Romain Abraham
email: romain.abraham@univ-orleans.fr

Aymen Bouaziz
email: bouazizaymen16@yahoo.com

Jean-Franc ¸ois ¸ois Delmas
email: delmas@cermics.enpc.fr

LOCAL LIMITS OF GALTON-WATSON TREES CONDITIONED ON THE NUMBER OF PROTECTED NODES

Keywords: Mathematics Subject Classification. 60J80, 60B10, 05C05 Galton-Watson tree, random trees, local limits, protected nodes

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated sizebiased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.

Introduction

In [START_REF] Kesten | Subdiffusive behavior of random walk on a random cluster[END_REF], Kesten proved that a critical or sub-critical Galton-Watson (GW) tree conditioned on reaching at least height h converges in distribution (for the local topology on trees) as h goes to infinity toward the so-called sized-biased tree (that we call here Kesten's tree and whose distribution is described in Section 3.2). Since then, other conditionings have been considered, see [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF][START_REF] Abraham | An introduction to Galton-Watson trees and their local limits[END_REF][START_REF] He | Local convergence of critical random trees and continuum condensation tree[END_REF] and the references therein for recent developments on the subject.

A protected node is a node that is not a leaf and none of its offsprings is a leaf. Precise asymptotics for the number of protected nodes in a conditioned GW tree have already been obtained in [START_REF] Devroye | Protected nodes and fringe subtrees in some random trees[END_REF][START_REF] Janson | Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton-Watson trees[END_REF] for instance. Let A(t) be the number of protected nodes in the tree t. Remark that this functional A is clearly monotone in the sense of [START_REF] He | Local convergence of critical random trees and continuum condensation tree[END_REF] (using for instance (13)); therefore, using Theorem 2.1 of [START_REF] He | Local convergence of critical random trees and continuum condensation tree[END_REF], we immediately get that a critical GW tree τ conditioned on {A(τ ) > n} converges in distribution toward Kesten's tree as n goes to infinity. Conditioning on {A(τ ) = n} needs extra work and is the main objective of this paper. Using the general result of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF], if we have the following limit [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] lim n→+∞ P(A(τ ) = n + 1)

P(A(τ ) = n) = 1,
then the critical GW tree τ conditioned on {A(τ ) = n} converges in distribution also toward Kesten's tree, see Theorem 5.1.

In fact, the limit (1) can be seen as a special case of a more general problem: conditionally given the tree, we mark the nodes of the tree independently of the rest of the tree with a probability that depends only on the number of offsprings of the nodes. Then we prove that a critical GW tree conditioned on the total number of marked nodes being large converges in distribution toward Kesten's tree, see Theorem 3.3.

The paper is then organized as follows: we first recall briefly the framework of discrete trees, then we consider in Section 3 the problem of a marked GW tree and the proofs of the results are given in Section 4. In particular, we prove the limit (1) when A is the number of marked nodes in Lemma 4.2 and we deduce the convergence of a critical GW tree conditioned on the number of marked nodes toward Kesten's tree in Theorem 3.3. We finally explain in Section 5 how the problem of protected nodes can be viewed as a problem on marked nodes and deduce the convergence in distribution of a critical GW tree conditioned on the number of protected nodes toward Kesten's tree in Theorem 5.1.

Technical background on GW trees

2.1. The set of discrete trees.

We denote by N = {0, 1, 2, . . .} the set of non-negative integers and by N * = {1, 2, . . .} the set of positive integers.

If E is a subset of N * , we call the span of E the greatest common divisor of E. If X is an integer-valued random variable, we call the span of X the span of {n > 0; P(X = n) > 0}.

We recall Neveu's formalism [START_REF] Neveu | Arbres et processus de Galton-Watson[END_REF] for ordered rooted trees. Let U = n≥0 (N * ) n be the set of finite sequences of positive integers with the convention (N * ) 0 = {∅}. For u ∈ U, its length or generation |u| ∈ N is defined by u ∈ (N * ) |u| . If u and v are two sequences of U, we denote by uv the concatenation of the two sequences, with the convention that uv = u if v = ∅ and uv = v if u = ∅. The set of ancestors of u is the set

An(u) = {v ∈ U; ∃w ∈ U such that u = vw}.
Notice that u belongs to An(u). For two distinct elements u and v of U, we denote by u < v the lexicographic order on U i.e. u < v if u ∈ An(v) and u = v or if u = wiu ′ and v = wjv ′ for some i, j ∈ N * with i < j. We write u ≤ v if u = v or u < v.

A tree t is a subset of U that satisfies:

• ∅ ∈ t.

• If u ∈ t, then An(u) ⊂ t.

• For every u ∈ t, there exists k u (t) ∈ N such that, for every i ∈ N * , ui ∈ t iff 1 ≤ i ≤ k u (t). The vertex ∅ is called the root of t. The integer k u (t) represents the number of offsprings of the vertex u ∈ t. The set of children of a vertex u ∈ t is given by:

(2) C u (t) = {ui; 1 ≤ i ≤ k u (t)}. By convention, we set k u (t) = -1 if u ∈ t. A vertex u ∈ t is called a leaf if k u (t) = 0. We denote by L 0 (t) the set of leaves of t. A vertex u ∈ t is called a protected node if C u (t) = ∅ and C u (t) L 0 (t) = ∅,
that is u is not a leaf and none of its children is a leaf. For u ∈ t, we define F u (t), the fringe subtree of t above u, as

F u (t) = {v ∈ t; u ∈ An(v)} = {uv; v ∈ S u (t)} with S u (t) = {v ∈ U; uv ∈ t}.
Notice that S u (t) is a tree. We denote by T the set of trees and by T 0 = {t ∈ T; Card(t) < +∞} the subset of finite trees.

We say that a sequence of trees (t n , n ∈ N) converges locally to a tree t if and only if lim n→∞ k u (t n ) = k u (t) for all u ∈ U. Let (T n , n ∈ N) and T be T-valued random variables. We denote by dist(T ) the distribution of the random variable T and write

lim n-→+∞ dist(T n ) = dist(T )
for the convergence in distribution of the sequence (T n , n ∈ N) to T with respect to the local topology.

If t, t ′ ∈ T and x ∈ L 0 (t) we denote by

(3) t ⊛ x t ′ = {u ∈ t} ∪ {xv; v ∈ t ′ }
the tree obtained by grafting the tree t ′ on the leaf x of the tree t. For every t ∈ T and every x ∈ L 0 (t), we shall consider the set of trees obtained by grafting a tree on the leaf x of t:

T(t, x) = {t ⊛ x t ′ ; t ′ ∈ T}.
2.2. Galton Watson trees. Let p = (p(n), n ∈ N) be a probability distribution on N. We assume that (4) p(0) > 0, p(0) + p(1) < 1, and µ :=

+∞ n=0 np(n) < +∞.
A T-valued random variable τ is a GW tree with offspring distribution p if the distribution of k ∅ (τ ) is p and it enjoys the branching property: for n ∈ N * , conditionally on {k ∅ (τ ) = n}, the subtrees (F 1 (τ ), . . . , F n (τ )) are independent and distributed as the original tree τ .

The GW tree and the offspring distribution are called critical (resp. sub-critical, super-critical) if µ = 1 (resp. µ < 1, µ > 1).

Conditioning on the number of marked vertices

3.1. Definition of the marking procedure. We begin with a fixed tree t. We add marks on the vertices of t in an independent way such that the probability of adding a mark on a node u depends only on the number of children of u. More precisely, we consider a mark function q : N -→ [0, 1] and a family of independent Bernoulli random variables (Z u (t), u ∈ t) such that for all u ∈ t:

P(Z u (t) = 1) = 1 -P(Z u (t) = 0) = q(k u (t)).
The vertex u is said to have a mark if Z u (t) = 1. We denote by M(t) = {u ∈ t; Z u (t) = 1} the set of marked vertices and by M(t) its cardinal. We call (t, M(t)) a marked tree.

A marked GW tree with offspring distribution p and mark function q is a couple (τ, M(τ )), with τ a GW tree with offspring distribution p and conditionally on {τ = t} the set of marked vertices M(τ ) is distributed as M(t). Remark 3.1. Notice that for A ⊆ N, if we set q(k) = 1 {k∈A} , then the set M(t) is just the set of vertices with out-degree (i.e. number of offsprings) in A considered in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF][START_REF] Rizzolo | Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set[END_REF]. Hence, the above construction can be seen as an extension of this case.

3.2. Kesten's tree. Let p be an offspring distribution satisfying Assumption (4) with µ ≤ 1 (i.e. the associated GW process is critical or sub-critical). We denote by p * = (p * (n) = np(n)/µ, n ∈ N) the corresponding size-biased distribution.

We define an infinite random tree τ * (the size-biased tree that we call Kesten's tree in this paper) whose distribution is described as follows:

There exists a unique infinite sequence

(v k , k ∈ N * ) of positive integers such that, for every h ∈ N, v 1 • • • v h ∈ τ * , with the convention that v 1 • • • v h = ∅ if h = 0. The joint distribution of (v k , k ∈ N *
) and τ * is determined recursively as follows. For each h ∈ N, conditionally given (v 1 , . . . , v h ) and {u ∈ τ * ; |u| ≤ h} the tree τ * up to level h, we have:

• The number of children (k u (τ * ), u ∈ τ * , |u| = h) are independent and distributed according to

p if u = v 1 • • • v h and according to p * if u = v 1 . . . v h . • Given {u ∈ τ * ; |u| ≤ h + 1} and (v 1 , . . . , v h ), the integer v h+1 is uniformly distributed on the set of integers {1, . . . , k v 1 •••v h (τ * )}. Remark 3.2.
Notice that by construction, a.s. τ * has a unique infinite spine. And following Kesten [START_REF] Kesten | Subdiffusive behavior of random walk on a random cluster[END_REF], the random tree τ * can be viewed as the tree τ conditioned on non extinction.

For t ∈ T 0 and x ∈ L 0 (t), we have:

P(τ * ∈ T(t, x)) = P(τ = t) µ |x| p(0) • 3.3. Main theorem.
Theorem 3.3. Let p be a critical offspring distribution that satisfies Assumption (4). Let (τ, M(τ )) be a marked GW tree with offspring distribution p and mark function q such that p(k)q(k) > 0 for some k ∈ N. For every n ∈ N * , let τ n be a tree whose distribution is the conditional distribution of τ given {M(τ ) = n}. Let τ * be a Kesten's tree associated with p. Then we have:

lim n→+∞ dist(τ n ) = dist(τ * ),
where the limit has to be understood along a subsequence for which P(M(τ ) = n) > 0.

Remark 3.4. If for every k ∈ N, 0 < q(k) < 1, then P(M(τ )) = n) > 0 for every n ∈ N, hence the above conditioning is always valid.

Proof of Theorem 3.3

Set γ = P(M(τ ) > 0). Since there exists k ∈ N with p(k)q(k) > 0, we have γ > 0. A sufficient condition (but not necessary) to have P(M(τ ) = n) > 0 for every n large enough is to assume that γ < 1 (see Lemma 4.3 and Section 4.4). Taking q = 1 A , see Remark 3.1 for 0 ∈ A ⊂ N implies γ = 1 and some periodicity may occur.

The following result is the analogue in the random case of Theorem 3.1 in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] and its proof is in fact a straighforward adaptation of the proof in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] by using:

(i) M(t) ≤ Card(t). (ii) For every t ∈ T 0 , x ∈ L 0 (t) and t ′ ∈ T, we have that M(t

⊛ x t ′ ) is distributed as M (t ′ ) + M(t) -1 {Zx(t)=1}
, where M (t ′ ) is distributed as M(t ′ ) and is independent of M(t).

n-→+∞ P(M(τ ) ∈ [n + 1, n + 1 + n 0 ) P(M(τ ) ∈ [n, n + n 0 )) = 1, (5) lim
we have:

lim n-→+∞ dist(τ |M(τ ) ∈ [n, n + n 0 )) = dist(τ * ).
Proof. According to Lemma 2.1 in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF], a sequence (T n , n ∈ N) of finite random trees converges in distribution (with respect to the local topology) to some Kesten's tree τ * if and only if, for every finite tree t ∈ T 0 and every leaf x ∈ L 0 (t), ( 6) lim n→+∞ P (T n ∈ T(t, x) = P τ * ∈ T(t, x) and lim n→+∞ P(T n = t) = 0.

Let t ∈ T 0 and x ∈ L 0 (t). We set D(t, x) = M(t) -1 {Zx(t)=1} . Notice that D(t, x) ≤ Card(t) -1. Elementary computations give for every t ′ ∈ T 0 that:

P(τ = t ⊛ t ′ ) = 1 p(0) P(τ = t)P(τ = t ′ ) and P(τ * ∈ T(t, x)) = 1 p(0) P(τ = t).
As τ is a.s. finite, we have:

P(τ ∈ T(t, x), M(τ ) ∈ [n, n + n 0 )) = t ′ ∈T 0 P(τ = t ⊛ x t ′ , M(τ ) ∈ [n, n + n 0 )) = t ′ ∈T 0 P(τ = t ⊛ x t ′ )P(M(t ⊛ x t ′ ) ∈ [n, n + n 0 )) = t ′ ∈T 0 P(τ = t)P(τ = t ′ ) p(0) P( M (t ′ ) + D(t, x) ∈ [n, n + n 0 )) = P(τ * ∈ T(t, x)) P( M(τ ) + D(t, x) ∈ [n, n + n 0 )).
Notice that:

P( M (τ ) + D(t, x) ∈ [n, n + n 0 )) = Card(t)-1 k=0 P( M (τ ) + D(t, x) ∈ [n, n + n 0 ) | D(t, x) = k) P(D(t, x) = k) = Card(t)-1 k=0 P(M(τ ) ∈ [n -k, n + n 0 -k)) P(D(t, x) = k).
Then we obtain using Assumption (5) that:

lim n-→+∞ P( M(τ ) + D(t, x) ∈ [n, n + n 0 )) P(M(τ ) ∈ [n, n + n 0 )) = 1, that is lim n-→+∞ P(τ ∈ T(t, x) | M(τ ) ∈ [n, n + n 0 )) = P(τ * ∈ T(t, x)).
This proves the first limit of (6).

The second limit is immediate since, for every n ≥ Card(t),

P(τ = t | M(τ ) ∈ [n, n + n 0 )) = 0.
The main ingredient for the proof of Theorem 3.3 is then the following lemma.

Lemma 4.2. Let d be the span of the random variable M(τ ) -1. We have

(7) lim n→+∞ P(M(τ ) ∈ [n + 1, n + 1 + d)) P(M(τ ) ∈ [n, n + d)) = 1.
The end of this section is devoted to the proof of Lemma 4.2, see Section 4.4, which follows the ideas of the proof of Theorem 5.1 of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF].

4.1.

Transformation of a subset of a tree onto a tree. We recall Rizzolo's map [START_REF] Rizzolo | Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set[END_REF] which from t ∈ T 0 and a non-empty subset A of t builds a tree t A such that Card(A) = Card(t A ). We will give a recursive construction of this map φ: (t, A) → t A = φ(t, A). We will check in the next section that this map is such that if τ is a GW tree then τ A will also be a GW tree for a well chosen subset A of τ . Figure 1 below shows an example of a tree t, a set A and the associated tree t A which helps to understand the construction.

For a vertex u ∈ t, recall C u (t) is the set of children of u in t. We define for u ∈ t:

R u (t) = w∈An(u)
{v ∈ C w (t); u < v} the vertices of t which are larger than u for the lexicographic order and are children of u or of one of its ancestors. For a vertex u ∈ t, we shall consider A u the set of elements of A in the fringe subtree above u:

(8) A u = A ∩ F u (t) = A ∩ {uv; v ∈ S u (t)}.
Let t ∈ T 0 and A ⊂ t such that A = ∅. We shall define t A = φ(t, A) recursively. Let u 0 be the smallest (for the lexicographic order) element of A. Consider the fringe subtrees of t that are rooted at the vertices in R u 0 (t) and contain at least one vertex in

A, that is (F u (t); u ∈ R A u 0 (t)), with R A u 0 (t) = {u ∈ R u 0 (t); A u = ∅} = {u ∈ R u 0 (t); ∃v ∈ A such that u ∈ An(v)} .
Define the number of children of the root of tree t A as the number of those fringe subtrees:

k ∅ (t A ) = Card(R A u 0 (t)). If k ∅ (t A ) = 0 set t A = {∅}. Otherwise let u 1 < . . . < u k ∅ (t A )
be the ordered elements of R A u 0 (t) with respect to the lexicographic order on U. And we define t A = φ(t, A) recursively by: (9)

F i (t A ) = φ (F u i (t), A u i ) for 1 ≤ i ≤ k ∅ (t A ).
Since Card(A u i ) < Card(A), we deduce t A = φ(t, A) is well defined and it is a tree by construction. Furthermore, we clearly have that A and t A have the same cardinal:

(10) Card(t A ) = Card(A). Let ((X i , Z i ), i ∈ N * ) be i.i.d. random variables such that X i is distributed according to p and Z i is conditionally on X i Bernoulli with parameter q(X i ). We define:

• G = inf{k ∈ N * ; k i=1 (X i -1) = -1}. • N = inf{k ∈ N * ; Z k = 1}.
• X a random variable distributed as 1+ N i=1 (X i -1) conditionally on {N ≤ G}. • Y a random variable which is conditionally on X binomial with parameter ( X, γ). We say that a probability distribution on N is aperiodic if the span of its support restricted to N * is 1. The following result is immediate as the distribution p of X 1 satisfies (4). Recall that for a tree t ∈ T 0 , we have:

(11) u∈t (k u (t) -1) = -1
and u∈t,u<v (k u (t)-1) > -1 for any v ∈ t. We deduce that G is distributed according to Card(τ ) and thus N is distributed like the index of the first marked vertex along the depth-first walk of τ . Then, we have:

(12) γ = P(N ≤ G).
Using Wald's equality and E[X i ] = 1, we get E N i=1 (X i -1) = 0 and thus using the definition of X as well as (12):

γE[ X] = γ + E N i=1 (X i -1)1 {N ≤G} = γ -E N i=1 (X i -1)1 {N >G} .
We have:

E N i=1 (X i -1)1 {N >G} = E G i=1 (X i -1)1 {N >G} + P(N > G)E N i=1 (X i -1) = -P(N > G) = γ -1,
where we used the strong Markov property of ((X i , Z i ), i ∈ N * ) at the stopping time G for the first equation, the definition of T and Wald's equality for the second, and (12) for the third. We deduce that E[Y ] = γE[ X] = 1, which ends the proof. 4.4. Proof of [START_REF] Neveu | Arbres et processus de Galton-Watson[END_REF]. According to Lemma 4.4 and (10), we have that M(τ 0 ) is distributed as the total size of a critical GW whose offspring distribution satisfies (4). The proof of Proposition 4.3 of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF] (see Equation (4.15) in [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF]) entails that if τ ′ is a critical GW tree, then, if d denotes the span of the random variable Card(τ ′ ) -1, we have

lim n→∞ P(Card(τ ′ ) ∈ [n + 1, n + 1 + d)) P(Card(τ ′ ) ∈ [n, n + d)) = 1.

Protected nodes

Recall that a node of a tree t is protected if it is not a leaf and none of its offsprings is a leaf. We denote by A(t) the number of protected nodes of the tree t. Theorem 5.1. Let τ be a critical GW tree with offspring distribution p satisfying (4) and let τ * be the associated Kesten's tree. Let τ n be a random tree distributed as τ conditionally given {A(τ ) = n}. Then:

lim n-→+∞ dist(τ n ) = dist(τ * ).
Proof. Notice that P(A(τ ) = n) > 0 for all n ∈ N. Notice that the functional A satisfies the additive property of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF], namely for every t ∈ T, every x ∈ L 0 (t) and every t ′ ∈ T that is not reduced to the root, we have (13) A(t ⊛ x t ′ ) = A(t) + A(t ′ ) + D(t, x)

where D(t, x) = 1 if x is the only child of its first ancestor which is a leaf (therefore this ancestor becomes a protected node in t ⊛ x t ′ ) and D(t, x) = 0 otherwise. According to Theorem 3.1 of [START_REF] Abraham | Local limits of conditioned Galton-Watson trees: the infinite spine case[END_REF], to end the proof it is enough to check that (14) lim n→+∞ P(A(τ ) = n + 1)

P(A(τ ) = n) = 1.
Notice that the protected nodes of τ are exactly the nodes of τ 0 N * on which we did not add leaves i.e. for which W (u) = 0. If we set M(τ 0 N * ) = {u ∈ τ 0 N * , W (u) = 0}, we have M(τ 0 N * ) = A(τ ). Using (15), we get that the corresponding mark function q is given by:

q(k) = p(k)(1 -p(0)) k-1 p N * (k) 1 {k≥1} .
As τ is distributed as τ 0 , we have: As P(A(τ ) = n) = P(A(τ ) = n|k ∅ (τ ) > 0)P(k ∅ (τ ) > 0) and P(A(τ ) = n|k ∅ (τ ) > 0) = P(A(τ 0 ) = n) for n ≥ 2, we obtain (14) and hence end the proof.

Proposition 4 . 1 .

 41 Let n 0 ∈ N ∪ {∞}. Assume that P(M(τ ) ∈ [n, n + n 0 )) > 0 for n large enough. Then, if

Figure 1 . 4 . 2 .

 142 Figure1. Left: A tree t and the set A. Center: The fringe subtrees rooted at the vertices in R u 0 (t). Left: the tree t A . The labels have no signification, they only show which node of t corresponds to a node of t A

Lemma 4 . 3 .

 43 The distribution of Y satisfies (4) and if γ < 1 then it is aperiodic.

  lim n→+∞ P(A(τ 0 ) = n + 1) P(A(τ 0 ) = n) = lim n→+∞ P(A(τ ) = n + 1) P(A(τ ) = n) = lim n→+∞ P(M(τ 0 N * ) = n + 1) P(M(τ 0 N * ) = n) • As τ 0 N * is a critical GW tree, we deduce from Lemma 4.2 that lim n→+∞ P(M(τ 0 N * ) = n + 1) P(M(τ 0 N * ) = n) = 1.

Acknowledgements. We would like to thank the referee for his useful comments which helped to improve the paper.

We denote by (τ 0 , M(τ 0 )) a random marked tree distributed as (τ, M(τ )) conditioned on {M(τ ) = ∅}. By construction, Card(τ 0 ) is distributed as G conditioned on {N ≤ G}. Lemma 4.4. Under the hypothesis of this section, we have that τ 0 M(τ 0 ) = φ(τ 0 , M(τ 0 )) is a critical GW tree with the law of Y as offspring distribution. Lemma 4.4. In order to simplify notation, we write τ for τ 0 M(τ 0 ) = φ(τ 0 , M(τ 0 )) and for u ∈ τ 0 , we set R u for R u (τ 0 ). Lemma 4.5. The random tree τ is a GW tree with offspring distribution the law of Y .

Proof of

Proof. Let u 0 be the smallest (for the lexicographic order) element of M(τ 0 ). The branching property of GW trees implies that, conditionally given u 0 and R u 0 , the fringe subtrees of τ 0 rooted at the vertices in R u 0 , (S u (τ 0 ), u ∈ R u 0 ) are independent and distributed as τ . Recall notation (8) so that the set of marked vertices of the fringe subtree rooted at u is M u (τ 0 ) = M(τ 0 ) F u (τ 0 ). Define Mu (τ 0 ) = {v; uv ∈ M u (τ 0 )} the corresponding marked vertices of S u (t). Then, the construction of the marks M(τ ) implies that the corresponding marked trees ((S u (τ 0 ), Mu (τ 0 )), u ∈ R u 0 ) are independent and distributed as (τ, M(τ )). Notice that for u ∈ R u 0 , the fringe subtree F u (τ 0 ) contains at least one mark iff u belongs to

Then by considering only the fringe subtrees containing at least one mark, we get that, conditionally on R

) are independent and distributed as (τ 0 , M(τ 0 )). We deduce from the recursive construction of the map φ, see (9), that τ is a GW tree. Notice that the offspring distribution of τ is given by the distribution of the cardinal of R M(τ 0 ) u 0

. We now compute the corresponding offspring distribution. We first give an elementary formula for the cardinal of R u (t). Let t ∈ T 0 and u ∈ t. Consider the tree t ′ = R u (t) {v ∈ t; v ≤ u}. Using (11) for t ′ , we get:

This gives Card(R u (t)) = 1 + v∈t; v≤u (k v (t ′ ) -1). We deduce from the definition of X that Card(R u 0 ) is distributed as X. We deduce from the first part of the proof that conditionally on Card(R u 0 ), the distribution of Card(R

) is binomial with parameter (Card(R u 0 (τ 0 )), γ). This gives that the offspring distribution of τ is given by the law of Y . Lemma 4.6. The GW tree τ is critical.

Proof. Since the offspring distribution is the law of Y we need to check that E[Y ] = 1 that is γE[ X] = 1 since Y is conditionally on X binomial with parameter ( X, γ).

Recall N has finite expectation as P(Z 1 = 1) > 0, is not independent of (X i ) i∈N * and is a stopping time with respect to the filtration generated by ((X i , Z i ), i ∈ N * ).

For a tree t = {∅}, let t N * = φ(t, t \ L 0 (t)) be the tree obtained from t by removing the leaves. Let τ 0 be a random tree distributed as τ conditioned to {k ∅ (τ ) > 0}. Using Theorem 6 and Corollary 2 of [START_REF] Rizzolo | Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set[END_REF] with A = N * (or Lemma 4.4 with q(k) = 1 {k>0} ), we have that τ 0 N * is a critical GW tree with offspring distribution:

Conditionally given {τ 0 N * = t}, we consider independent random variables (W (u), u ∈ t) taking values in N * whose distributions are given for all u ∈ t by P(W (u) = 0) = 0 for k u (t) = 0 and otherwise for k u (t) + n > 0 (remark that p N * (k u (t)) > 0), by

In particular for k u (t) > 0, we have:

Then, we define a new tree τ by grafting, on every vertex u of τ 0 N * , W (u) leaves in a uniform manner, see Figure 2.

We also consider a family (i 1 , . . . , i k ) of integervalued random variables such that (i 1 , i 2 -i 1 , . . . , i k -i k-1 , W (u)+k+1-i k ) is a uniform positive partition of W (u) + k + 1. Then, for every j ≤ k such that j ∈ {i 1 , . . . , i k }, we set k j (τ ) = 0 i.e. these are leaves of τ . For every 1 ≤ j ≤ k, we set ψ(j) = i j and we apply to them the same construction as for the root and so on.

Lemma 5.2. The new tree τ is distributed as the original tree τ 0 . Proof. Let t ∈ T 0 . As P(τ = {∅}) = 0, we assume that k ∅ (t) > 0. Let t N * be the tree obtained from t by removing the leaves. Using (11), we have:


Rabah Dahmani

Yasmina Layouni

Vincent Semet

Michel Cabrera
email: michel.cabrera@univ-lyon1.fr

Micro EDM Milling with Low Energy Discharges and Thin Microtool

Keywords: Micro EDM, Micro EDM milling, Microtool, Tool wear, Stainless steel

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

Introduction

Micro electrical discharge milling is a process of growing importance, which allows machining of complex parts made of electrically conductive material by controlling the three dimensional path of a thin cylindrical microtool [START_REF] Pham | Micro-EDM-recent developments and research issues[END_REF]. Contrary to micromilling based on chips formation, micro EDM milling is a non-contact process so that ultrathin microtools can be in theory used. However, the process is still challenging as regards many aspects (discharge energy, vibrations,…) [START_REF] Egashira | Electrical discharge machining of submicron holes using ultrasmall-diameter electrodes[END_REF][START_REF] Egashira | Electrical discharge machining with ultralow discharge energy[END_REF]. Therefore, the aim of this paper is to report on the machining with ultrathin and high aspect ratio microtools and on the use of small energy discharges.

Experimental set up

This work is based on the 2 nd generation micro EDM milling machine, which has been entirely developed in our laboratory. Among other features, the microtool manufacturing is made by electrochemistry. The machine has also a generator allowing generating low energy discharges.

The machine has basically two functions:

-To etch cylindrical ultrathin microtools by electrochemical dissolution of tungsten rods in the microtool manufacturing station (5Mol/L NaOH).

-To machine parts by micro EDM milling with these tools in the micro EDM station.

The machine (Fig. 1) is equipped with translation stages with stepper motors (0.3125 µm microstep), which allow moving the two stations and the microtool respectively in the horizontal XY and vertical Z directions. The gap between the tool and the part is adjusted with a piezoelectric actuator (90 µm range; 1.8 nm resolution) and the tool wear is compensated with a stick slip actuator. A side view camera, with a Navitar x12 zoom, allows viewing the tool (camera 1). A top view camera with a second Navitar x12 zoom allows viewing the part (camera 2). All components are installed on a vibration isolation workstation. The 1 st generation of micro EDM milling machine and the microtool manufacturing station have been already described [START_REF] Layouni | Onboard Electrochemical Fabrication of Microelectrodes for Micro EDM Milling[END_REF][START_REF] Girardin | Micro EDM with the in situ electrochemical fabrication and regeneration of the tungsten microelectrode tool[END_REF][START_REF] Girardin | Développement d'un procédé d'usinage par micro-électroérosion[END_REF]. Therefore, we shall report here only briefly on the new features of the 2 nd generation machine, which are mainly related to the electronics.

This one is based on a parallel architecture (Fig. 2) with 3 microcontrollers (µC) in order to (1) monitor the process, (2) detect events which may happen simultaneously and (3) trigger actions with a fast response time, such as: -Movements of the microtool and of the part.

-Discharge generation (see RC generator below).

-Discharge characterization with the help of a series of voltage comparators [START_REF] Girardin | Développement d'un procédé d'usinage par micro-électroérosion[END_REF].

-Detection of the contact between the tool and the part (see below).

-Retraction of the microtool in case of contact.

-Regulation of the gap with the piezoelectric actuator.

-Tool wear compensation with the stick slip actuator.

In this paper, we shall demonstrate machining with a 9 µm diameter microtool. No spindle is used to rotate the tool. The first challenge is to avoid the destruction of the tool during a possible collision. The second challenge is to reduce the discharge energy in order to improve machining resolution. In its simplest form, the generator of electrical discharges is of RC type with a resistor RC and a working capacitance CC. In addition, a ballast resistor Rb has been introduced in series with the microtool in order to reduce the energy of the discharges (Fig. 3). The part is at positive polarity.

A

Machining of stainless steel

Electrochemical etching of the microtool

The machine allows electrochemical etching of tungsten rods with 250 µm initial diameter down to 4-10 µm diameter with very high aspect ratio (see ref [START_REF] Layouni | Onboard Electrochemical Fabrication of Microelectrodes for Micro EDM Milling[END_REF][START_REF] Girardin | Micro EDM with the in situ electrochemical fabrication and regeneration of the tungsten microelectrode tool[END_REF][START_REF] Girardin | Développement d'un procédé d'usinage par micro-électroérosion[END_REF] for details, including repeatability of the tool realization). Fig. 4 shows a view of a tungsten microtool with 9 µm average diameter (0.66 µm STD) along the 500 µm working length as imaged with camera 1 and characterized with a Mitutoyo LSM-500S laser profilometer (working range: 5 µm -2 mm; precision: 10 nm). The aspect ratio of the microtool is about 50.

The microtool is used for machining stainless steel substrates in deionized water. To prepare the substrates, a sheet with mirror like aspect (ref 559-206 from Radiospares) is cut with plasma into 20x20x0.6 mm pieces. All substrates are polished.

Machine set up

The initial gap between the microtool and the part is set up without deionized water (Fig. 5): the tool is slowly translated downward until electrical contact is reached and moved upward of 5 µm. Then a goniometer allows adjusting the gap during the lateral movement of the part so that it remains in the 5+/-1 µm range for 500 µm XY translations (as checked with camera 1). Then deionized water is added and machining can be started. The channel shown in Fig. 6a is machined with the 500x100 µm XY rectangular tool path of Fig. 7 at 156 µm/s speed. The tool always moves down of one microstep h (0.3125µm) when coming back at the same corner. Then the machining cycle is repeated again. A total of 160 layers is machined so that the theoretical height Hc of the channel is 50 µm.

A voltage comparator allows comparing the voltage across CC with a threshold value (typically 0.5 V). When the voltage is below this threshold, the tool is retracted and flushing is performed long enough in order to recover the gap isolation. Then the tool is moved down to its original position and machining is started again.

A series of channels have been machined at different values of Rb (5 to 500 Ω) with 50 V VEE, 100 Ω Rc and 1 nF Cc. The initial gap was checked for each channel. After each machining, the tool wear was measured with camera 1. The parts were also molded with polydimethylsiloxane PDMS (Fig. 6b).

Characterization of discharge energy

Characterization of machined microchannels

Fig. 10 shows top views of the machined channels recorded with camera 1. The images tend to show that the higher the Rb, the sharper the channel edges. However, views with a scanning electron microscope (SEM) do not confirm this statement (Fig. 11).

A regards the molded part, Fig. 12 is a side view with camera 1 and Fig. 13 is a top view with the SEM. They show similar surface roughness for all values of Rb. Fig. 14 shows the depth H and width W of the molded channels at different Rb as measured with the SEM and camera 1. As molding with PDMS is very precise [START_REF] Xia | Soft Lithography[END_REF], the geometrical measurements characterize mainly micro-EDM. So, thanks to the use of Rb, machining has been demonstrated with an energy as low as 29 nJ/discharge at 1 nF capacitance and 50 V operating voltage. In the literature, the use of such small energy discharges has been rarely reported and is performed in general with the stray capacitance of the machine at low operating voltage [START_REF] Egashira | Electrical discharge machining of submicron holes using ultrasmall-diameter electrodes[END_REF][START_REF] Egashira | Electrical discharge machining with ultralow discharge energy[END_REF]. This is a disadvantage since the performance of the machine is less predictable: the stray capacitance may vary from one machine to another and the operating voltage is close to the polarization voltage of the components and the electrical noise.

Removal rate and tool wear

The Ø 9 µm microtool is not damaged during the machining. This demonstrates that the electronics of the machine (Fig. 2) protects the microtool in an efficient manner.

Fig. 15 shows the final gap after the machining of every channel. The worn length of the tool is deduced by comparison with the initial gap (Fig. 5). This allows estimating (Table 1): -The tool removal rate (TRR), which is the ratio of the worn volume of the tool with machining time.

-The material removal rate (MRR) which is the ratio of the volume removed from the part with machining time.

-The relative tool wear RTW which is the ratio of TRR with MRR.

RTW decreases when Rb increases. For Rb greater than 100 Ω, RTW below 0.01% is obtained.

Machining resolution

Other attempts of machining have been performed successfully with a Ø 3 µm microtool without damaging the tool. However, the channel width is not significantly smaller than with the Ø 9 µm one. This is attributed to vibrations which induce +/-5 µm oscillations of the apex of the tool tips (as imaged with a slow motion camera at 2000 images/s). Vibrations are present for all tips.

Conclusions

With the 2 nd generation micro EDM milling machine developed in our laboratory, it has been possible to fabricate ultrathin (Ø 9 µm) high aspect ratio (~ 50) tungsten microtool and machine stainless steel channels without damaging the microtool. Different possibilities to improve machining resolution have been explored.

The first one was to reduce the energy per discharge. This has been performed with a ballast resistor. Parts have been successfully machined with 29 nJ/discharges using a 500 Ω ballast resistor at 50 V and 1 nF working capacitance. Operating at such a medium range voltage without relying on the machine stray capacitance is an advantage. With a ballast resistor of 100 Ω, the relative tool wear was below 0.01%. This is the most significant result of this work since tool wear in µEDM milling is known to affect part accuracy so that complex strategies are proposed to overcome this problem [START_REF] Pham | Micro-EDM-recent developments and research issues[END_REF]. Although no significant result has been shown on the surface characteristics of the channels, future work will evaluate if the reduction of the tool wear allows improving the machining accuracy. However, depending on the value of the ballast resistor Rb, the channel width is about 31-36 µm with aspect ratio of 1.4-1.8.

The second possibility to improve machining resolution was to use microtools with 3 µm diameter. Successful attempts have been performed without damaging the tool. However, no improvement was found in terms of machining resolution. Therefore, the mechanical stability of the machine is currently being improved in order to reduce the tip oscillations.

Fig. 1 .

 1 Fig. 1. 2 nd generation Micro EDM milling machine: (1) Z translation stage; (2) Y translation stage; (3) X translation stage; (4) Piezoelectric actuator; (5) Stick slip actuator; (6) Microtool manufacturing station; (7) Micro EDM station; (8) Camera 2; (9) Camera 1.

Fig. 2 .

 2 Fig. 2. Parallel architecture of the electronics of the machine.

Fig. 3 .

 3 Fig. 3. RCCC discharge generator with Rb ballast resistor operating at VEE voltage.

Fig. 4 .

 4 Fig. 4. Microtool with 9 µm diameter and aspect ratio of 50.

  5µm

Fig. 5 .

 5 Fig. 5. Initial gap between the microtool and the part (direct image and image reflected by the substrate).

Fig. 6 .Fig. 7 .

 67 Fig. 6. (a) Dimensions of the machined part. (b) Molded part.

Fig. 8

 8 Fig. 8 shows typical waveforms of voltage across the capacitance CC and current discharge at different Rb. The discharge energy is calculated by integrating the recorded voltage and current and subtracting the energy dissipated in Rb. Typical discharge energies vary from 4.1 µJ for Rb = 5 Ω to 29 nJ for Rb = 500 Ω (Fig. 9).

Fig. 8 .

 8 Fig. 8. Voltage and current during a discharge at Rb = 5, 20, 100, 200 and 500 Ω (VEE = 50 V, CC = 1 nF, RC = 100 Ω, initial gap = 5 µm).

Fig. 9 .

 9 Fig. 9. Variation of the discharge energy with the value of the ballast resistor.

Fig. 10 .

 10 Fig. 10. Top views of the channels machined in stainless steel (camera 2).

Fig. 11 .

 11 Fig. 11. Top view of the channels machined in stainless steel (SEM).

Fig. 12 .Fig. 13

 1213 Fig. 12. Side view of the channels molded with PDMS (camera 1) with mirrored image.

Fig. 14 .

 14 Fig. 14. Depth and width of the molded channels at different values of ballast resistor.

Fig. 15 .

 15 Fig. 15. Gap after the machining of the channels showing tool wear (direct image and image reflected by the substrate).

Table 1 .

 1 Material removal rate, tool removal rate, relative tool wear and energy / discharge at different values of ballast resistor.

	Rb (Ω) MRR (µm 3 /min)	TRR (µm 3 /min)	RTW (%) E (µJ)
	5	6.2 x 10	4	30	0.05	4.1
	20	9.1 x 10	4	24	0.03	2.1
	100	9.4 x 10	4	< 1	< 0.01	0.45
	200	8.7 x 10	4	< 1	< 0.01	0.068
	500	9.0 x 10	4	< 1	< 0.01	0.029

Acknowledgements

This work was financed by the FUI ConProMi.


Claire Bidart

Narratives and Traces « Trans-» Quantitative and Qualitative Methods for Bridging the Gap between Objective and Subjective Data by

Keywords: Méthodes mixtes, Analyses quantitatives, Analyses qualitatives, Innovation méthodologique, Processus temporels, Parcours de vie Mixed Methods, Quantitative Analysis, Qualitative Analysis, Methodological Innovation, Temporal Processes, Life Courses

Dans cette brève introduction nous rappelons les objectifs de ce double numéro : explorer de nouvelles voies dans la combinaison des méthodes qualitatives et quantitatives. La dimension temporelle et l'articulation des échelles sont présentes également dans l'ensemble de ces travaux. Ici la focale est portée plus précisément sur les articulations entre données objectives et subjectives.

Debates between qualitative and quantitative methods have too often created the illusion they covered a division between objective data and subjective data. But one cannot forget that "objective" data come from human beings who kindly answered to questionnaires, and that responses were given through the double filter of their representations (when selecting a response) and of those of the researchers (when designing the survey, and analyzing and interpreting the results). One also underestimates that traces left by individual paths, as crystallized and institutionalized as they can be, are the results of complex social processes of choice, diffusion, validation, labelling and temporal configurations [START_REF] Grossetti | Status and Formal Conception of Individual Social Trajectories in Sociological Analysis[END_REF].

This issue of the BMS is in continuation with and following the aims of the number 124 issue. Let us recall that they are to provide exploratory paths between qualitative and quantitative methods. Issue number 124 presented three papers from the same research project (BIPAJE, ANR-09-BLAN-0301-01) focused on bifurcations during youth life courses. Those papers crossed and combined methods applied to materials that are not a priori suited for them. Quantitative analyzes were made of life stories and qualitative analyzes were carried out on questionnaires. But in all those contributions, qualitative and quantitative methods were closely articulated (in relation to eachother) together, processing differently the same initial materials and displaying how they can fit together in innovative approaches.

The three papers presented in this issue number 125 continue with this approach and with another common point which is the importance of the temporal dimension. The phenomena analyzed here are processes which are treated as such. This implies that sequences of events, regimes or phases are used to capture reality and help in its interpretation; the authors tell us stories that involved different elements over time. This also means taking into account the fact that social time is not linear; it includes moments of different intensity with events, turning points, changes in activity regimes and various processes of emergence [START_REF] Abbott | Time Matters -On Theory and Method[END_REF][START_REF] White | Identity and Control: How Social Formations Emerge[END_REF]. Each of the three articles presented here tells such stories, each one in its own framework: biographical, organizational, or, even more broadly, at the level of a scientific field. But in each of these articles, these frameworks are also articulated between themselves: the broad historical events and the succession of generations are involved in the qualification of the different sequences of a life course; a change in activity regime, during a process of business creation, crosses individual paths with markets. Finally, the global circulation of an invention results from occasional meetings of certain persons, from the institutional location of pioneering work, and from irregular processes of diffusion in geographical space.

These three papers contribute largely to the discussion of the dividing line between subjective data and objective data. They criticize the illusion of objectivity conveyed by standardized records and show the importance of recognizing their subjective component and, moreover, the importance of highlighting it. In doing so, they build innovative links between narratives and traces, between accounts and calendars of events, between life stories and status changes, and between careers and publications.

In their article, Eva Lelièvre and Nicolas Robette attempt to collect the individual perceptions of well-being through the qualification by respondents of their own life periods as hard times or good times. The authors thus elaborate a subjective narrative which, following its specific trajectory, complements the factual recording of events and sheds new light on the life course.

The article by Jean-François Barthe, Nathalie Chauvac and Michel Grossetti combines economic and organizational data with biographical accounts, in the analysis of the various activity regimes which follow each other in the process of business creation.

The article by Marion Maisonobe shows that the emergence of a scientific specialty (DNA repair) cannot be assessed only through the observation of bibliometrics, but is built through processes involving individual actors, institutions and complex regulations.

These three articles provide original intermediate tools for performing a temporal placement and sequencing of periods, regimes and milestones that punctuate the story under

Claire Bidart, Aix-Marseille Université, CNRS, Laboratoire d'Economie et de Sociologie du Travail, UMR7317, 35 Avenue Jules Ferry, 13626, Aix en Provence, France. Email: claire.bidart@univ-amu.fr

Arnaud Dupray, Céreq, 10 Place de la Joliette BP 21321, 13567 Marseille Cedex 02. Email: dupray@cereq.fr

study. This sequencing sometimes allows the construction of typologies combining objective and subjective dimensions. In doing so, these analyses manage to get closer to "real life" with events and experiences that drove the process in different frameworks (biographical, organizational, scientific). They also contribute to overcoming the singularities of individual experience and to drawing typical patterns and sequences. These can then be tested in other fields and possibly generalized.

Finally, this double issue of the BMS allows experimentations with new avenues of research concerning the interactions of qualitative and quantitative methods. These original papers lead to more flexible boundaries between these methods. Their attention to the temporal dimension, to the sequencing and identification of key points and bifurcations, which is also another of their common characteristics, will hopefully renew studies life course, organizational pathways and careers.


Roxane Bertrand
email: roxane.bertrand@lpl-aix.fr

Béatrice Priego-Valverde
email: bea.priego-valverde@lpl-aix.fr

Does prosody play a specific role in conversational humor?

In this paper we use Conversation Analysis (CA) to investigate conversational humor in talkin-interaction. We attempt to better understand how the latter is produced and co-constructed by participants in accounting for the devices used by participants in their sequential environment. The framework of CA enables us to take into account the various means available to speakers to communicate, orient to the others, etc. From our data, reported speech, confirmation request/answer and repetitions appear as the main discursive devices to create humor (presented as the result of the appearance of the incongruity). We focused on prosodic cues that are strongly involved in these devices and then contribute to the humorous tonality of the talk. Finally, the co-construction of conversational humor is mainly described through the notion of orientation and prosodic orientation.

CONVERSATIONAL HUMOR

Conversational humor is the main focus of this article. Following [START_REF] Attardo | Linguistic theories of humor[END_REF] or [START_REF] Priego-Valverde | L'humour dans la conversation familière: description et analyse linguistiques[END_REF] among others, we know that it is a very complex phenomenon and that its boundaries are very difficult to delimitate principally because it is a multiform phenomenon. It can be short and correspond to a single utterance or it can be very long like the canned jokes or a whole sequence co-constructed by the participants. Its "tonality" is also very difficult to determine. Conversational humor may be simple jokes, teasing, wit, sarcasm, word play and so on. And most of the time, it may be several things at the same time (teasing jokes, friendly wit etc). Thus we can say that humor is often both friendly and aggressive. Because of this ambivalence, the interlocutors have to be very close to be able to perceive and/or appreciate humor. But humor, even between close friends, have its own boundaries or more precisely, humor has the boundaries fixed by the participants themselves which correspond to social, moral and of course personal limits. Thus, the subjects have to be on the same wavelength to make humor work. In our data, thanks to their close relationship, the two speakers are on the same wavelength and they laugh about the same things. Thus, each speaker chimes in on the other's purpose until creating a whole humorous sequence. One thing is sure: conversational humor is a very interactive phenomenon. Usually, it appears as a lightning thanks to a thing done or said by the interlocutor and on which the humorist speaker chimes in or continues until it is turned into derision. Conversational humor is thus directly linked to the context. And in a conversational setting, the context has to be understood in a broad sense: the situation hic et nunc, the participants' conversational history, their relationship and their shared knowledge. Using the framework of Conversational Analysis (henceforth CA) [START_REF] Couper-Kuhlen | Prosody in conversation[END_REF][START_REF] Couper-Kuhlen | Sound Patterns in Interaction[END_REF] we attempt to better understand what is precisely conversational humor and how participants co-construct it. Among the available resources, participants can use prosodic cues. Prosodic means are intended here as the whole phenomenon related to pitch level, pitch register, syllable duration, accentuation, tempo, but also the 'phonetic details' [START_REF] Local | Phonetic detail in talk-in-interaction: on the deployment and interplay of sequential context and phonetic resources[END_REF]) as vocal quality for instance which are relevant to account for turn-taking in talk-in-interaction [START_REF] Ogden | Non-modal voice quality and turn-taking in Finnish[END_REF]. Following Local for which prosodic cues must be analyzed in situ, we claim that conversational humor combines several criteria which make it difficult to describe it without taking into account the context and the combination of prosodic cues with the others resources (like the lexicon for instance). According to us, the last point explains why it is not surprisingly that there is no ironical tone of voice (Bryant & Fox-Tree 2005). Indeed, our conception implies that there is no biunivocal form-function relation between a linguistic device and a prosodic form [START_REF] Hirschberg | The pragmatics of Intonational meaning[END_REF][START_REF] Cristo | Outils prosodiques et analyse du discours[END_REF]). We shall see in this paper that if prosodic cues do exist in our humorous sequences and can play a crucial role, they are not humorous per se but one part among others of a congruence of elements that make humor appear [START_REF] Attardo | Multimodal markers of irony and sarcasm[END_REF].

CORPUS CID

General presentation

The CID, audio-video taped recorded data elaborated in the LPL1 [START_REF] Bertrand | [END_REF]) is a unique resource for the analysis of speech in interaction. It is constituted by 8 hours audio-video recorded dialogues. The interactions are transcribed and annotated to be useful for different fields: phonetics, prosody, syntax, discourse and non-verbal analysis (see the ToMA project (OTIM in French, Blache et al 2009). This goal required a particular protocol of experiment, which allowed us to obtain dialogues interesting both for the interactional level and the phonetic and prosodic approach.

Device and protocol

At this point, the CID is composed by 8 one hour interactions in French2 . The dialogues between two interlocutors are based on instructions given by the experimenters before the recording. 2 series of recordings were done with two different instructions: (i) according to the first one, the subjects had to speak about professional conflicts (ii) in the second one, they had to talk about unusual situations they had experienced. These two instructions were presented by the experimenters as a topic support allowing the subjects to begin quickly the interaction but they were only a support because the experimenters said right away to the subjects that they could digress from these topics if they wanted it.

The participants and their conditions of recording

10 females and 6 males are implicated in non mixed and face to face interactions. The 16 participants are French native and half of them come from Southern France. At the moment of the recording, all were living in Southern France for at least several months and knew very well the place where the recordings took place. This last aspect was a necessary condition to minimize the stress such a condition of recording (video taped) entails. All the participants were thus selected among the members of the laboratory but nobody knew the reason why they were recorded. Otherwise, all were chosen according to their high level of acquaintance. This condition guaranteed the fact they shared a "conversational history" [START_REF] Golopentja | Interaction et histoire conversationnelles[END_REF]. Such a sharing favored more spontaneous and fruitful exchanges and a certain ability to digress from the instructions of the beginning.

Each time, the two speakers were recorded in an anechoic room. They had a headphone in order to record them on separate tracks. They were also video recorded.

Conversational characteristics of the data

If all the subjects tried to respond to the instructions, they frequently digressed by speaking more freely, creating thus some conversational sequences. The CID is an intermediate corpus between authentic and natural data and orientated task data [START_REF] Anderson | The HCRC map task corpus[END_REF]. Otherwise, all the dialogs presented in the CID are very close to conversational interactions. Among the different characteristics, only the external goal due to the instructions distinguishes them from real conversations that have an inward goal, centred on contact [START_REF] Vion | La communication verbale. Analyse des Interactions[END_REF]. Except this criterion, as in real casual conversations, we notice: -a total symmetry between the participants in terms of status and positions. Theoretically, all of them have the same rights and the same duties, especially those to be alternately speaker and hearer [START_REF] Sacks | A simplest systematics for the organization of turn-taking for conversation[END_REF]:

-an apparent informality that carries on as much in the discourse (the two participants can speak about all and nothing, in a spontaneous way, without a precise goal) as in the interaction itself. Indeed, as [START_REF] Schegloff | Opening Up Closings[END_REF] showed, in a conversation, no explicit rules exist concerning the order of speech turns, their duration, all of that being determined progressively; -the absence of a tierce person to distribute the speech turn, like in a debate for instance. Finally, if the dialogs of the CID are globally conversational, they reflect also a heterogeneity of the various discursive sequences present in every interaction. So, here also, we can notice numerous narrations (due to the instructions), but also some descriptive, argumentative or explaining sequences.

TYPE OF HUMOR IN THE CID

In this preliminary study, we will analyze 1-hour dialogue with the first instruction. The choice of this dialogue is based on the relationship between A and L who exhibit a close acquaintance between them. This dialogue seemed to be the closer to real conversations previously analyzed in [START_REF] Priego-Valverde | L'humour dans la conversation familière: description et analyse linguistiques[END_REF]. Thus we can say that, even if this corpus was constituted in a laboratory and with an instruction, we observed the same kind of humor as in authentic conversations3 . The only difference is the fact that here, humor usually appears in narrative sequences. The humorous schema observed here is most of the time the same, classic schema: (i) presence of a "connector" [START_REF] Greimas | Sémantique structurale[END_REF]) that carries two possible and different scripts [START_REF] Raskin | Semantic Mechanisms of Humor, D[END_REF]. The first is logical and expected and the second is illogical, sometimes absurd, unrealistic and of course unexpected; (ii) the "disjunctor" [START_REF] Morin | L'histoire drôle[END_REF]) actualizes the second script and the gap between what is expected and what really happens creates the humorous effect. This gap is the origin of what we usually call the "incongruity" [START_REF] Bariaud | La genèse de l'humour chez l'enfant[END_REF]. One of the issues in this paper will be to show whether and how the prosodic level contributes to mark this gap between what is expected and what is really produced. This gap may be created by several devices and we shall see here two of them. The first one is reported speech that more or less explicitly actualizes two different voices speaking through the speaker, even if the reported voice may be fictitious. The second one implies confirmation answer/request through lexical repetitions allowing speaker to create, step by step, a distance between himself and his own discourse. In this way, humor appears here as a case of "double voicing" (in Bakhtin's sense) during which the speaker actualizes a second voice telling unrealistic and humorous purposes. We shall see that according to the discursive device, the connector and the disjunctor can be merged (in the same words or expressions for reported speech) or can be separate (after a space of some words or turns for repetitions). Both kinds of humor have in common that they are accepted by the interlocutor who laughs and who enters with the speaker in a humorous mode of communication until creating a "joint fantasy" [START_REF] Kotthoff | Oral genres of humor: On the dialectic of genre knowledge and creative authoring[END_REF] where each participant chimes in on the other's purpose. We described this through the notion of interactional orientation that refers in CA to the fact that 'throughout the course of a conversation (…) speakers display in their sequentially "next" turns an understanding of what the "prior" turn was about' (Hutchby & Wooffitt 1998: 15).

In the next section, we will show how speakers create humorous sequences in adapting the classic schema mentioned above (connector/disjonctor) according to whether the speaker uses reported speech or confirming turns. We will observe how vocal cues make humor hearable to the interlocutor in contributing to mark the gap between what is expected and what appears in these specific devices but also how vocal cues enable the interlocutor to orient to the main speaker's discourse.

ANALYSIS

The humorous reported speech

When speakers report the speech or thought of another person through reported speech the main question addressed is who is speaking and for what reason. According to the type of function assumed by reported speech, the main speaker may or may not be precise about the origin of the words. Reported speech is not only used to report words but also to enable speakers to convey their assessment of the utterance while reproducing it (Holt 2000), or to effect other functions such as increasing one's standing or saying something without really assuming it (given the duality of the other voice) (Bertrand 2003: 3). We will analyze here its use in interaction (as Holt, 2000) which is the best way to show functions other than a simple reproduction. Reported speech indeed is a very relevant discursive device that benefits from a diversity of means. Among others, vocal means can be used to make explicit the other voice (e.g., [START_REF] Couper-Kuhlen | Coherent On prosody in conversational reported speech[END_REF][START_REF] Klewitz | Quote-Unquote? The Role of Prosody in the Contextualization of Reported Speech Sequences[END_REF][START_REF] Bertrand | Voice diversity in conversation : a case study[END_REF]. Because of the design and the instruction of the CID, it was not surprising to find numerous sequences of reported speech used, for example, to describe situations in which participants were involved or to illustrate a relationship between the different protagonists in the unusual situations described in the CID storytelling framework. But cases of reported speech used to create humor apart from the cases described and directly linked to the stories were unusual. However, we found several instances of this phenomenon.

In our examples, only recipients make use of reported speech to create a humorous utterance.

The following example (1) is the only one where the main speaker A of the narrative sequence produces the humorous utterance in order to ease the nature of his own previous talk. This reported speech appears in a typical sequence called « aside » [START_REF] Selting | The construction of 'units' in conversational talk[END_REF] which corresponds to a parenthetical sequence in a narrative sequence.

(1)

 Tu es un vilain / You're a bad boy4 The two subjects are speaking about the way pornography is perceived in Saudi Arabia, a country were L lived few years ago. Digressing around this topic, they evoke the case of a man who, apparently, had big problems with the justice because he would have seduced a woman via internet. The reality of the situation they evoke seems relatively serious and dramatic. But the nature of his discourse (too dramatic) and the vocal cues of the utterance in 257 strongly contrast with the previous story told by A. In this previous discourse, we can distinguish a first phase in which A tells the story using emotional words with emphatic accents. When L reacts to this phase in an unexpected mode (in laughing and in expressing his doubt (178-179), AP expresses (255, in overlapping) a mitigation in a typical delivery (increasing loudness associated to laugh in the voice). This mitigation is followed by a reported speech that also functions as mitigation ( 257) and eases this previous story. The 'voice' is assignable to an abstract figure: the voice of the 'Saudi justice'. The reported speech used refers to an invented reported speech (as defined by [START_REF] Vincent | Le discours rapporté au quotidien, Québec, Nuit Blanche éditeur[END_REF]. For authors, reported speech is not limited to a strict definition (report word) but also concerns utterances that have never been expressed. Such a reported speech is usually neither anchored in the past nor the present time. Conversely here the reported utterance is strongly anchored in the reporting situation. Moreover, even if AP uses the design of reported speech (prefatory, introductory verb) both the incongruity of his utterance and of his source clearly ensure that words exhibited as repeating words could not have been delivered

The humorous aspect of the utterance gpd 257 is due to the use of the terms "bad boy"/"vilain"an infantile and derisory term that is animated through Saudi Arabian law expressed with a teacher voice -(overarticulation, lower pitch register, and an intonation risefall contour on 'vilain^'). This utterance contrasts with the last one. All these parameters (invented reported speech anchored in the past, infantile term in the Saudi law voice and vocal cues) contribute to reinforce the gap between what is expected and what really happens and constitute the incongruity of the utterance. Otherwise we notice that the connector and the disjonctor seem to be merged. Indeed, the terms "bad boy"/"vilain" seem to function both as the connector and the disjonctor. If the term carries out a punishing script (by the Arab law in the expected script), it also infers an infantile script and it refers more to the moment the children are on recess and when they told off by the teacher. This turn is followed by a long pause. After this pause the speaker comes back to his previous story with his own voice and AP concludes repeating "it was hot"/"c'était chaud". As at the beginning, the two expressions function as boundaries of the aside sequence. Moreover, this sequence provides an example of metapragmatic function of humor that is used by the speaker to save his own face (Priego-Valverde 2003).

In the following example (2), we observe the similar situation with only the change of the animator (i.e the interlocutor).

(2)  Le film porno / the pornografic movie LJ gpd_198 ils =z= ont PAssé un FILM crypté de Canal PLUS, un FILM de CUL AP gpd_265a @ AP gpd_265b @ LJ gpd_199 <<f> euh @ <<l> mais aTTENDS, mais c'était inciDENT, AP gpd_265c @ ça i(l) dire AP gpd_266

§ <<f> c'est du TErroRRISME^ euh euroPEEN^ § LJ gpd_200 non c'es-c'es-c'-r-rigole pas c'était incident diplomatique euh limite quoi

In this excerpt, LJ goes on telling about the years he spent in Saudi Arabia. He evokes a French TV channel that broadcasts French programs or programs in French language outside of France. During his narration, he explains that, one time, the channel made a mistake and broadcasted a pornographic movie. Despite AP's laughs, LJ tries to explain him how serious this mistake was and how this mistake was quite a diplomatic incident (199,200: "don't laugh"/"rigole pas"). Despite the serious tone and his laughter, LJ wants to go on explaining his story. AP laughs and produces (266) an utterance expressed in a typical vocal quality (extreme harschness) associated with high pitch register and increased loudness. Contrary to (1), there is no the explicit design of a reported speech but prosodic cues mark this utterance as belonging to another voice. Once again, the reported speech is not assignable to a real character and may correspond to the Arabian authorities. The humorous aspect of the utterance becomes an exaggeration as "European terrorism" is used to qualify a simple movie. Simultaneously, the overlapping (199-266) contributes to show the clash between two modes of communication: LJ's wants to stay in a bona fide communication [START_REF] Raskin | Semantic Mechanisms of Humor, D[END_REF] and AP wants to switch into a humorous mode. The speaker's comment ("don't laugh"/"ne rigole pas") provides evidence that LJ understands the humorous tonality but he does not want to switch into it even if he laughs.

The two following examples (3 and 4) illustrate instances of humor designed to comment on a protagonist in the narrative. The recipient's design in this utterance shows his orientation to the speaker's storytelling. The main speaker produced sufficient information enabling the listener to chime in on the caricature of one of the story's protagonist. We observe that both participants converge in producing simultaneously reported speech, the first in an illustrative function and the second in a humoristic function. Contrary to the previous example, the overlapping reported speech contributes here to enhance the strong convergence between the two participants to create a joint fantasy.

( In this excerpt, AP talks about his first meeting with his cousin's boyfriend. The meeting was not a good memory because the boyfriend was a very shy man and did not speak at all, which is not a good way to begin a relationship with someone. This narration evokes to LJ some memories about a man he met when he was doing some archaeological dig. Therefore, LJ explains that he met a man he considered strange because he did not drink coffee or alcohol nor eat meat. In this context, the two participants produce simultaneously an instance of reported speech: the reported speech of AP (390) completes the last utterance of LJ, and in fact is partially stated in overlap. The delivery of AP is very specific: the first part of its turn is accented and lengthened but in a low register and loudness. Then AP pauses and in this pause LJ hesitates as if waiting for the following utterance. AP then produces the second part of the turn in a specific form: the inversion of the subject (very rare in French spoken language), the conditional associated to the prosodic delivery (lengthening of the subject 'you' and the high rise contour on the last syllable of "vegetarian") and the absurdity of the question create humor. LJ simultaneously produces a reported speech dialogue (not designed to make anyone laugh) in which he contrasts 2 figures (the other man animated by AP and himself) with a different prosodic device. The following utterance in 392 also overlapping laughing with high register and loudness. Both utterances (308 and 392) exhibit a crescendo in the high register and loudness. This congruence of activities (simultaneous reported speech, overlap) illustrates the strong cooperation and the alignment between participants that is one of the criteria of conversational humor. Its success depends on the participants' shared knowledge/representations of the prototypical figure mentioned here (491). The shift in vocal cues comments on representations one can have about vegetarians (weird people) but is also a comment on AP's part on the figure he animates in this utterance. Otherwise, the reported speech is typically a display of orientation to LJ's discourse: it comments on the discourse by chiming in on it.

(4)

 Noir bizarre / black weird Here, LJ tells how he met with a very strange and scary man during his archaeological prospection. It is by focusing on this scary aspect that the two interlocutors will develop a humorous sequence exaggerating the man's scary aspect: "the rapist of the wood"/"le violeur des bois"; "black weird"/"noir bizarre".

In 490 AP already grunted (with its laugh and "mh") announcing his following turn in 491 in a low pitch register and in descending line. We also heard a hoarse voice in this utterance that will increase in the next utterance (499) which is the vocal animation of the figure presented in 491. We observe a clear mapping between enunciative and prosodic devices: both interlocutors make use of reported speech and express the same voice in the overlap phase. We notice a higher pitch register and loudness, a typical vocal quality (hoarse voice) which corresponds to the representations shared by the interlocutors to refer to the character in question in the narrative. The vocal cues enable the participants to typify a character that is shared by a community [START_REF] Klewitz | Quote-Unquote? The Role of Prosody in the Contextualization of Reported Speech Sequences[END_REF] more specifically here by both interlocutors. More globally in the last examples, humor is inferable from the prosodic cues that refer more or less explicitly to another voice. But it is not the vocal cues per se but the vocal cues which enable us to say that this utterance is a case of double-voicing that is destined to provoke laughter.

( In this excerpt, LJ evokes the time when he went with a friend to a coffee shop to drink a cup of hot chocolate, in which he found a cockroach.

Here, the fictive reported speech (142) seems to be an internal reflexion that LJ shares with the interlocutor. It is self-disparaging kind of humor (Priego-Valverde, 2007) which allows to mock himself and the situation. He produces two voices (himself and probably, the friend of AP): the two first parts of LJ's turn refers to his own voice in the reported speech. They are a confirmation request without answer but an assertion that follows in the sense of the previous question. Prosodically they are both expressed in the same pitch register, and with a rising contour of question for the first part and a rise-fall contour for the second (with the higher pitch peak on "si" / "if", 142). This prosodic device contrasts with the final part of LJ's turn that exhibits a flat melodic configuration and in a pianissimo voice. The two voices are prosodically contrastive. We confirm that when more than one figure is reported, separate prosodic marking may coincide with the different voices (Klewittz & Couper-Kuhlen 1999: 466). This example clearly illustrates yet another case of strong agreement between participants. In reporting here a narrative sequence in which he has never been but as if he were a protagonist of the story (instead of AP), LJ displays an orientation to the prior discourse of AP: not only protagonist but also the reported situation itself is fictitious.

Summary

In this previous sub-section, we showed the relevance for participants in talk-in-interaction to use the design of reported speech and/or to make hearable another voice in their discourse in order to create humoristic sequences. We showed that multiple reported speech could appear simultaneously in both the discourse of the main speaker and the recipient, with an illustrative function for the former and a humorous function for the latter. The using of the same devices at the same time for both participants reveals a strong agreement between them. It can be interpreted in the general framework of the interactional orientation [START_REF] Hutchby | Conversation Analysis. Principles, Practices and Applications[END_REF]. With his reported speech, the recipient chimes in on the previous talk exhibiting then his strong orientation to it. We also showed that the reported speech analyzed here is invented and precisely anchored in the reporting situation and in the time, conversely to what Vincent & Dubois (1997: 60) have described under this term. This point contributes to the incongruity of the humorous utterance ("repeated" words have never been delivered). In the same way, the second important point is the abstract or fictitious figure in the reported speech that is animated with unexpected words associated to a typical delivery. Vocal cues are indeed strongly involved in the sequences through different shifts that do not serve to delimit boundaries but rather play a deictic role (Couper-Kuhlen 1999). We confirmed that vocal cues are crucial to make the other voice hearable and understandable. They enable the speaker to typify a character on which members of the same community share knowledge and typical representations [START_REF] Klewitz | Quote-Unquote? The Role of Prosody in the Contextualization of Reported Speech Sequences[END_REF]. All these parameters contribute to the creation of conversational humor by reinforcing the gap between what is expected and what really happens.

Repetitions and confirmation requests/answers

In the following sub-section, we observe a few examples of humor that appear in a sequential environment involving repetition sequences and confirmation request and answer. These phenomena are crucially interactional in explicitly showing the co-construction of the discourse by participants [START_REF] Perrin | Pragmatic functions of local diaphonic repetitions in conversation[END_REF]. Moreover, we show that prosodic cues are involved in these devices and strongly contribute to the interactional orientation (defined above) and more specifically to the prosodic orientation [START_REF] Reed | Prosodic orientation in English Conversation[END_REF]. Finally, we show that the connector and the disjonctor here are produced separately by using different types of shift in prosodic cues.

( The two interlocutors are speaking about a third person that LJ met when he was at an archaeological dig. LJ describes him as a tall and brown man. AP proceeds to produce humor ("The tall brown with a white shoe"/"Le grand brun avec une chaussure blanche"). The humorous aspect of this sentence is based on an "intertextual" reference [START_REF] Kristeva | Sèmiotikè[END_REF]) to a very famous French movie called "The tall blond with a black shoe". From 312 to 314, LJ expresses his story in a narrow span and flat configuration. The second part of the "connector" initiated in 314 is the subject of a confirmation request expressed by AP (in 396) with a rise on "BRUN / Brown" followed by "là / here" with a high plateau. LJ produces his confirming answer in a flat configuration. Then AP produces the disjonctor that consists of two parts. The first one ( 397) is a backchannel that confirms and supports the previous confirmation answer. The second one ("chaussures blanches"/"white shoes") refers to the intertextual reference. It is interesting to note that each part corresponds to an intonative unit. The first is produced in the same configuration of the previous utterance of the main speaker (flat configuration, level pitch), while the second is ends in a rise movement for which it is difficult to decide if it is a final or a continuation rise. After the overlap LJ alludes to AP's story by repeating his utterance (316) with an accentual arch (Di Cristo 2000) which prosodically merge the three words in a single unit and makes it emphasised ("DEUX chaussures NOIRES"/"TWO BLACK shoes"). Finally, AP ends (398) this parallel sequence by orienting to the story of LJ and repeating the same utterance with a prosodic mapping (Szcezpeck-Reed 2006) through the accentual arch. Finally in 398, his discourse functions as a backchannel repetition (called complex backchannel, Laforest 1992) with an assessment of the content but more specifically an assessmentor at least a real take into accountof the tonality (humor) which ends in a rise-fall pitch contour. The laughter in his voice also supports this point. The final sequence exhibits another repetition with a prosodic mapping (flat configuration) ("deux chaussures noires"/"two black shoes") and AP finally ratifies and concludes this sequence (in 399) also with a flat configuration on the whole followed by a long pause (0.5 s). This sequence highlights the co-construction of a parallel sequence involving humorous tonality. The linguistic devices are confirmation request, answer and repetition with lexical and prosodic mapping.

In the following example, contrary to the previous example in which we showed an orientation through the use of lexical repetitions, we show an instance of prosodic orientation only through a copy of the prosodic cues. This excerpt appears in the same narrative sequence as the example 6. In this sequence, the two participants are speaking about the archaeological dig LJ did some years ago. This excerpt is a part of a long narrative sequence about this topic in which he tells several anecdotes. He evokes one person who is working to the I.R.A.A. i.e. the Research Institute of Antique Archaeology. LJ pronounces the two A's (the corpus shows it) but of course, AP elaborates on this acronym which is phonetically very close to I.R.A., the Irish terrorist organization. In other words, by elaborating on this acronym, AP makes it the polysemic connector that allows him to switch from a bona fide communication about the archaeological searching into a non bona fide communication about terrorism. The acronym I.R.A. becomes thus the disjonctor actualising the second script, a humorous and fictitious script about terrorism. The switch from a serious to a humorous mode of communication is totally accepted by LJ who does not seem to be disturbed by the interruption of his narration. On the contrary, he shows his complete adherence to the humorous mode of communication beginning by laughing and then producing an utterance about terrorism ("he blows up"/"il plastique"). From 321 to 323, we have a short joint fantasy where each speaker overbids on the other's sentence. The end of this short sequence is due to the instigator himself (AP: "that's not funny"/"c'est pas drôle"). To end this sequence, he produces a metacommunicative utterance evaluating as not funny (despite the laugh) what was just said. Once again, LJ follows his interlocutor and goes back to a serious mode of communication to give the real signification of the acronym. His narration can proceed hereafter (324). The disjonctor (the first time where IRA appears in the discourse of A) belongs to a turn composed of two parts (an adjacency pair). The first part is a kind of an information request (with a slightly rise on IRA in order to ask for confirmation: does he work at IRA?) while the second part corresponds to the response of the same speaker in his own discourse (ah with a rise-fall pitch contour, of course no, it is the bad word! with a strong fall). However, after this turn, AP seems to use another voice that precisely participates in the first part of the utterance by adopting the point of view rejected immediately afterwards. AP animates this voice by increasing loudness, and emphasizing the utterance notably through the words "travaille"/"work" and "IRA". We also noted a hoarse voice associated or linked to the laughing heard in the voice. This shift is indeed heard by LJ who also laughs (in 321). LJ shows his prosodic orientation to the humorous tonality of AP by expressing a complement's turn in which prosodic cues exhibit the same pattern as AP. Among the different prosodic orientation, Szcezpeck- [START_REF] Reed | Prosodic orientation in English Conversation[END_REF] mentions the case in which interactants may display prosodic orientation by matching the previous speaker's prosodic design. There is also the case in which the interactants may complement a prior turn with a second structurally related prosodic design or they may continue previously unfinished prosodic patterns. The example examined here refers to a case of mapping prosody that consists of the activity of copying a previous speaker's prosodic pattern. The following turns from LJ and AP display the same prosodic pattern such as a strong lengthening of the last syllable or on the additional "euh", a very little modulation and slight falling pitch on the whole utterance, a similar duration of these successive utterances and laughter in their voices. If we look at the example (3), we can also describe it as a typical case of prosodic orientation where the reported speech completes the prior turn of LJ, -even if vocal cues are emphasized to create a prosodic chiming in-. In the present example, the utterance in 404 shifts again to another person (himself) with a metacommunicative signal and a denial of the humorous nature of this previous sequence. Similarly LJ moves back to the serious mode in explaining once again the IRAA acronym (by overarticulating the different letters). In the next turn, AP moves again on the acronym (by emphasizing 'IRAA' in a "saoudite manner"). ( 8 In this excerpt, LJ tells a story of a salesperson who came to sell door to door some cards. LJ explains that if he were alone, he would have sent him back but he did not dare to do it because he was with his girlfriend. This point is the moment when AP begins to produce his humorous utterances. He begins to tease LJ about his girlfriend asking about the color of her hair. This attempt to a humorous digression fails because LJ chooses to answer AP's question seriously about his girlfriend and goes on with his story. When LJ specifies that the sellers was also of Arabic descent, AP proceeds with this topics and begins to play with the double meaning of "beur" in French that means both Arab and butter. This time, LJ enters in the game and develops with AP a short humorous sequence where they play on this double meaning. "Beur" is thus the polysemic connector which has two disjonctors. The first one is "beurre/butter" i.e. a phonic disjonctor and the second, more explicit is "des petits Lu"6 that presents the second unrealistic absurd and humorous meaning, i.e. the fact that two cookies are salespersons.

In this example, we show another case of co-construction by participants of a humorous sequence. The device used is exactly the same as in the previous example. The interlocutor (AP) comments again on the word produced by LJ by expressing a form of confirmation request (with a rise on BEU and a level pitch on "e" as a kind of suspension between the request and a proposition) overlapping with the turn of LJ on "et::::" . But here it is not AP but LJ who gives an answer in repeating once again "deux beurs" (with a flat configuration) but a very long pause (> 1 second) after he proposes his second script in a specific prosodic frame: lengthening on "des::::"des petits lus", with also a specific vocal quality (breathy on "pe" de petit and "lu"). AP follows with an increase of loudness, an overarticulation of each syllable ("c'est pas possible"!/ "it is not possible"!) and a rising-falling contour with a peak on the penultimate which sounds strange here (shift!). LJ repeats in overlap (877 "et" / "and" 927) his utterance in attempting to imitate a foreign accent (maghrebin yet). In 928, the laughter in the voice of AP can be heard and he also repeats the utterance by imitating the foreign accent. Finally, both interlocutors laugh and, once again, LJ and AP repeat the utterance with the same accent and the same prosodic delivery. This example illustrates a case of prosodic orientation with a prosodic matching in the last repetitions.

( In the last excerpt, LJ is speaking about something that happened to him when he was doing an archaeological dig. He explains that at the beginning, he had a difficult relationship with another man and that, as the days passed, their relationship improved. The expression "normal relationship"/rapports normaux" is the connector from which LJ proceeds to modify and switch into a humorous mode of communication. Indeed, "normal relationship" may be interpreted with two different meanings. In other words, it potentially carries two different scripts: (i) the logical and expected script, i.e., a professional and/or friendly relationship with a colleague, (ii) a sexual script. AP proceeds to use the connector to render the second script unexpected and humorous (A 328, 329). If "condom" / "preservatif" is the explicit disjonctor that allows to reveal the second script AP's first attempt already produced when he repeated the term "normal". This attempt was successful as LJ's laughter shows. Even if LJ laughs, at the end, he produces a metacommunicative comment ("T'es dedans là" / "you're in") showing that if he had perceived humor he probably would not want to develop it too much. AP produces backchannel signals in 326 and 327. Then, when he takes the floor in 328 on a similar level pitch, LJ can expect that it is a true confirmation request. But AP insists, pauses and complements his first pair part with the explicit disjonctor that carries the pitch contour described above (rise followed with high plateau) to characterize the confirmation request that functions as humorous. Even if the prosodic device is less glaring here, LJ can hear the humorous tonality (laughing). Moreover, it seems that the shift between lexical and prosodic cues precisely contributes to create humor. In this way, this last example differs from all the others observed in this paper. The non marked connector could be interpreted in the same way as the reported speech displaying no marked prosody. As Klewitz and Couper-Kuhlen explain (1999: 474) the absence of mark where it is expected is significant in the reported speech, and we claim that humor emerges in this specific context thanks to the absence of explicit prosodic marks.

CONCLUSION

In this paper, we focused on conversational humor in talk-in-interaction in order to understand how speakers co-construct and switch into a humorous mode of communication. Consequently, we wondered what exactly the role of prosodic cues was in conversational humor by analyzing them in their context, i.e., in their sequential environment. We could extract two discursive devices in which humor appears: reported speech and repetitions. In the former, we highlighted that the connector and the disjonctor could appear as simultaneously while they appear rather separately in the second device through successive repetitions that function as confirmation request and/or answer. Several studies have shown the role of prosodic cues in these devices independently of the humor phenomenon. Concerning the reported speech, we confirm here the importance of prosodic cues to animate different figures in speakers' discourse and more specifically to typify a character. This latter appears as the peak of humor sequence co-constructed gradually with several reported speeches from both interlocutors. Concerning the request/answer confirmation in the repetition's device, we highlight the role of the prosodic orientation (through different prosodic means) to show the successful co-construction of the humorous sequence. Finally, we demonstrate that not all the prosodic cues are specific to humor but participate in a congruence of cues that allow the creation of humor. (Selting 2004)

Conventions of transcription

  @ et i(l) → POSE euh les détonaTEURS euh :: LJ gpd_323 ouai-→ IL est horloGER à l'I.R.A. : @ AP gpd_404 ah c'est pas drôle ça c'est pas LJ gpd_323a non c'est I.R.A.A. A. euh Archite-euh Institut d(e) recherch(e) s-sur l'Architecture Antique AP gpd_405 I.R.A ah AP gpd_405a oh putain

  AG_gpd_252 et je crois p-AP gpd_253!PAS ! dire de !CO !nneRIES, je crois qu'i(ls) di-i(ls) tii(ls) l'ont !RUINE! le MEC, ou ils l'ont !TUE ! enfin je sais PAS y a eu vraiment un TRUC, euh

	LJ gpd_178a @
	AP gpd_254	<<f> <<h> OUAIS^ c'était CHAUD^ enfin
	AP gpd_255	@ peut-être ça a rien à voir euh >>
	LJ gpd_179	ah ouais m'enfin bon bon Roxane, nous dira ::
	AP gpd_256	<<h> euh le MEC, euh ::
	AP gpd_257	ils lui ont DIT § tu es =z= un VILAIN^ §>> (0.6)
	AP gpd_258	mais ouais (en)fin bon je sais plus c'était chaud quoi et
		i(l) me disait que les euh i(ls) filtraient carrément tout
		l-
	LJ gpd_180	mh mh

This corpus, named Corpus d'Interactions Dialogales (Corpus of Interactional Data) was collected by the authors of this paper at the LPL (Laboratoire Parole et Langage).

The final goal is to obtain 20 dialogs.

Because of the instruction of the beginning, we have a lot of narrative jokes in this corpus.

The translation of the examples is at the end of this paper.

(…) refers to a truncation of the example into two excerpts.

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