) be respectively the pre-g a process and the first excursion of X -I that reaches height a and killed at time l. According to Theorem 4.3,

E ↑ x F ( ← - X )G( - → X • k l ) = lim a→∞ E x F ( ← - X )G( - → X • k l ) τ a ≤ T 0 .
Using (43), since F and G are bounded and since ← -X = X a 1 and -→ X • k l = X a 2 on {τ a > g}, we have

E ↑ x F ( ← - X )G( - → X • k l ) = lim a→∞ E x F ( ← - X )G( - → X • k l )1 {τa>g} τ a ≤ T 0 = lim a→∞ E x F (X a 1 )G(X a 2 )1 {τa>g} τ a ≤ T 0 = lim a→∞ E x [ F (X a 1 )G(X a 2 )| τ a ≤ T 0 ] .
As already said in the computation of (35) in the proof of Theorem 4.3, under P x ( • |τ a < T 0 ), the two processes X a 1 and X a 2 are independent. Moreover γ a = I ga is an exponential r.v. with parameter v(a) = N (sup H > a) and independent from the process X a 1 . Then, we have

E ↑ x F ( ← - X )G( - → X • k l ) = lim a→∞ E x [ F (X a 1 )| τ a ≤ T 0 ] • N [G(ε • k l )| sup H(ε) ≥ a] . (44)
Furthermore, under P x ( • |τ a < T 0 ), X a 1 has the same distribution as X (under P x ), that do not reach height a and killed at the hitting time of the level I ga , that is, We now compute the law of I ga under P x ( • |τ a < T 0 ). For 0 ≤ y ≤ x, using that under P, -I ga is exponentially distributed with parameter v(a), P x (I ga > y|τ a < T 0 ) = P(-I ga < xy) P(-I ga < x) = 1 -e -(x-y)v(a)

1 -e -xv(a) .

Then, and under P ↑ x , the pre-minimum process has the law of X under P x killed when it reaches an uniform level in [0, x].

Finally, according to (41), the second term in the r.h.s. of (44) converges as a → ∞ to P ↑ (G(X • k l ) because the convergence in (41) is stronger than the convergence in law in D l .

Letting a → ∞ in (44), we obtain that under P ↑ x , ← -X and -→ X are independent, and that the law of -→ X is P ↑ and the proof is complete.

The path decomposition given in Proposition 4.8 enables us to prove the following convergence result.

Theorem 4.9. The family of measures (P ↑

x , x > 0) converges on the Skorokhod space to P ↑ .

Proof. To prove this convergence, we follow the proof of [START_REF] Chaumont | On Lévy processes conditioned to stay positive[END_REF]Thm 2]. According to it, thanks to the path decomposition obtained in Proposition 4.8, to prove the convergence of P ↑ x to P ↑ as x → 0 on the Skorokhod space, it is sufficient to prove that for C > 0, both P ↑

x (η > C) and P ↑

x (S η > C) vanish as x → 0 (we recall that S is the past supremum process associated with X).

First, using Proposition 4.8, under P ↑ x , the pre-minimum process has the law of X under P x , killed when it reaches an uniform level in [0, x]. Then, for x < C,

P ↑ x (S η > C) = P ↑ x T [C,∞) < η ≤ P x T [C,∞) < T 0 = P(T [C-x,∞) < T -x
). According to [START_REF] Bertoin | Lévy processes[END_REF]Thm VII.8], there exists a continuous increasing function W : [0, ∞] → [0, ∞] (called scale function) such that

P(T [C-x,∞) < T -x ) = 1 - W (C -x) W (C)
and this quantity goes to 0 as x → 0 since W is continuous. We now want to prove that P ↑ x (η > C) vanishes as x → 0. We first investigate the Laplace transform of η under P ↑

x . For λ > 0, using again (43), we have

E ↑
x e -λη = lim a→∞ E x e -λga τ a < T 0 .

Using the same notation and doing similar computations as those in (35), we have E x e -λga τ a < T 0 = E e -λY a (γa) 1 {γa<x} P x (τ a < T 0 ) = v(a) v(a) + ϕ a (λ)

1 -e -x(v(a)+ϕ a (λ))

1 -e -xv(a) .

Then, as a → ∞, we have

E ↑ x e -λη = 1 -e -xϕ(λ) xϕ(λ) -→ x→0 1.
Hence, by Markov inequality, we have

P ↑ x (η > C) ≤ E ↑ x 1 -e -λη
1 -e -ληC -→ x→0 0 and that ends the proof.

Theorem 2 . 3 .

 23 (i) Assume that b := Λ(R + ) is finite and m = 1 and (0,∞)

  the k-th individual of B has a Poissonian number (with parameter T k b) of non-marked daughters which are born independently and uniformly on [0, T k ].

  -1 (a), B(t, a) = E e -λS L -1 (a) e αa 1 {Lt>a} e -λ St-S L -1 (a) e α(Lt-a) • L -1 (a) 0 d r S r e -αLr + t L -1 (a) d r S r e -αLr

e αLu and f 2

 2 (u) = E e -λSu e αLu u 0 d r S r e -αLr ≤ E e -λSu S u e αLu

E 0 P

 0 x [ F (X a 1 )| τ a ≤ T 0 ] = x x (-I ga ∈ dy|τ a ≤ T 0 )E x f (X • k Ty )|T y < τ a .

E x [ F (X a 1 )

 1 | τ a ≤ T 0 ] = x 0 v(a)e -(x-y)v(a) dy 1 -e -xv(a) E x f (X • k Ty )|T y < τ a f (X • k Ty )

  Thus, according to Lemmas 4.5 and 4.6, the r.h.s. of (29) is integrable. Then, applying the dominated convergence theorem, we have proved that ( ν t∧θ , e α , t ≥ 0) is a martingale under P 0 with respect to H. Furthermore, ν t∧θ , e α = (I t + x)1 {It>-x} + ρt , e α 1 {t≤θ} = (x + I t + ρt , e α ) 1 {t≤T -x }

							Hence,
	|M a,K t	| ≤ (I t + x)1 {It>-x} + ρt , e α +	t	αe αHs ds +	t	C 1 e αHs + C 2 a e αHs -1	2 ds
			0		0
		t
		≤ x + ρt , e α +	AC 2 + (α + C 1 )e αHs + AC 2 e 2αHs ds	(29)
		0
	if a belongs to the compact set [0, A].

January 30, 2012

Acknowledgments

This work was supported by project MANEGE ANR-09-BLAN-0215 (French national research agency). I want to thank my supervisor, Amaury Lambert, for his very helpful advice.

My thanks also to an anonymous referee for his/her careful check of this manuscript and helpful remarks.


Françoise Bodénan

Pascale Michel

Lise Cary

Aurélien Leynet

Patrice Piantone

Environmental surveillance of incinerators: 2006-2009 data on dioxin/furan atmospheric deposition and associated thresholds

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Dioxin/furan (PCDD/F) emission into the atmosphere has clearly diminished over about the last 20 years with the closure of non-standard municipal solid waste (MSW) incineration plants and also the installation of more powerful smoke treatment systems in most industrial facilities. In France the reduction between 1990 and 2008 has been -94%, with total industrial emissions in 2008 estimated at 101 g TEQ-NATO [START_REF] Citepa | French national emission register[END_REF] . Since 28 November 2005 2 , all MSW incinerators have to respect the limitation of 0.1 pg TEQ-NATO 89 /Nm [START_REF] Fiedler | [END_REF] where PCDD/F emission is concerned.

French regulations also impose environmental monitoring in the vicinity of incinerators. This is translated by the introduction of surveillance plans since 2006 and, more particularly, of gauge monitoring to measure total wet and dry atmospheric PCDD/F deposition. Analysis of these data is limited due to of the lack of recent reference values. The currently available references widely used in France are those published by Fiedler [START_REF] Fiedler | [END_REF] in 1996 (based on a 1993 German compilation). Other similar references are also available (1991) 4 . All these reference values, however, should now be considered as obsolete in view of the changes in European emission regulations, including those for MSW incineration plants which are a major source of dioxins.

The collection of environmental monitoring data provides a large number of new data that need to be gathered and suitably processed. The present article summarises all the processed data, describes a treatment method and suggests value ranges and thresholds regarding environmental deposition.

Materials and methods

The approach used in constructing this new French frame of reference on dioxin deposition in the environment is based on four years (2006-2009) of sampling atmospheric deposition with Owen type (dust and precipitation) gauges at 1113 collection points in the vicinity of 49 MSW incineration plants. The standard NF X 43-014 was applied; the gauges measured total wet and dry deposition during period from 1 to 2 months. When selecting statistically processed TEQ data from the French and German laboratories, preference was given to TEQ calculated according to the NATO 1989 and then WHO 1998 methods that show little divergence on profiles dominated by the OCDD congener with very low TEF, and preferably taking detection limits into account (i.e. nd=DL). Of the various data-processing methods found in the literature 5,[START_REF] Cary | French Geological Survey public report[END_REF] , the most suitable is that of Lepeltier (1969) [START_REF] Lepeltier | [END_REF] applied to deconvoluting the distribution of non-Gaussian populations.

Results and discussion

The compiled data range between <0.01 and 115.5 pg TEQ/m 2 /d (Scatterdiagram, Figure 1) and their distribution into 40 classes falls into maximum 5-6 main groups (Figure 1). Comparison with the pg TEQ/m 2 /d intervals suggested by Fiedler (1996) Figure 2 -TEQ deposition data treatment: graphical representation of Henry's line method 5,[START_REF] Cary | French Geological Survey public report[END_REF][START_REF] Lepeltier | [END_REF] .

Applying the Lepeltier method (Figure 2), data can be deconvoluted in three populations limited by inflexion pointsof the curve the function reverses the Gaussian one applied to the logarithm of the distribution of the population-: two log normal population and one smaller population -around 2% of the values-with a very large value distribution. Up to 85% of the data population is lower than 5 pg/m 2 /d; these data can be considered as urban and industrial background values of atmospheric deposition. Then the 85-98% population are in the 5-16 pg/m 2 /d interval, which can be considered as values more impacted by human activities. Finally 2% of the population gives values >16 pg/m 2 /d, which can be considered as an anthropic signature in relation with a nearby emission source. The limiting values of 5 and 16 pg TEQ/m 2 /d concern TEQ data assigned to population quartiles; they are consistent with the points distribution (Figure 1).

These new data give thresholds than can be used in air quality surveillance. Being aware of these data when monitoring surveillance plans should make it possible to investigate anomalous situations where dioxin/furan deposition is more than 16 pg TEQ/m 2 /d. After confirming the measurement (for example at several points downwind during a same visit), a new gauge and/or emission control should be considered, as well as a detailed study of the congener distribution (profiles) to try and assess the potential source(s) and suggest actions where appropriate. Between 5-16 pg TEQ/m 2 /d data are also to be discussed with respect to the potential surrounding emissions. Below 5 no additional study is required.

Profiles

The plot of a few data profiles with values varying from 2 to 65 pg TEQ-NATO/m 2 /d (Figure 3) confirms the prevalence of the most chlorinated congener of dioxins, as it is known from elsewhere. The differences noted for 2-3 congeners cannot be completely explained at this stage; reduction of the OCDD part, which is often evoked for high TEQ values, is confirmed only for TEQ 65.5 and not for the already high values of 18 and 37 pg TEQ-NATO/m 2 /d. Finally one can insist on the interest of integrated monitoring method such as Owen gauges to measure atmospheric pollution over a long period and detect anomalous situations; not always detectable during spot analysis at the source.

Figure 1 -

 1 Figure 1 -TEQ dioxin/furan deposition data: scatter diagram (left) and distribution (right)

Figure 3 -

 3 Figure 3 -Example of profiles sorted by increased TEQ values (six analyes) Conclusion In a context of installations complying with current regulations, which means low emission values from industrial plants, this article presents a compilation of recent data (2006-2009) obtained from French MSW incinerator surveillance plans with respect to atmospheric dioxin/furan deposition. Two thresholds are provided that could be used in air quality surveillance as discussed above: 5 and 16 pg TEQ/m 2 /d.Finally one can insist on the interest of integrated monitoring method such as Owen gauges to measure atmospheric pollution over a long period and detect anomalous situations; not always detectable during spot analysis at the source.

Acknowledgements:

The authors thank the French Ministry of the Environment for financial support and Regional agencies (DREAL) for access to data.


Jean-Luc Rouas
email: rouas@limsi.fr

Mayumi Beppu
email: beppu@ks.cs.titech.ac.jp

Martine Adda-Decker

Comparison of spectral properties of read, prepared and casual speech in French

 on Japanese speaking styles, with the difference that we here focus on French. Using Nakamura's method, we use classical speech recognition features, MFCC, and try to represent the effects of the speaking styles on the spectral space. Two measurements are defined in order to represent the spectral space reduction and the spectral variance extension. Experiments are then carried on to investigate if indeed we find some differences between the three speaking styles using these measurements. We finally compare our results to those obtained by Nakamura on Japanese to see if the same phenomenon appears.

Introduction

This work follows the work of [START_REF] Nakamura | Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance[END_REF] on Japanese speaking styles, with the difference that we here focus on French. We first conduct the same experiments as Nakamura and then propose further analysis. Motivations for this work are expressed in section 2.. The section 3. describes the analysis method. In section 4., we recall the experiments and the results obtained by Nakamura on Japanese. The next section, section 5., details the French corpora we are using. The experiments achieved on this French data are detailed in section 6.. In the last section, we compare the results between Japanese and French data whenever they may be compared, and we propose further analyses.

Motivations

After having focused mainly on read speech, the automatic speech transcription systems also achieve nowadays very good performances on broadcast news data (usually around 10-15% of Word Error Rate [START_REF] Fousek | Transcribing Broadcast Data Using MLP Features[END_REF]). These systems benefit of years of research focused only on the automatic transcription of read speech and broadcast news data, which can be attested by looking at the numerous NIST evaluations carried on during the last decades. However, when coping with spontaneous speech, even the best actual systems achieve quite poor performances, with an accuracy that may be as low as 40%. These results emphasise the fact that spontaneous speech and read speech are indeed very different both acoustically and linguistically [START_REF] Furui | Recent advances in spontaneous speech recognition and understanding[END_REF]. One of the remaining challenges for automatic speech transcription is to give the systems the ability to deal with every kind of input data, including spontaneous speech. To fulfil this goal, it is crucial to analyse what are the most noticeable differences between speaking styles using features traditionally used by automatic speech transcription systems. On the acoustic point of view, it is known that the spectral distribution of continuously spoken vowels or syllables is much smaller than that of isolated spoken vowels or syllables. This phenomenon is sometimes called spectral reduction. Similar reduction have been observed in the parametric space for spontaneous speech compared to read speech, using formant frequency data measured from one speaker [START_REF] Van Son | An acoustic description of consonant reduction[END_REF]. A study on Japanese confirmed this observation using large corpora [START_REF] Nakamura | Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance[END_REF]. Considering French, we know that read speech and spontaneous speech are structurally different. Complex syllables tend to be simplified, and deletion of the word-final consonants and vowels of unstressed syllables frequently occurs in French spontaneous speech [START_REF] Adda-Decker | Investigating syllabic structures and their variation in spontaneous french[END_REF]. This phenomenon may be responsible for some of the automatic transcription errors on spontaneous speech. That is why we propose to study it in this paper. Another phenomenon that may characterise spontaneous speech is the spectral variance extension. This phenomenon has been studied in [START_REF] Nakamura | Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance[END_REF] on Japanese data. In the following section, we present the approaches used to compute means to assess these phenomena.

Acoustic analysis method

3.1. Features Utterances are digitalised using 16kHz sampling, and segmented by silences of 400 ms duration or longer. The speech waveform are converted to 12 dimensional MFCC vectors using a 25 ms length window shifted every 10 ms. The MFCC vectors are augmented with their first and second derivatives, together with the first and second differential log-energy, resulting in 38dimensional vectors.

Spectral reduction ratio

In order to characterise the spectral reduction phenomenon, we must use a reference corpus. This corpus, hereafter denoted R, is in our case a read speech corpus. The spectral space extent is estimated by calculating the difference between the mean of the MFCC features for a given phoneme p and the averaged value over all the phonemes of the same corpus. The ratio is then computed by dividing the spectral space extent for the corpus X with the one measured for the reference corpus R. The distances used in this paper are euclidean distances. In formal terms, the spectral reduction ratio can be expressed by the following formula:

red p (X) = µ p (X) -Av(µ p (X)) µ p (R) -Av(µ p (R)) (1)
with µ p being the mean vector of phoneme p uttered with the speaking style X, µ p (R) is the mean vector for phoneme p for read speech, and Av indicates the average value.

Spectral variance extension ratio

Using a similar method, we now define the variance extension ratio, still using the reference corpus R.

The spectral variance is estimated as being the sum of the variances obtained for each MFCC coefficient for all the realisations of a phoneme p.

The spectral variance extension ratio is thus defined as being the ratio between the spectral variance of the phoneme p on the corpus X and the spectral variance of the same phoneme on the corpus R.

The formalised method used to compute this ratio is:

ext p (X) = K k=1 σ 2 pk (X) K k=1 σ 2 pk (R) (2)
with K being the dimension of the MFCC vector, and σ 2 pk (X) the kth dimensional element of the variance vector of MFCC for phoneme p uttered with speaking style X.

Acoustic characteristics of Japanese

speaking styles [START_REF] Nakamura | Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance[END_REF] In [START_REF] Nakamura | Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance[END_REF], various speaking styles are analysed using large-scale Japanese speech corpora. This work focused mainly on the analysis of the differences between acoustic characteristics for spontaneous and read speech.

The data used in these experiments come from the Corpus of Spontaneous Japanese (CSJ) [START_REF] Maekawa | Corpus of spontaneous japanese: its design and evaluation[END_REF] and the Japanese Newspaper Article Sentence (JNAS) [START_REF] Itou | Jnas: Japanese speech corpus for large vocabulary continuous speech recognition research[END_REF]. The CSJ corpus is composed of 660 hours of speech and was recorded from 1999 to 2004. The JNAS corpus consists in 60 hours of speech. It was recorded from 1995 to 1997.

These corpus covers several conditions, including monologue, dialogue and read speech. For the purpose of the following experiments, only some situations, considered most representative of a speaking style, are utilised. These are, ranking from the most formal to the most informal: Read speech, Academic presentations, Extemporeaneous presentations (informal presentations), Dialogue.

The experiments carried on using this framework show that reduction of MFCC space is observed for almost all the phonemes in the three speaking styles in comparison with read speech. This phenomenon is most noticeable for the dialogue utterances.

Results obtained using the spectral variance extension ratio indeed show that an extension of the variance is observed for almost all the phonemes in the three speaking styles.

French corpora

Three speech corpora with read, prepared and casual spontaneous speaking styles were used. The reference read speech corpus is the French BREF [START_REF] Lamel | Bref, a large vocabulary spoken corpus for french[END_REF]

Experiments

The first experiment aims at measuring the durations of the automatically aligned phonemes for the three corpora. This is done in order to visualise the distribution of phoneme durations for each speaking style and to select the phonemes according to their duration.

The result of this experiment is drawn in figure 1.

As seen on this figure, the distributions are quite similar for BREF and ESTER, with a slight shift to the left for ESTER. This shift may be due to the effect of speaking rate, which should be faster for prepared journalistic speech than for read speech. 0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,00 0,05 0,10 0 The shape of the distribution for the NCCFr corpus is however very different. We can observe a peak for phonemes of 30 ms duration, which is the shorter duration allowed by the automatic alignment. This may be an effect of conversational spontaneous speech, for which some phonemes might be subject to deletion (see [START_REF] Adda-Decker | Contributions du traitement automatique de la parole à l'étude des voyelles orales du français[END_REF] for more details on this phenomenon).

Although the shapes clearly differs, the median value for each distribution is almost the same (between 60 and 70 ms). Thus, we will first consider only "normal" length segments, which have a duration between 40 and 120 ms.

Phoneme reduction ratio & variance extension

The phoneme reduction ratio and variance extension ratio are computed on our data using the same formula respectively described in sections 3.2., equation 1 and 3.3., equation 2. The reference corpus R, used for all the following experiments, is the BREF corpus. As we can observe on the first two columns of figure 2, the spectral space reduction is observed mainly for the ESTER corpus. For this corpus, almost all the phonemes have a reduced spectral space, except /i/ and /y/. However, for NCCFr, the spectral reduction phenomenon is not as obvious. In fact, the spectral space seems almost identical to that of read speech, with a few exceptions. The results on the spectral variance extension, shown on the last two columns of figure 2, are in accordance with our expectations. The spectral variance is indubitably increased for ESTER, and even more for ERNESTUS.

Influence of segment duration

As the results from the previous section are a bit surprising, we have therefore investigated if the observed reduction phenomenon may be emphasised or not for short or long segments, short segments being segments

ESTER red

NCCFr red Figure 3 shows the spectral reduction ratio for the ES-TER and NCCFr corpora compared to BREF for both duration conditions. We observe that for short segments, especially for the NCCFr corpus, the spectral space is quite reduced. Even given the short duration of those segments, this phenomenon should not be neglected as short segments are indeed frequent in our conversational speech corpus. We can note however that the spectral space reduction is not very important for longer segments.

BREF ESTER p 1 1,14 w 1 1,02 h 1 1,07 j 1 1,1 r 1 1,07 1 1,15 N 1 1,2 n 1 1,1 m 1 1,04 Z 1 0,98 z 1 0,99 v 1 1,07 S 1 1,17 s 1 1,17 f 1 1,5 g 1 0,95 d 1 1,06 b 1 1,13 k 1 1,
The variance extension ratio has also been computed for both duration conditions (see figure 4). The figures show that the variance of phonemes are extended for both conditions. The extension appears to be stronger for short segments of conversational speech as opposed to long segments. This phenomenon is observed for both speaking styles.

ESTER vowels

NCCFr vowels

ESTER consonants NCCFr consonants

Influence of lexical content

In this section, we will focus of acoustic differences that might be observed while considering the role of the words. We have distinguished function and content words, based on the word frequency in each corpus. First, words are ranged in decreasing frequencies. Function words are then isolated using the following paradigm: they are the 100 most frequent words, as long as their cumulated count is less than 50% of all the words in the corpus. Some content words occuring in this list are manually removed: as we deal here with french news corpora (BREF and ESTER), some of the most frequent words are for example "France", "Monsieur", "président". Moreover, function words do not cover all the french phonemes equally. We have thus removed from the analysis the phonemes that occur less than 1000 times. Those omitted phonemes are : /c o @/ and /b g S v N r h w/. The results of the analysis are shown in figure 5 for the spectral space reduction, and in figure 6 for the variance extension. Figure 5 shows that the spectral space is always more conserved for the content words than for the function words. The variance extension ratio is larger for the vowels contained in function words than for content words.

Discussion

Nakamura, in [START_REF] Nakamura | Differences between acoustic characteristics of spontaneous and read speech and their effects on speech recognition performance[END_REF] seems to provide results showing quite clearly the effects of spectral space reduction and spectral variance extension in Japanese. On his data, the spectral space is more reduced when the speaking style gets more spontaneous. Similarly, the spectral variance is increased for spontaneous data.

In our experiments on French data, we have been able to observe the same phenomenon for the spectral variance. The tests carried on "normal" duration phonemes do not however show the same behaviour for the spectral reduction. The spectral space is more reduced for We have nevertheless been able to observe a strong spectral space reduction for short segments, quite frequent in our spontaneous speech data.

Considering the role of words, we have seen that function words tend to be more subject to spectral reduction than content words. This is also true for the variance extension phenomenon, which is more present on function words.

However interesting these results may seem, they have to be taken with some caution: the features are indeed extracted at the middle of each phonetic segment, which might lead to false results in some cases, especially when considering consonantal phonemes.

Nevertheless, this approach provides a simple way of representing variations occurring between speaking styles, and we plan to carry on further experiments using different features, amongst them prosodic features.

Figure 1 :

 1 Figure 1: Duration distribution for each corpus. Xaxis : duration in seconds, Y-axis : estimation of the probability

Figure 2 :

 2 Figure 2: spectral reduction ratio (red) and variance extension (ext) for the ESTER and NCCFr corpora compared with BREF

Figure 4 :

 4 Figure 4: spectral variance extension ratio for the ESTER and NCCFr corpora compared with BREF for different phoneme durations.first line : short phonemes, second line : long phonemes (>120ms)

Figure 6 :

 6 Figure 6: spectral variance extension ratio for the ESTER and NCCFr corpora compared with BREF for different word classes. first line : function words, second line : content words