Jean-St Éphane Dhersin

Fabian Freund
email: ffreund@uni-hohenheim.de

Arno Siri-J Égousse

Linglong Yuan
email: yuan@math.univ-paris13.fr

ON THE LENGTH OF AN EXTERNAL BRANCH IN THE BETA-COALESCENT

Keywords: Mathematics Subject Classification. 60J70, 60J80, 60J25, 60F05, 92D25 Coalescent process, Beta-coalescent, external branch, block counting process, recursive construction

In this paper, we consider Beta(2 -α, α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the n-coalescent, we prove the convergence of n α-1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of collisions which occur in the n-coalescent until the end of the chosen external branch, and for the block counting process associated with the n-coalescent.

1. Introduction 1.1. Motivation and main results. In modern genetics, it is possible to sequence whole genomes of individuals. In order to put this information to maximal use, it is important to have well-fitting models for the gene genealogy of a sample of individuals. The standard model for a gene genealogy of a sample of n individuals is Kingman's n-coalescent (see [START_REF] Kingman | The coalescent[END_REF], [START_REF] Kingman | On the genealogy of large populations[END_REF]). Kingman's n-coalescent is a continuous-time Markov process with state space P (n) , the set of partitions of {1, . . . , n}. The process starts in the trivial partition ({1}, . . . , {n}) and transitions are only possible as mergers of exactly two blocks of the current state. Each such binary merger occurs with rate 1. These mergers are also called collisions.

For many populations, Kingman's n-coalescent describes the genealogy quite well. Kingman showed in [START_REF] Kingman | On the genealogy of large populations[END_REF] that the ancestral trees of a sample of size n in populations with size N evolving by a Wright-Fisher model will converge weakly to Kingman's n-coalescent for N → ∞ (after a suitable time-change). This result is relatively robust if population evolution deviates from the Wright-Fisher model (see [START_REF] Kingman | On the genealogy of large populations[END_REF] or [START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF]). However, there is evidence that there are populations where the gene genealogy of a sample is not described well by Kingman's n-coalescent. Examples of such populations can be found in maritime species, where one individual can have a huge number of offspring with non-negligible probability (see [START_REF] Árnason | Mitochondrial cytochrome b variation in the high-fecundity atlantic cod: trans-atlantic clines and shallow gene genealogy[END_REF], [START_REF] Boom | Mitochondrial DNA variation in introduced populations of Pacific oyster, Crassostrea gigas, in British Columbia[END_REF] [START_REF] Hedgecock | Does variance in reproductive success limit effective population sizes of marine organisms?[END_REF], [START_REF] Gnedin | On the number of collisions in Λ-coalescents[END_REF] and [START_REF] Eldon | Coalescent processes when the distribution of offspring number among individuals is highly skewed[END_REF]).

A whole class of potential models for the gene genealogy of a sample was introduced independently by Pitman and Sagitov (see [START_REF] Pitman | Coalescents with multiple collisions[END_REF] and [START_REF] Sagitov | The general coalescent with asynchronous mergers of ancestral lines[END_REF]): The class of n-coalescents with multiple collisions. A n-coalescent with multiple collisions is a continuous-time Markov process with state space P (n) , where all possible transitions are done by merging two or more blocks of the current state into one new block. Every n-coalescent Π (n) is exchangeable, meaning τ •Π (n) d = Π (n) for every permutation τ of {1, . . . , n}. The transition rate of a merger/collision of k of b present blocks is given by [START_REF] Árnason | Mitochondrial cytochrome b variation in the high-fecundity atlantic cod: trans-atlantic clines and shallow gene genealogy[END_REF] λ b,k = for a finite measure Λ on [0, 1] (this definition is due to Pitman [START_REF] Pitman | Coalescents with multiple collisions[END_REF]). Since the process is characterized by the measure Λ, it is also called a Λ-n-coalescent. Note that Kingman's n-coalescent is a Λ-n-coalescent with Λ being the Dirac measure δ 0 in 0. It is possible to define a continuous-time process Π with state space P, the partitions of the natural numbers N, whose restriction on {1, . . . , n} is a Λ-n-coalescent for all n ∈ N. Such a process is called a Λ-coalescent.

An important subclass of Λ-n-coalescents are Beta n-coalescents characterized by Λ being a Beta distribution, especially for the choice of parameters 2 -α and α for α ∈ (0, 2). The class of Beta(2 -α, α)-n-coalescents appears as ancestral trees in various settings. They appear in the context of supercritical Galton-Watson processes (see [START_REF] Schweinsberg | Coalescent processes obtained from supercritical Galton-Watson processes[END_REF]), of continuous-state branching processes (see [START_REF] Birkner | Alphastable branching and beta-coalescents[END_REF]) and of continuous random trees (see [START_REF] Berestycki | Beta-coalescents and continuous stable random trees[END_REF]). They also seem to be a class where suitable models for the ancestral tree can be found for samples from species who do not fit well with the Kingman-setting (see [START_REF] Birkner | Computing likelihoods for coalescents with multiple collisions in the infinitelymany-sites model[END_REF]). Note that for α → 2, the rates of the Beta(2 -α, α)-n-coalescent converge to the rates of Kingman's n-coalescent. In this sense, Kingman's n-coalescent can be seen as a border case of this class of Beta n-coalescents.

For α = 1, Beta(2 -α, α) is the uniform distribution on [0, 1]. The corresponding ncoalescent is the Bolthausen-Sznitman n-coalescent. It appears in the field of spin glasses (see [START_REF] Bolthausen | On ruelle's probability cascades and an abstract cavity method[END_REF], [START_REF] Bovier | Much ado about derrida's grem[END_REF]) and is also connected to random recursive trees (see [START_REF] Goldschmidt | Random recursive trees and the Bolthausen-Sznitman coalescent[END_REF]).

Let us denote by Π (n) = (Π

(n)
t ) t≥0 a n-coalescent. In this paper, we are interested in three functionals of n-coalescents

• the length T (n) of a randomly chosen external branch ;

• the number σ (n) of collisions which occur in the n-coalescent until the end of a randomly chosen external branch ;

• the block counting process R (n) = (R (n) t ) t≥0 : R (n) t = |Π (n) t | is the number of blocks of Π (n) t .
Note that T (n) can also be characterized as the waiting time for the first collision of a randomly chosen individual and σ (n) as the number of collisions we have to wait to see the randomly chosen individual merge. For i ∈ {1, . . . , n} define

T (n) i := inf t| {i} / ∈ Π (n) t
as the length of the ith external branch and

σ (n) i := inf {k|{i} / ∈ π k }
as the number of collisions until the end of the ith external branch, where π k is the state of the n-coalescent after k jumps. Due to the exchangeability of the n-coalescent, we have

T (n) d = T (n) 1 and σ (n) d = σ (n)
1 . Since we are only interested in distributional results, for the remainder of the article we will identify

T (n) with T (n) 1 and σ (n) with σ (n) 1 .
If the n-coalescent is used as a model for an ancestral tree of a sample of individuals/genes, the functionals T (n) and σ (n) can be interpreted biologically. The length of an external branch measures the uniqueness of the individual linked to that branch compared to the sample, since it gives the time this individual has to evolve by mutations that do not affect the rest of the sample (see the introduction of [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF] for more information). It was first introduced by Fu and Li in [START_REF] Fu | Statistical tests of neutrality of mutations[END_REF], where they compare mutations on external and internal branches of Kingman's n-coalescent in order to test for the neutrality of mutations.

The functional σ (n) was first introduced in [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF], though n-σ (n) was also analyzed in [START_REF] Blum | Minimal clade size and external branch length under the neutral coalescent[END_REF] as the level of coalescence of the chosen individual with the rest of the sample. In both articles, the functionals were defined for Kingman's n-coalescent.

For the biological interpretation of σ (n) , we see the n-coalescent as an ancestral tree of a sample of size n. Each collision in the n-coalescent then resembles the emergence of an ancestor of the sample. σ (n) -1 is the number of ancestors of the sample which emerge before the most recent ancestor of the randomly chosen individual/gene emerges (recall that time runs backwards in the n-coalescent). In this line of thought, σ (n) gives the temporal position of the first ancestor of the chosen individual/gene among all ancestors of the sample, which are the τ (n) collisions in the n-coalescent. Thus, σ (n) τ (n) gives the relative temporal position of the first ancestor of the chosen individual/gene among all ancestors of the sample (until the most recent common ancestor). In this sense, we interpret σ (n) τ (n) as a measure of how ancient the chosen individual/gene is compared to the rest of the sample.

In this article, we focus on the asymptotics of T (n) , σ (n) and R (n) for n → ∞. The asymptotics of these functionals are already known for some Λ-n-coalescents. For T (n) , we have

• Λ = δ 0 (Kingman's coalescent): nT (n) d
→ T , where T has density t → 8 (2+t) 3 (see [START_REF] Blum | Minimal clade size and external branch length under the neutral coalescent[END_REF], [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF], [START_REF] Janson | The external lengths in kingman's coalescent[END_REF]),

• Λ = Beta(1, 1) (Bolthausen-Sznitman coalescent): log(n)T (n) d
→ Exp(1) (see [START_REF] Freund | On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process[END_REF]), [START_REF] Möhle | Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent[END_REF], see also [START_REF] Gnedin | On asymptotics of exchangeable coalescents with multiple collisions[END_REF])

• Λ with µ -1 = 1 0 x -1 Λ(dx) < ∞: T (n) d → Exp(µ -1 ) (see
for n → ∞. For σ (n) , we have [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF]), [START_REF] Freund | On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process[END_REF]), [START_REF] Gnedin | On asymptotics of exchangeable coalescents with multiple collisions[END_REF]) for n → ∞, where Geo(p) is the geometric distribution on N with parameter p.

• Λ = δ 0 : σ (n) /n d → Beta(1, 2) (see
• Λ = Beta(1, 1): log(n) n σ (n) d → Beta(1, 1) (see
• Λ with µ -2 = 1 0 x -2 Λ(dx) < ∞: σ (n) d → Geo( µ -1 µ -2 ) (see
There are some known results for the asymptotics of the block counting process R (n) . Define R = (R t ) t≥0 by setting R t := |Π t |, the number of blocks of a Λ-coalescent Π = (Π t ) t≥0 . Note that R = lim n→∞ R (n) , if the block counting process R (n) is defined for the restriction of Π to {1, . . . , n} for each n ∈ N. For the small-time behaviour of the block counting process R, the following results are known:

• If Λ(dx) = f (x)dx, where f (x) ∼ Ax 1-α for 1 < α < 2 and ∼ means that the ratio of the two sides tends to one as x → 0+. Then

(2) lim

t→0+ t 1/(α-1) R t = ( α AΓ(2 -α)
) 1/(α-1) a.s.

(see [START_REF] Berestycki | Beta-coalescents and continuous stable random trees[END_REF] and [START_REF] Berestycki | Small-time behavior of beta coalescents[END_REF])

• If [ǫ,1] x -2 Λ(dx) = ǫ -α L ǫ
with ǫ ∈ (0, 1) and L ǫ slowly varying as ǫ → 0. Denote by -α) , in Probability.

g ǫ := L -1 ǫ ǫ α-1 , then lim ǫ→0+ ǫR gǫ = (Γ(2 -α)) 1/(1
(see [START_REF] Bertoin | Stochastic flows associated to coalescent processes. III. Limit theorems[END_REF]).

Notice that both cases are complementary. The block counting process of Kingman coalescent gets similar limit value almost surely by taking α = 2, (see [START_REF] Berestycki | Beta-coalescents and continuous stable random trees[END_REF], page 216).

• For any Λ coalescent that comes down from infinity(that means for any t > 0, R t < ∞ almost surely), there exists a deterministic positive function v t on (0, ∞), such that:

lim t→0+ R t v t = 1, a.s.
For explicit form of v and finer results, we refer to [START_REF] Berestycki | The lambda-coalescent speed of coming down from infinity[END_REF]. This result is much more general than the two former ones. We will analyze the asymptotics for T (n) , σ (n) and R (n) for Λ-n-coalescents with Λ fulfilling

ρ(t) = C 0 t -α + O(t -α+ζ ), t → 0 for some C 0 > 0, α ∈ (1, 2) and ζ > 1 -1/α, where ρ(t) = 1 t x -2 Λ(dx).
Note that this class of n-coalescents includes all Beta(a, b)-n-coalescents with parameters a ∈ (0, 1) and b > 0. In this class of n-coalescents, we have the following asymptotics for T (n) , σ (n) and R (n) :

• σ (n) n(α-1) d -→ σ, • n α-1 T (n) d → 1 C 0 Γ(2-α) ((1 -σ) 1-α -1), • for any t 0 > 0, ε > 0, P( sup 0≤t≤t 0 |n -1 R (n) tn 1-α -(1 + C 0 Γ(2 -α)t) -1/(α-1) | > ε) → 0, for n → ∞ , where σ d = Beta(1, α). Remark that if Λ(dx) = f (x)dx, where f (x) ∼ Ax 1-α for 1 < α < 2, we get that for fixed t > 0, lim n→∞ 1 n (1 + AΓ(2 -α) α t) 1/(α-1) R (n)
tn 1-α = 1 in probability, which is to be compared to the following a.s. convergence implied by (2):

lim n→∞ 1 n ( AΓ(2 -α) α t) 1/(α-1) R tn 1-α = 1.
The different scaling comes from the fact that we do not consider the block counting process R = lim n→∞ R (n) of a Λ-coalescent Π, but instead the time-changed, scaled limit lim n→∞

1 n R (n)
tn 1-α . To our best knowledge, there does not seem to be a direct link between these two results.

Note that if we see the Bolthausen-Sznitman n-coalescent as a Beta(1, 1)-coalescent and Kingman's n-coalescent as the borderline case of a Beta distribution with parameter α → 2, the convergence results for σ (n) shows a nice continuity in the parameters of the limit distributions in the range of Beta(2 -α, α)-n-coalescents with α ∈ [START_REF] Árnason | Mitochondrial cytochrome b variation in the high-fecundity atlantic cod: trans-atlantic clines and shallow gene genealogy[END_REF][START_REF] Berestycki | The lambda-coalescent speed of coming down from infinity[END_REF]. Our convergence result itself is even somewhat true in the border cases 1 and 2 (if one wages (α -1) -1 → ∞ for α → 1 against log(n) → ∞ for n → ∞ for the Bolthausen-Sznitman n-coalescents). Also note that T obtained as the limit variable of nT (n) in Kingman's case has the same law as 2((1-σ) -1 -1), which gives again a nice continuity in results. The continuity also appears for the block counting process: replacing α by 2 in the formula for the Beta(2-α, α)-n-coalescent gives the formula for Kingman's, this result will be proved in the sequel.

Finally we can remark that together with the known asymptotics τ (n) n d → α -1 for this class of n-coalescents (see [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], [START_REF] Gnedin | On the number of collisions in Λ-coalescents[END_REF] and [START_REF] Iksanov | On the number of jumps of random walks with a barrier[END_REF]), we have σ (n) /τ (n) d → σ. To prove these results, we will exploit some techniques from [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF]. In [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], they were used to analyze the asymptotics of a part of the length of a n-coalescent and the number of collisions in a n-coalescent for the same class of n-coalescents as analyzed in the present paper. For the convergence result for T (n) , we present two proofs. One mimics the approach in [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF] and [START_REF] Janson | The external lengths in kingman's coalescent[END_REF], using the convergence result for σ (n) and T

(n) = σ (n)
i=1 T i , where T i is the waiting time between the (i -1)th and ith collision/jump of the n-coalescent. The other proof is based on the representation of T (n) as the first jump time of a Cox process driven by a random rate process which depends only on the block counting process associated with the remaining individuals labelled {2, 3, . . . , n}. We use a recursive construction suitable for any Λ-n-coalescent: This construction consists in adding individual i to a coalescent process constructed by individuals from 1 to n except i such that consistence relationship is fulfilled.

1.2. Organization of the paper. In section 2, we recall some known technical results which can all be found in [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF]. In section 3, we obtain the asymptotic result about σ (n) and also about the ratio between σ (n) and τ (n) . Section 4 studies the small time behavior of the block counting process R (n) . Depending on the property of R (n) , our first method taking T (n) as the first jump time of a Cox process gives the asymptotic behavior of T (n) in section 5. In section 6, another method is provided by taking into account the fact that T (n) is the sum of σ (n) initial waiting times for the coalescent process Π (n) to jump from one state to the following.

Preliminaries

In this Section, we recall some results from [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF]. Consider a n-coalescent with multiple collisions characterized by a finite measure Λ on

[0, 1]. Let ν(dx) = x -2 Λ(dx) and ρ(t) = ν[t, 1].
When the process has k blocks, the next coalescence event comes at rate g k given by ( 3)

g k = k-1 ℓ=1 k ℓ + 1 λ k,ℓ+1 = (0,1) 1 -(1 -x) k -kx(1 -x) k-1 Λ(dx) x 2 .
For n ≥ 1, x ∈ (0, 1), let B n,x be a binomial r.v. with parameter (n, x). Recall that for 1 ≤ k ≤ n, we have ( 4)

P(B n,x ≥ k) = n! (k -1)!(n -k)! x 0 t k-1 (1 -t) n-k dt.
Use the first equality in (3) and (4) to get

g n = 1 0 n k=2 n k x k (1 -x) n-k ν(dx) = 1 0 P(B n,x ≥ 2)ν(dx) = n(n -1) 1 0 (1 -t) n-2 tρ(t) dt.
All along this paper, the following hypothesis will be assumed

(5) ρ(t) = C 0 t -α + O(t -α+ζ )
for some C 0 > 0, α ∈ (1, 2) and ζ > 1 -1/α. Under this hypothesis, Lemma 2.2 of [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF] gives us that, for n ≥ 2, (6) 1) ).

g n = C 0 Γ(2 -α)n α + O(n α-min(ζ,
Recall that we call τ (n) the number of coalescence events until reaching the common ancestor of the initial population (of size n). For k ≥ 0, denote by

Y (n) k the number of blocks remaining after k jumps. Notice that Y (n) is a decreasing Markov chain with Y (n) 0 = n and Y (n) k = 1 for k ≥ τ (n) . Let X (n) k = Y (n) k-1 -Y (n) k
be the number of blocks we lose during the kth coalescence event. We write X (n) 0 = 0. The Markov property makes that the law of the first jump X (n) 1 will be of much interest. We will look at some properties of

X (n) 1 . Notice that (7) P(X (n) 1 = k) = 1 g n 1 0 P(B n,x = k + 1)ν(dx)
and thus ( 8)

P(X (n) 1 ≥ k) = 1 0 P(B n,x ≥ k + 1)ν(dx) g n = (n -2)! k!(n -k -1)! 1 0 (1 -t) n-k-1 t k ρ(t) dt 1 0 (1 -t) n-2 tρ(t) dt .
Under the same assumptions on ρ(t), setting ε 0 > 0 and ( 9) [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF] tells us there exists a constant C 10 s.t. for all n ≥ 2, we have

ϕ n =      n -ζ if ζ < α -1, n 1-α+ε 0 if ζ = α -1, n 1-α if ζ > α -1, Lemma 2.3 of
(10) E[X (n) 1 ] - 1 α -1 ≤ C 10 ϕ n .
Moreover, from Lemma 2.4 of [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], there exists a constant C 11 s.t. for all n ≥ 2, we have

(11) E X (n) 1 2 ≤ C 11 n 2 g n .
We consider φ n the Laplace transform of

X (n) 1 : for u ≥ 0, φ n (u) = E[e -uX (n) 1 ]
. Assume that hypothesis (5) holds true. Let ε 0 > 0. Recall ϕ n given by ( 9). Then we have (see [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], Lemma 2.5) , for n ≥ 2, [START_REF] Boom | Mitochondrial DNA variation in introduced populations of Pacific oyster, Crassostrea gigas, in British Columbia[END_REF] φ

n (u) = 1 - u α -1 + u α α -1 + R(n, u), where R(n, u) = uϕ n + u 2 h(n, u) with sup u∈[0,K],n≥2 |h(n, u)| < ∞ for all K > 0.
Moreover, if we assume that ζ > 1 -1/α and set η ≥ 1 α , then (from [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], Lemma 3.2) there exist ε 1 > 0 and C 13 (K) a finite constant such that for all n ≥ 1 and u ∈ [0, K], a.s. with

a n = n -η , ( 13
) τn i=1 R(Y (n) i-1 , ua n ) ≤ C 13 (K)n -ε 1 .
We will also use the following result : Let V = (V t , t ≥ 0) be a α-stable Lévy process with no positive jumps (see chap. VII in [START_REF] Bertoin | Lévy processes[END_REF]) with Laplace exponent ψ(u) = u α /(α -1): for all u ≥ 0, E[e -uVt ] = e tu α /(α-1) . We assume that ρ(t) = C 0 t -α + O(t -α+ζ ) for some C 0 > 0 and

ζ > 1 -1/α. Recall that τ (n)
is the number of coalescing events in the n-coalescent until reaching its absorbing state. Let

V (n) t = n -1/α ⌊nt⌋∧τ (n) k=1 (X (n) k - 1 α -1 )
for t ∈ [0, α -1), and

V (n) α-1 = n -1/α τ (n) k=1 (X (n) k - 1 α -1 ) = n -1/α n -1 - τ (n) α -1 . Then, ( 14
) (V (n) t , t ∈ [0, α -1]) → (V t , t ∈ [0, α -1]
) in the sense of convergence in law of the finite-dimensional marginals (see [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], Corollary 3.5, see also [START_REF] Gnedin | On the number of collisions in Λ-coalescents[END_REF][START_REF] Iksanov | On the number of jumps of random walks with a barrier[END_REF]).

The number of collisions in an external branch

Consider a n-coalescent with multiple collisions characterized by a finite measure Λ on [0, 1]. Recall that ν(dx) = x -2 Λ(dx) and ρ(t) = ν[t, 1] which is assumed to satisfy [START_REF] Bertoin | Lévy processes[END_REF]. A n-coalescent takes its values in P (n) , the set of partitions of {1, . . . , n}. For i ≥ 0, let

π i = π (n) i
be the state of the process after the ith coalescence event.

Pick at random an individual from the initial population and denote by T (n) the length of the external branch starting from it. Because of exchangeability, T (n) has the same law as the length

T (n) 1
of the external branch starting from the initial individual labelled by {1}. A quantity of interest will be σ (n) , the number of coalescence events we have to wait to see the randomly chosen external branch merging. Again because of exchangeability, σ (n) has the same law as σ

(n) 1 = inf{i > 0, {1} / ∈ π i },
the time we have to wait to see the external branch linked to individual 1 merging. We can write ( 15)

T (n) 1 = σ (n) 1 i=1 e i g Y (n) i-1
where the e i 's are i.i.d. exponential random variables with mean 1. Note that the formula also holds true for T (n) and σ (n) (just omit the subscripts). For the remainder of the chapter, we will identify σ (n) with σ

(n) 1 .
In this section, we will determinate the asymptotic law of σ (n) for a class of coalescents containing the Beta-coalescent with α ∈ (1, 2).

Theorem 3.1. We assume that ρ(t) = C 0 t -α + O(t -α+ζ ) for some C 0 > 0, α ∈ (1, 2) and ζ > 1 -1/α. Then (16) σ (n) n(α -1) d → σ, for n → ∞, where σ d = Beta(1, α).
Recall that in this class of n-coalescents, we also have τ (n) /n d → α -1 (from [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF], see also [START_REF] Gnedin | On the number of collisions in Λ-coalescents[END_REF] and [START_REF] Iksanov | On the number of jumps of random walks with a barrier[END_REF]) for n → ∞. Slutsky's theorem gives a convergence result for σ (n) /τ (n) , which measures how ancient the chosen individual is compared to the rest of the sample Corollary 3.2. We assume that ρ

(t) = C 0 t -α + O(t -α+ζ ) for some C 0 > 0, α ∈ (1, 2) and ζ > 1 -1/α. Then σ (n) τ (n) d → σ, for n → ∞, where σ d = Beta(1, α). proof of Theorem 3.1. For convenience, we set (1) a b = 0, if 0 ≤ a < b, a ∈ Z + , b ∈ Z + , Z + = {0, 1, 2, ...}. (2) log(0) = -∞.
Notice that σ (n) ≤ τ (n) . Let Y = (Y k , k ≥ 0) denotes the filtration generated by Y (n) . For any t ≥ 0, we have

P(σ (n) > nt|Y) = P(σ (n) > nt, τ (n) > nt|Y) = ⌊nt⌋∧τ (n) i=1 P({1} ∈ π i |{1} ∈ π i-1 , Y) = ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -1 X (n) i +1 Y (n) i-1 X (n) i +1 = ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -(X (n) i + 1) Y (n) i-1 . Notice that X (n) i +1 Y (n) i-1 < 1 if i = τ (n)
, and

X (n) i +1 Y (n) i-1 = 1 if i = τ (n) .
We can hence write

log P(σ (n) > nt|Y) = ⌊nt⌋∧τ (n) i=1 log 1 - X (n) i + 1 Y (n) i-1
and proceed to a power series expansion :

log P(σ (n) > nt|Y) = I (1) nt + I (2)
nt , with

I (1) nt = - ⌊nt⌋∧τ (n) i=1 X (n) i + 1 Y (n) i-1 -1
, and

I (2) nt = ⌊nt⌋∧τ (n) i=1 (log(1 - X (n) i + 1 Y (n) i-1 ) + X (n) i + 1 Y (n) i-1
),

where I

nt can be -∞ if

X (n) i +1 Y (n) i-1 = 1. Let us look further at I (1)
nt . The idea is to replace

X (n) i
by the limit of its expectation.

I (1) nt = J (1) nt + J (2)
nt , with

J (1) nt = - ⌊nt⌋∧τ (n) i=1 1 α -1 +1 Y (n) i-1 -1 , and
J (2) nt = - ⌊nt⌋∧τ (n) i=1 X (n) i - 1 α -1 Y (n) i-1 -1
.

We will use three lemmas whose proofs are given in the rest of the Section. Lemma 3.3, with η = 1, tells us that, when 0 < t < α -1 ( 17)

J (1) nt P → - α α -1 t 0 1 - x α -1 -1 dx = α log 1 - t α -1 . Lemma 3.4 gives, for 0 < t < α -1, (18) J
(2)

nt = -n 1/α-1 n 1-1/α ⌊nt⌋∧τ (n) i=1 X (n) i - 1 α -1 Y (n) i-1 -1 P -→ 0.
Finally, Lemma 3.5 gives, for t < α -1,

(2) nt P -→ 0.

Adding [START_REF] Freund | On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process[END_REF], ( 18) and ( 19), we get that for t < α -

1 log P(σ (n) > nt|Y) P → α log 1 - t α -1 ,
and thus

P(σ (n) > nt|Y) P → 1 - t α -1 α .
While we know that P(σ (n) > nt|Y) ≤ 1,then

P(σ (n) > nt) = E[P(σ (n) > nt|Y)] → 1 - t α -1 α .
We thus obtain that, for x ∈ (0, 1),

P(σ (n) > n(α -1)x) → (1 -x) α . and then that σ (n) n(α -1)
converges in distribution to a Beta(1, α) law.

Lemma 3.3. We set ν η (t) = t 0 1 -x α-1 -η dx,η ∈ R. We assume that ρ(t) = C 0 t -α + O(t -α+ζ ) for some C 0 > 0, α ∈ (1, 2
) and ζ > 1 -1/α. For any 0 < t < α -1 and η ∈ R, we have (1) Let t 0 ∈ [0, α -1) and δ > 0. The following convergence in probability holds when n → ∞:

n (α-1)/2-δ sup 0≤t≤t 0 |n η-1 ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -η -ν η (t)| -→ 0.
(

) Let t ∈ [0, α -1) 2
. The following convergence in distribution holds when n → ∞:

n η-1/α ( ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -η -n 1-η ν η (t)) -→ η t 0 dr(1 - r γ ) -η-1 V r .
Proof. The case η = α -1 is given by Theorem 5.1 in [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF]. Following the same arguments, it is easy to get the general result.

Lemma 3.4. For any t < α -1, the following convergence in distribution holds :

n 1-1/α ⌊nt⌋∧τ (n) i=1 X (n) i - 1 α -1 Y (n) i-1 -1 d -→ (v α (t)) 1/α V 1 ,
where (V t ) t≥0 is an α-stable Lévy process with no positive jumps and v α (t) =

t 0 1 -x α-1 -α dx. Proof. Let δ ∈ (0, α -1), t 0 = α -1 -δ and t ∈ [0, t 0 ]. Let ε ∈ (0, 1 -t α-1 ) and β = 1 -t α-1 -ε > 0. We have n 1-1/α ⌊nt⌋∧τ (n) i=1 X (n) i - 1 α -1 Y (n) i-1 -1 = A nt + B nt ,
with

A nt = n 1-1/α ⌊nt⌋∧τ (n) i=1 X (n) i - 1 α -1 Y (n) i-1 -1 1 {Y (n) i-1 ≥nβ} , and
B nt = n 1-1/α ⌊nt⌋∧τ (n) i=1 X (n) i - 1 α -1 Y (n) i-1 -1 1 {Y (n) i-1
<nβ} . We will show that B nt converges to 0 in probability and that A nt weakly converges to

(v α (t)) 1/α V 1 as n → ∞. Convergence of A nt . Let Z (n) i = n Y (n) i-1 -1 1 {Y (n) i-1 ≥nβ} . We have that sup n,i≥1 Z (n) i ≤ β -1 a.s..
By using [START_REF] Boom | Mitochondrial DNA variation in introduced populations of Pacific oyster, Crassostrea gigas, in British Columbia[END_REF], it is enough to prove that

E[exp(-uA nt )] ---→ n→∞ e vα(t)u α /(α-1) ,
for any u positive, where e vα(t)u α /(α-1) is the Laplace transform of (v α (t))

1/α V 1 . Taking uZ (n) i as Z (n)
i , we shall only consider the case u = 1.

Let us consider n

-1 ⌊nt⌋∧τ (n) i=1 Z (n) i α = n α-1 ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -α 1 {Y (n) i-1 ≥nβ} . We have, because the process (Y (n) i , i ≥ 0) is decreasing, that P(n -1 ⌊nt⌋∧τ (n) i=1 Z (n) i α = n α-1 ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -α ) = P(∃i; Y (n) i-1 < nβ) ≤ P({Y (n) (⌊nt⌋∧τ (n) )-1 < nβ) ≤ P(n -1 (⌊nt⌋∧τ (n) )-1 j=1 (X (n) j - 1 α -1 ) ≥ ε).
Use [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF] to get that the right-hand side of the last inequality converges to 0 as n goes to infinity. Using also Lemma 3.3 with η = α, we have that

n α-1 ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -α P -→ v α (t),
as n → ∞. We can thus deduce that ( 20)

n -1 ⌊nt⌋∧τ (n) i=1 Z (n) i α P -→ v α (t),
as n → ∞.

For a > 0, we set

M (a) n,k = exp k i=1 -n -1/α aZ (n) i X (n) i -log φ Y (n) i-1 (n -1/α aZ (n) i ) .
The process (M = 0 for i > τ (n) , we also have

M (a) n,k = exp   k∧τ (n) i=1 -n -1/α aZ (n) i X (n) i -log φ Y (n) i-1 (n -1/α aZ (n) i )   .
Using R(n, u) defined in [START_REF] Boom | Mitochondrial DNA variation in introduced populations of Pacific oyster, Crassostrea gigas, in British Columbia[END_REF], we get that :

M (a) n,⌊nt⌋ = exp   - ⌊nt⌋∧τ (n) i=1 n -1/α aZ (n) i (X (n) i - 1 α -1 ) - ⌊nt⌋∧τ (n) i=1 n -1 aZ (n) i α α -1 - ⌊nt⌋∧τ (n) i=1 R(Y (n) k-1 , n -1/α aZ (n) i )   = exp (-aA nt ) exp   -n -1 ⌊nt⌋∧τ (n) i=1 aZ (n) i α α -1 - ⌊nt⌋∧τ (n) i=1 R(Y (n) k-1 , n -1/α aZ (n) i )   . Let Λ n = -n -1 ⌊nt⌋∧τ (n) i=1 Z (n) i α α -1 + v α (t) α -1 ,
and write

E [exp (-A nt )] = A 1 + A 2 ,
with A 1 = E e -Ant 1 -e Λn and A 2 = E e -Ant e Λn . First of all, let us prove that A 1 converges to 0 when n tends to ∞. Recall that the r.v Z (n) i are uniformly bounded by β -1 a.s.. Thanks to (13), we have

E[e -2Ant ] = E   M (2) n,⌊nt⌋ exp   n -1 ⌊nt⌋∧τ (n) i=1 2Z (n) i α α -1 + ⌊nt⌋∧τ (n) i=1 R(Y (n) k-1 , 2n -1/α Z (n) i )     ≤ M,
where M is a finite constant which does not depend on n. By Cauchy-Schwarz' inequality, we get that

(21) (A 1 ) 2 ≤ E e -Ant 1 -e Λn 2 ≤ E e -2Ant E 1 -e Λn 2 ≤ M E 1 -e Λn 2 .
The quantity Λ n is bounded and goes to 0 in probability when n goes to infinity (see [START_REF] Gnedin | On the number of collisions in Λ-coalescents[END_REF]). Therefore, the right-hand side of ( 21) converges to 0. This implies that lim n→∞ A 1 = 0.

Let us now consider the convergence of A 2 . Remark that

A 2 = E   M (1) n,⌊nt⌋ exp   v α (t) α -1 + ⌊nt⌋∧τ (n) k=1 R(Y (n) k-1 , n -1/α Z (n) i )     .
Recall that E[M

n,⌊nt⌋ ] = 1. Using ( 13), we get

exp -C 13 (β -1 )n -ε 1 + v α (t) α -1 ≤ A 2 ≤ exp C 13 (β -1 )n -ε 1 + v α (t) α -1 .
We get that lim n→∞ A 2 = e vα(t)/(α-1) , which achieves the proof.

Convergence of B nt . Here, we will use a similar approach as the one we used on the first half of p.11. The process (Y

(n) i , i ≥ 0) is decreasing. So if for some i ≤ ⌊nt⌋, Y (n) i-1 < nβ, then we have Y (n) ⌊nt⌋-1 < nβ. Thus we get B nt = B nt 1 {Y (n) (⌊nt⌋∧τ (n) )-1 <nβ} . Moreover, {Y (n) (⌊nt⌋∧τ (n) )-1 < nβ} ⊂ {n -1 (⌊nt⌋∧τ (n) )-1 j=1 (X (n) j - 1 α -1 ) ≥ ε},
and then for any ε ′ > 0

P(|B nt | ≥ ε ′ ) = P(1 {Y (n) (⌊nt⌋∧τ (n) )-1 <nβ} |B nt | ≥ ε ′ ) ≤ P({Y (n) (⌊nt⌋∧τ (n) )-1 < nβ) ≤ P(n -1 (⌊nt⌋∧τ (n) )-1 j=1 (X (n) j - 1 α -1 ) ≥ ε).
Use [START_REF] Caliebe | On the length distribution of external branches in coalescence trees: genetic diversity within species[END_REF] to get that the right-hand side of the last inequality converges to 0 as n goes to infinity.

Now we deal with I

(2)

nt .

Lemma 3.5. We assume that ρ(t) = C 0 t -α + O(t -α+ζ ) for some C 0 > 0 and ζ > 1 -1/α. Then, for any t < α -1, we have

|I (2) nt | = ⌊nt⌋∧τ (n) i=1 -(log(1 - X (n) i + 1 Y (n) i-1 ) + X (n) i + 1 Y (n) i-1 ) P -→ 0,
when n → ∞.

Proof. Let 0 ≤ t < α -1. First of all, remark that:

(22) Y (n) ⌊nt⌋ n P -→ α -1 -t α -1 .
Indeed,

Y (n) ⌊nt⌋ n = n -(⌊nt⌋)/(α -1) n - ⌊nt⌋ k=1 (X (n) k -1/(α -1)) n ,
and we conclude using ( 14) and the convergence of P(τ (n) > ⌊nt⌋) to 1. Let us write

|I

(2)

nt | = A n + B n , with A n = |I (2) nt |1 Y (n) ⌊nt⌋ <(1-t/(α-1))n/2 and B n = |I (2) nt |1 Y (n) ⌊nt⌋ ≥(1-t/(α-1)
)n/2 . The convergence [START_REF] Hedgecock | Does variance in reproductive success limit effective population sizes of marine organisms?[END_REF] implies that A n tends to 0 in probability. To prove the convergence of B n , let us first notice that for a ∈ (0, 1), there exists a constant C(a) such that, if B n,x is a binomial r.v. with parameter (n, x), then [START_REF] Iksanov | On the number of jumps of random walks with a barrier[END_REF] 0 < -

1 0 E 1 {2≤Bn,x≤(1-a)n} (log(1 - B n,x n ) + B n,x n ) ν(dx) ≤ C(a).
Indeed, there exists a constant C ′ (a) such that for u ∈ (0, 1-a), 0 7), ( 23) and ( 6), we get that

< -ln(1-u)-u ≤ C ′ (a)u 2 . Hence, 0 < - 1 0 E 1 {2≤Bn,x≤(1-a)n} (log(1 - B n,x n ) + B n,x n ) ν(dx) ≤ C ′ (a) 1 0 E 1 {2≤Bn,x≤(1-a)n} ( B n,x n ) 2 ν(dx) ≤ C ′ (a) 1 0 E ( B n,x n ) 2 ν(dx) ≤ 2C ′ (a) 1 0 E B n,x (B n,x -1) n 2 ν(dx) = 2C ′ (a) 1 0 n(n -1)x 2 ν(dx) n 2 ≤ 2C ′ (a)Λ([0, 1]) =: C(a). Let us set a = (1 -t/(α -1))/2. Hence B n = |I (2) nt |1 Y (n) ⌊nt⌋ ≥an . Notice that if n is large enough such that an ≥ 2, then if Y (n) ⌊nt⌋ ≥ an we have τ (n) > nt. Moreover, if Y (n) ⌊nt⌋ ≥ an, for i ≤ nt, we have Y (n) i ≥ an ≥ aY (n) i-1 and X (n) i = Y (n) i-1 -Y (n) i ≤ (1 -a)Y (n) i-1 < (1 -a/2)Y (n) i-1 . Using (
E[B n ] ≤ ⌊nt⌋ i=1 E[E[-(log(1 - X (n) i + 1 Y (n) i-1 ) + X (n) i + 1 Y (n) i-1 )1 {1≤X (n) i ≤(1-a)Y (n) i-1 } 1 {Y (n) i-1 ≥an} |Y (n) i-1 ]] ≤ ⌊nt⌋ i=1 E[-E[ 1 0 1 {2≤B Y (n) i-1 ,x ≤(1-a/2)Y (n) i-1 } 1 {Y (n) i-1 ≥an} 1 g Y (n) i-1 (log(1 - B Y (n) i-1 ,x Y (n) i-1 ) + B Y (n) i-1 ,x Y (n) i-1 )ν(dx)|Y (n) i-1 ]] ≤ C(a/2)nt g an → 0,
when n tends to ∞. This achieves the proof of the Lemma.

A result on small-time behavior of the block process

We now turn to the study of the length of an external branch picked at random, denoted by T (n) . For any integer k between 1 and τ (n) , define A (n) k as the time when the kth jump is achieved. This variable can be expressed as a sum of k independent exponential random variables. More precisely,

A (n) k = k∧τ (n) i=1 e i g Y (n) i-1 ,
where the e i 's are independent standard exponential variables. Notice that

T (n) = A (n) σ (n) .
We will first study asymptotics of A (n) k . For this, we use a two-step approximation method close to Section 4 of [START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF]. Define first

Ã(n) k = k∧τ (n) i=1 1 g Y (n) i-1
, obtained replacing the e i 's by their mean,and

Â(n) k = 1 C 0 Γ(2 -α) k∧τ (n) i=1 (Y (n) i-1 ) -α ,
obtained replacing g b by its equivalent in [START_REF] Bertoin | Stochastic flows associated to coalescent processes. III. Limit theorems[END_REF]. Proposition 4.1. We assume that ρ(t) = C 0 t -α +O(t -α+ζ ) for some C 0 > 0 and ζ > 1-1/α. Then, for any t < α -1, we have

n α-1 A (n) ⌊nt⌋ P → 1 C 0 Γ(2 -α) ((1 - t α -1 ) 1-α -1), when n → ∞.
The proof is a straight consequence of Lemma 3.3 with η = α and the following Lemmas 4.2 and 4.3.

Lemma 4.2. Under the assumptions of Proposition 4.1, we have

n α-1 ( Ã(n) ⌊nt⌋ - Â(n) ⌊nt⌋ ) P → 0,
when n → ∞.

Proof. Use (6) to get

Ã(n) ⌊nt⌋ - Â(n) ⌊nt⌋ = ⌊nt⌋∧τ (n) i=1 Y (n) i-1 -α O Y (n) i-1 -min(ζ,1)
.

The result then follows from Lemma 3.3 with η = α + min(ζ, 1).

Lemma 4.3. Under the assumptions of Proposition 4.1, we have

n α-1 (A (n) ⌊nt⌋ - Ã(n) ⌊nt⌋) ) P → 0, when n → ∞.
Proof. Recall that Y = (Y k , k ≥ 0) denotes the filtration generated by Y . Conditionally on Y, the random variables

e i -1 g Y (n) i-1
are independent with zero mean. We deduce that

E sup t≥0 (n α-1 (A (n) ⌊nt⌋ - Ã(n) ⌊nt⌋ )) 2 |Y = n 2α-2 E   sup t≥0   ⌊nt⌋∧τ (n) i=1 e i -1 g Y (n) i-1   2 |Y    ≤ 4n 2α-2 ⌊nt⌋∧τ (n) i=1   1 g Y (n) i-1   2 ,
where we used Doob's inequality for the inequality. Thanks to (6) and Lemma 3.3 with

η = 2α, we get the 4n 2α-2 ⌊nt⌋∧τ (n) i=1 1 g Y (n) i-1 2
converges to 0 in probability.

Heuristically, combining Theorem 3.1 and Proposition 4.1, we should get that n α-1

T (n) = n α-1 A (n) σ (n) converges in law to 1 C 0 Γ(2-α) ((1 -σ) 1-α -1)
. This line of proof will be followed in the last section. However, in the next section we will first present another way to prove this result with a method based on the consistency property of exchangeable coalescents. As a first step to this approach, we end this session with a result about small-time behavior of the block-counting process.

Let R

(n) t denote the number of blocks of the n-coalescent Π (n) at time t. The initial value R (n) 0 is n. We show that the limit law of the process R (n) is deterministic under a certain time rescaling Theorem 4.4. We assume that ρ(t) = C 0 t -α + O(t -α+ζ ) for some C 0 > 0 and ζ > 1 -1/α. For any t 0 > 0, ε > 0 , we have [START_REF] Janson | The external lengths in kingman's coalescent[END_REF] P( sup

0≤t≤t 0 |n -1 R (n) tn 1-α -(1 + C 0 Γ(2 -α)t) -1/(α-1) | > ε) → 0,
when n → ∞.

Proof. Let 0 < r < α -1, we have the following relation :

R (n) A (n) ⌊nr⌋ = Y (n) ⌊nr⌋ = n - ⌊nr⌋∧τ (n) j=1 X (n) j
Let t ∈ [0, t 0 ], and define ( 25)

r(t) = (α -1)(1 -(1 + C 0 Γ(2 -α)t) -1/(α-1) ), on [0, t 0 ]. Notice that 1 C 0 Γ(2 -α) ((1 - r(t) α -1 ) 1-α -1) = t.
Then thanks to Proposition 4.1, n α-1 A

(n) ⌊nr(t)⌋ converges in probability to t. Using the remark at the beginning of the proof in Lemma 3.5, we get the convergence

n -1 R (n) A (n) ⌊nr(t)⌋ = Y (n) ⌊nr(t)⌋ n P → (1 - r(t) α -1 ) = (1 + C 0 -α)t) -1/(α-1) , when n → ∞. Moreover, since R (n) t
is decreasing, then for any 0 < δ < 1,

lim n→∞ P(R (n) A (n) ⌊nr(t-δt)⌋ ≤ R (n) tn 1-α+1 ≤ R (n) A (n) ⌊nr(t+δt)⌋ ) = 1.
The constant δ being arbitrary, we thus obtain the convergence in probability of n 1) . We obtain (24) using again the fact that R (n) is a decreasing process.

-1 R (n) tn 1-α to (1 + C 0 Γ(2 -α)t) -1/(α-
In fact, the asymptotic result concerning block counting process of Kingman's coalescent is also valid. The method is almost identical to that employed in the above Theorem. In the context of Kingman's coalescent, we use the same notations Π (n) , A

i , R (n) . Theorem 4.5. In the setting of the Kingman's coalescent, for any t 0 > 0, ε > 0 , we have [START_REF] Kingman | The coalescent[END_REF] P( sup

0≤t≤t 0 |n -1 R (n) tn -1 -(1 + t/2) -1 | > ε) → 0 when n → ∞.
Remark that this Theorem shows a nice continuity from Beta-coalescent(the process that we consider is more general but contains Beta-coalescent) to Kingman's coalescent., setting α = 2 in Theorem 4.4.

Proof. Recall that A (n) i
is the time when ith jump is achieved. When Π (n) has b individuals at some time t, then the process encounters the following coalescence at rate b 2 where two randomly chosen individuals will be coalesced. Π (n) remains 1 when all individuals are coalesced.

For 0 < t < 1, we have

A (n) ⌊nt⌋ = n k=n-⌊nt⌋+1 e k k 2
where e i s are i.i.d unit exponential variables. Notice that

E[nA (n) ⌊nt⌋ ] = n k=n-⌊nt⌋+1 n k 2 = 2( 1 n -⌊nt⌋ - 1 n )n → 2( 1 1 -t -1),
as n tends to ∞. There exist a constant K > 0, such that, Var(nA

(n) ⌊nt⌋ ) = n k=n-⌊nt⌋+1 n 2 ( 1 k 2 ) 2 ≤ K n .
So we deduce that

nA (n) ⌊nt⌋ L 2 → 2( 1 1 -t -1) = 2t 1 -t := f (t)
as n converges to ∞. We denote by f -1 (t) := t/(t + 2) the inverse function of f (t).

Similarly, R (n) is decreasing, so

P(R (n) A ⌊nf -1 (t-δ) (n) ⌋ ≤ R (n) tn -1 ≤ R (n) A ⌊nf -1 (t+δ) (n) ⌋ ) → 1,
as n tends to ∞ for any 0 < δ < t.

So

R (n) tn -1 n - R (n) A (n) ⌊nf -1 (t)⌋ n d → 0. Furthermore, R (n) A (n) nf -1 (t) n = Y (n) ⌊nf -1 (t)⌋ n = n -⌊nf -1 (t)⌋ n → 1 -f -1 (t) = 1 1 + t/2 , as n tends to ∞. So R (n) tn -1 n d → 1 1+t/2
. Using again the decreasing property of R (n) t , we finish the proof.

The length of an external branch picked at random

Dynamics of any exchangeable coalescent with multiple mergers are characterized by rates λ b,k which suit a consistent relationship (this is Pitman's structure theorem, see [START_REF] Pitman | Coalescents with multiple collisions[END_REF], Lemma 18): [START_REF] Kingman | On the genealogy of large populations[END_REF] λ b,k = λ b+1,k+1 + λ b+1,k .

This relationship comes from the fact that k given merging blocks among b can coalesce in two ways while revealing an extra block : either the coalescence event implies the extra block (and then k + 1 blocks will merge) or not. Thus we get a recursive construction of the n-coalescent process Π (n) . Let us define Π (n,2) as the coalescent process of individuals labelled from 2 to n. Now we consider the individual labelled by 1. The lineage of this individual can be 'connected' to Π (n,2) • either at any of its jump times, in which case block {1} participates to a multiple merger implying at least 3 blocks, and we call this collision "Type 1" (see Figure 1), • or at any other time to one of the present blocks and then participates to a binary collision, and we call it "Type 2" (see Figure 2). From now on, our analysis is conditional on Π (n,2) . Between two jump times of Π (n,2) , assuming that there are b blocks in Π (n,2) , the extra block coalesces at rate bλ b+1,2 . If the extra block remains unconnected just before a coalescence event implying k blocks among b, then it will participate to this event with probability (28)

1 0 x k+1 (1 -x) b-k ν(dx)/g b 1 0 x k (1 -x) b-k ν(dx)/g b = 1 - λ b+1,k λ b,k .
This equality comes from [START_REF] Kingman | On the genealogy of large populations[END_REF]. Let us see how to get the law of T (n) , the coalescence time of individual 1. We define by R (n,2) the block counting process of Π (n,2) . Notice that it has the same law as R (n-1) . We introduce

• T (n) c
the first jump time of a Poisson process η

(n) c
directed by the measure ν

(n) c = R (n,2) t λ R (n,2) t +1,2 dt; • T (n) d
the time of the first appearance of 'Head' in the following coin flip, independent of η (n) c : at each jump time t of R (n,2) t , we toss a coin, and get 'Head' with probability

1 - λ R (n,2) t- +1,R (n,2) t- -R (n,2) t +1 λ R (n,2) t-,R (n,2) t--R (n,2) t +1
and 'Tail' with probability

λ R (n,2) t- +1,R (n,2) t- -R (n,2) t +1 λ R (n,2) t-,R (n,2) t--R (n,2) t +1
(see [START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF]).

Then, conditionally on Π (n,2) , T (n) and T

(n) c ∧ T (n) d
have the same law.

Remark 5.1. A more formal way to interpret T (n) is as follow. Let ξ (n) be Cox process directed by random measure ν

(n) c + ν (n) d , where ν (n) d = {t is a jump time} λ R (n,2) t-+1,R (n,2) t--R (n,2) t +1 λ R (n,2) t-,R (n,2) t--R (n,2) t +1
δ t , and δ t is the Dirac measure in t (see [25, p.226]). Then T (n) has the same law as the first jump time of ξ (n) Let us now give our main result Theorem 5.2. The following convergence holds :

n α-1 T (n) d → T = 1 C 0 Γ(2 -α) ((1 -σ) 1-α -1),
for n → ∞. The density function of T is

f T (t) = αC 0 Γ(2 -α) α -1 (1 + C 0 Γ(2 -α)t) -α α-1 -1 , t ≥ 0.
In particular, in the Beta(2 -α, α) case, the density is

f T (t) = 1 (α -1)Γ(α) (1 + t αΓ(α) ) -α α-1 -1 , t ≥ 0.
Proof. For the sake of simplicity, we will make the proof only in the Beta(2 -α, α) case. The proof can be extended to the more general case where ( 5) is satisfied with the details omitted here. In this special case, C 0 = (αΓ(α)Γ(2 -α)) -1 and dynamics are given by

λ b,k = B(k -α, b -k + α) B(α, 2 -α) , where
= n -1. Let X (n,2) i = Y (n,2) i-1 -Y (n,2) i
be the number of blocks we lose during the ith coalescent event.

We write X (n) 0 = 0. Notice that (Y (n,2) , X (n,2) ) has the same law as ((Y (n-1) , X (n-1) )) . Using the description given above, we have

P(n α-1 T (n) > t) = E[P(n α-1 (T (n) c ∧ T (n) d ) > t|Π (n,2) )] = E[P(n α-1 T (n) c > t|Π (n,2) )P(n α-1 T (n) d > t|Π (n,2) )] = E[exp(- t 0 1 0 n 1-α R (n,2) sn 1-α x 2 (1 -x) R (n,2) sn 1-α -1 ν(dx)ds) r (n) t i=1 λ 1+Y (n,2) i-1 ,1+X (n,2) i λ Y (n,2) i-1 ,1+X (n,2) i ] = E[exp(- t 0 n 1-α R (n,2) sn 1-α B(2 -α, R (n,2) sn 1-α + α -1) B(2 -α, α) ds) r (n) t i=1 Y (n,2) i-1 -X (n,2) i + α -1 Y (n,2) i-1
].

We decompose the term in the expectation into two parts: the exponential on one side and the product on the other.

Let us first look at the exponential term. Using Stirling's formula we get that, for 0 ≤ s ≤ t,

n 1-α R (n,2) sn 1-α B(2 -α, R (n,2) sn 1-α + α -1) B(2 -α, α) = n 1-α (R (n,2) sn 1-α ) α-1 Γ(α) + ( R (n,2) sn 1-α n ) α-1 f (R (n,2) sn 1-α ),
where f = f (t) {t≥0} is a deterministic function which converges to 0 as t converges to ∞.

The sequence (R (n,2) sn 1-α , n ≥ 2) is decreasing so, thanks to Theorem 4.4, we deduce that sup 0≤s≤t (

R (n,2) sn 1-α n ) α-1 f (R (n,2)
sn 1-α ) converges in probability to 0 as n tends to ∞. Consequently, using again Theorem 4.4, we get that [START_REF] Pitman | Coalescents with multiple collisions[END_REF] exp(-

t 0 n 1-α R (n,2) sn 1-α B(2 -α, R (n,2) sn 1-α + α -1) B(2 -α, α) ds) P → (1 + t αΓ(α) ) -α , n → ∞.
Convergence of the product term is obtained by the same method as in proof of Theorem 3.1, combined with the convergence in [START_REF] Möhle | Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent[END_REF]. To avoid showing almost the same reasoning, we leave the details to readers. This way, we have ( 31)

r (n,2) t i=1 Y (n,2) i-1 -X (n,2) i + α -1 Y (n,2) i-1 P → (1 + t αΓ(α) ) -α(2-α) α-1 , n → ∞.
The product of ( 30) and ( 31) then converges in probability to (1 + t αΓ(α) ) -α/(α-1) . Since this product is bounded, we get that

P(n α-1 T (n) > t) → (1 + t αΓ(α) ) -α α-1 , n → ∞.
We achieve the proof.

As a consequence of Theorem 5.2, we can get an asymptotic result on the size of the population at the moment of collision of individual 1.

Corollary 5.3. The following convergence holds :

n -1 Y (n) σ (n) d → (1 + C 0 Γ(2 -α)T ) -1/(α-1) = 1 -σ
for n → ∞. Moreover, the density function of this limit is αx α-1 1 {0≤x≤1} .

Proof. In terms of block counting process, we have Y

(n) σ (n) = R (n) T (n) . Notice that R (n) T (n) = R (n) n 1-α (n α-1 T (n) )
. Using Theorem 5.2, we known that n α-1 T (n) converges in distribution to T . Hence, if t 0 > 0, we deduce from Theorem 4.4 that

1 {n α-1 T (n) <t 0 } R (n) n 1-α (n α-1 T (n) ) n d → 1 {T <t 0 } (1 + C 0 Γ(2 -α)T ) -1/(α-1) .
This achieves the proof.

6. An alternative proof for Theorem 5.2

In this section, we present an alternative proof for Theorem 5.2 using the convergence results for σ (n) from Theorem 3.1. First, we need a stronger version of Proposition 4.1 which gives weak convergence in the path space. Recall that

A (n) k = k∧τ (n) i=1 e i g Y (n) i-1 ,
where the e i 's are independent standard exponential variables. Proposition 6.1. We assume that ρ(t) = C 0 t -α +O(t -α+ζ ) for some C 0 > 0 and ζ > 1-1/α. Then, for any t < α -1, we have [START_REF] Schweinsberg | Coalescent processes obtained from supercritical Galton-Watson processes[END_REF] (n α-1 A

(n) ⌊ns⌋ ) s≤t d → ( 1 C 0 Γ(2 -α) ((1 - s α -1 ) 1-α -1)) s≤t ,
in the sense of convergence in the path space D[0, t] for n → ∞.

Proof. Note that Theorem 4.1 states

n α-1 A (n) ⌊ns⌋ P → ( 1 C 0 Γ(2 -α) ((1 - s α -1
) 1-α -1)), for 0 < s < α -1 and n → ∞. So for every fixed s ∈ [0, t], we have pointwise convergence in probability in [START_REF] Schweinsberg | Coalescent processes obtained from supercritical Galton-Watson processes[END_REF]. This implies weak convergence of all finite dimensional distributions due to the subsequence criterion for weak convergence. In order to show weak convergence in the path space, we will show tightness for the distributions from [START_REF] Schweinsberg | Coalescent processes obtained from supercritical Galton-Watson processes[END_REF]. Since the limit process is continuous, it suffices to show that the condition (i) of [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 7.3] and condition (7.12) from [START_REF] Billingsley | Convergence of probability measures[END_REF]Corollary 7.4] are fulfilled (see [START_REF] Billingsley | Convergence of probability measures[END_REF]Corollary 13.4]). For the present processes, these conditions translate to showing that for every ǫ > 0 and η > 0, (i) there exists a > 0 s.t. P (n α-1 A

(n) ⌊0⌋ ≥ a) ≤ η for n big enough and (ii) there exists a 0 < δ < 1 so that δ -1 P n α-1 (A (n) ⌊n•min(t 1 +δ,t)⌋ -A 1 -s α-1 ) 1-α -1). Note that P (f (min(t 1 + δ, t)) -f (t 1 ) ≥ ǫ) ≤ P (f (t) -f (t -δ) ≥ ǫ) ∈ {0, 1}. Since f is continuous, you can now choose δ small enough that f (t) -f (t -δ) < ǫ and then n big enough to fulfill (ii). Thus, we have shown tightness of the distributions in [START_REF] Schweinsberg | Coalescent processes obtained from supercritical Galton-Watson processes[END_REF] which establishes the desired weak convergence Now we come to the alternative proof of Theorem 5.2.

Alternative proof of Theorem 5.2. Fix t ∈ [0, α -1). We have (σ (n) /(n(α -1)), (n α-1 A

(n) ⌊ns⌋ ) s≤t ) d → (σ, ( 1
C 0 Γ(2 -α) ((1 - s α -1 ) 1-α -1)) s≤t ),
for n → ∞. Due to Skorohod-coupling, we can assume that this convergence also holds almost surely. Since s → (

1 C 0 Γ(2-α) ((1 -s α-1 ) 1-α -1)
) is continuous on [0, t], the almost sure convergence of (n α-1 A (n) ⌊nt⌋ ) s≤t in D[0, t] is even almost sure uniform convergence on [0, t] (see [7, p. 124]). For any series (x n ) n∈N on [0,t] with x n → x, we thus have

n α-1 A (n) ⌊nxn⌋ → 1 C 0 Γ(2 -α) ((1 - x α -1 ) 1-α -1)
almost surely for n → ∞. The only problem left is that σ (n) , σ may take values in [0, α -1) and not only in some subset [0, t]. To remedy this, note that if we restrict all random variables on {σ ≤ α -1 -2 k } for k ∈ N, we have σ (n) (ω)/n ≤ α -1 -1 k for n = n(ω) big enough for almost all ω ∈ {σ ≤ α -1 -1 k }. Thus, by using the Skorohod-coupling for the series (σ (n) /(n(α -1)), (n α-1 A (n) ⌊ns⌋ ) s≤α-1-1 k ), we have

n α-1 A (n) ⌊n•σ (n) /n⌋ d → 1 C 0 Γ(2 -α) ((1 - σ α -1
) 1-α -1), almost surely on {σ ≤ α -1 -2 k } for the coupled versions of these random variables (note that σ (n) ≤ τ (n) ). Since σ is Beta-distributed, we have, for k → ∞,

P ({n α-1 A (n) ⌊n•σ (n) /n⌋ ∈ •} ∩ {σ ≤ α -1 - 2 k }) ∼ P (n α-1 A (n) ⌊n•σ (n) /n⌋ ∈ •). This shows n α-1 T (n) = n α-1 A (n) σ (n) d → 1 C 0 Γ(2 -α) ((1 - σ α -1
) 1-α -1).

The only thing left to prove is that T has density f T . This is done by computing the distribution function

P( 1 C 0 Γ(2 -α) ((1 - σ α -1 ) 1-α -1) ≤ t) = P(σ ≤ 1 -(1 + C 0 Γ(2 -α)t) 1 α-1 ) = 1-(1+C 0 Γ(2-α)t) 1 α-1 0 α(1 -x) α dx = 1 -(1 + C 0 Γ(2 -α)t) -α α-1 .
and finally by differentiating.

  , k ≥ 1) is a bounded martingale w.r.t. the filtration Y. Notice that E[M (a) n,k ] = 1. As X n i = 0 and Z (n) i

Figure 1 . n = 5 .

 15 Figure 1. n = 5. Individual 1 is chosen. Type 1: individual 1 encounters a multiple collision.

Figure 2 .

 2 Figure 2. n = 5. Individual 1 is chosen. Type 2: individual 1 encounters a binary collision

1 C

 1 1 ⌋ ) ≥ ǫ ≤ η,for n big enough and any t 1 ∈ [0, t]. Condition (i) is trivially fulfilled, for condition (ii) we can use Theorem 4.1 to show that for n → ∞,P n α-1 (A (n) ⌊n•min(t 1 +δ,t)⌋ -A (n) ⌊n•t 1 ⌋ ) ≥ ǫ → P (f (min(t 1 + δ, t)) -f (t 1 ) ≥ ǫ),where f (s) := 0 Γ(2-α) ((

  B(a, b) is a Beta function of parameters a and b.

	Define r t (n,2) straightforward consequence of Proposition 4.1 that as the number of jumps of the process Π (n,2) up to time n 1-α t. It is a
	(29)	r	(n,2) t n
	(n,2) i i jumps which equals 1 from the time all individuals are coalesced to 1, and Y the number of blocks remaining after (n,2) 0

P

→ r(t), n → ∞ for t → ∞, where r(t) is defined in

[START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]

.

For i ≥ 0, in the process Π

(n,2)

, we denote by Y

Acknowledgments. Jean-Stéphane Dhersin and Linglong Yuan benefited from the support of the "Agence Nationale de la Recherche": ANR MANEGE (ANR-09-BLAN-0215). Arno Siri-Jégousse and Fabian Freund would like to thank the Haussdorf Research Institute for Mathematics for its support. Fabian Freund would like to thank Luca Ferretti for a discussion concerning the interpretation of σ (n) .


Gilles Olive
email: gilles.olive@excite.com

Xavier Rozanska

Wilfred Smulders

Alain Jacques

Anton German

Control of the Radical Polymerization by 2,2,15,15-tetramethyl-1aza-4,7,10,13-tetraoxacyclopentadecan-1-oxyl and its Sodium Salt

Keywords: crownether, ESR/EPR, nitroxide, polystyrene, radical polymerization

The control of the radical polymerization of styrene by 2,2,15,15-tetramethyl-1aza-4,7,10,13-tetraoxacyclopentadecan-1-oxyl is reported here in bulk at 90 °C, 120 °C and in miniemulsion. Similarly the control by its sodium complex is reported in bulk at 90 °C.

Introduction

Thanks to its interesting properties, the conventional free radical polymerization is applied in order to produce many commercial polymers. Among its advantages are the possibility to polymerize a wide range of monomers [START_REF] Moad | The chemistry of Free Radical Polymerization[END_REF] (e. g. acrylates, acrylamides, styrenes and vinyls), the compatibility with many solvents and functional groups [START_REF] Moad | The chemistry of Free Radical Polymerization[END_REF] (e. g. NR 2 , COOH, OH and NCO) and convenient conditions under which this reaction can be performed. [START_REF] Moad | The chemistry of Free Radical Polymerization[END_REF] The temperature range extends from 0 to 150 °C in bulk, in solution, in suspension or in emulsion. Although a radical polymerization requires oxygen free conditions, it can be carried out in water. [START_REF] Matyjaszewski | [END_REF]3] However, its structure control is less efficient when compared to ionic polymerization in terms of molecular weight distribution [START_REF] Moad | The chemistry of Free Radical Polymerization[END_REF] , end-group functionalities and chain branching [START_REF] Matyjaszewski | [END_REF]4] . This caveat is related to the reaction mechanism in itself. In consequence to these difficulties, a polydispersity (I = M w /M n ) lower than 1.5 can hardly be reached. [5] Since the 1980s, several techniques were developed in order to alleviate this drawback. All these enhancements rely on obtaining dormant species in equilibrium with an active form from propagating chain radicals (Scheme 1).

Insert Scheme 1

Nowadays, five methods using a reversible termination or a reversible chain transfer as depicted in Scheme 1 have been designed. [6] They have been named "living" or "controlled" free radical polymerization processes. They involve nitroxide [5,7] , cobalt/porphyrin complexes [8] , or copper(I)/bipyridine complexes [9,10] ( (reversible termination, equation (1) in Scheme 1). As an alternative, an "iodine transfer" [11][12][13] , a reversible addition-fragmentation chain transfer (RAFT) [14][15][16] ,

, o or r a a macromolecular design via interchange of xanthate (MADIX) [17] can be applied ( (reversible chain transfer, equation ( 2) in Scheme 1).

Others techniques do exist but they are seldom used or have only been recently

devised. [18][19][20][21][22][23][24] Insert Scheme 2 2,2,6,6-tetramethyl piperidinyl-1-oxy 1 (TEMPO) is one of the most commonly used compound in a living free radical polymerization process based upon nitroxides (Scheme 2). Similarly, ditertiobutylnitroxide 2 has recently been applied (Scheme 2) and its use in radical processes is growing. [25,26] Two severe drawbacks of TEMPO namely are a long induction time (around 10 h with styrene at 120 °C) [5] and a working temperature above 120 °C, preventing a polymerization to be carried out in aqueous dispersed systems. [3] However, a polymerization with TEMPO can be obtained under high pressure in an aqueous medium. Unfortunately some autopolymerization of styrene cannot be neglected at such a high temperature, leading to an increase of the polydispersity. [3,27] In order to avoid the limitations associated with 1, several groups have synthezised and tested nitroxides allowing lower temperature conditions. [START_REF] Matyjaszewski | [END_REF]4,25,26,[28][START_REF] Bon | DEBUT Collected Studies on Nitroxide-Mediated Controlled Radical Polymerization[END_REF][START_REF] Chong | [END_REF][31][32][33][34] Hereunder we will report the styrene polymerization in bulk at 90 and at 120 °C and in miniemulsion at 90 °C in the presence of 2,2,15,15-tetramethyl-1-aza-4,7,10,13-tetraoxacyclopentadecan-1-oxyl 3 (Scheme 3) and in the presence of its sodium complex. We will focus on the linear evolution of M n as a function of the conversion. Styrene was used as the reference monomer and all data were compared with TEMPO as the counter-radical.

Experimental part

Materials

Styrene (STY) was purchased from Aldrich and stored at -4 °C. The included 4tert-butylcatechol inhibitor was removed either by using an inhibitor-remover packaging (from Aldrich) or by styrene distillation under reduced pressure.

Benzoyl peroxide (BPO), TEMPO, hexadecane were used as purchased from Aldrich, as well as all the other compounds involved in the synthesis of 3, 3Li and 3Na. All solvents were purchased from Biosolve. Dowfax 8390, an anionic surfactant, was purchased from Dow Chemical Company. Chromatographic separations were performed on Aldrich Silicagel 60 (230-400 mesh) for column chromatography. Bulk Na: m BPO = 0.041 g, m 3Na = 0.100 g, m STY = 3.334 g.

Measurements

TEMPO 90: m BPO = 0.658 g, m TEMPO = 0.378 g, m STY = 50.016 g.

MINIEMULSION

The miniemulsion was prepared as follows: the monomer, the nitroxide, the BPO and the hexadecane were poured into water and a surfactant. They were mixed by using a high shear mixer (Ystral X1020) for 5 minutes at room temperature. An

Results and discussion

Synthesis and ESR

The synthesis of 3 has already been described elsewhere. [35,36] It is summarized in Scheme 3.

Insert Scheme 3

The synthesis of 4 has been carried out according to ref. [37] followed by a reduction by AlLiH 4 . The oxidation of 5 was difficult to achieve and several experiments had to be performed. A 30 % aqueous hydrogen peroxide solution with some sodium tungstate as a catalyst only gives a yield of 11 % after 40 h. It reached 26 % after 70 hours. The reaction conditions have been kept constant at room temperature. Over-oxidation occurred when the reaction involved meta chloroperbenzoic acid (m-CPBA). Ultimately, a 70 % yield was obtained by means of a proportion modified Brik procedure [38] and a flash chromatography with dichloromethane/ethanol 950/50 V/V as eluent. The Brik procedure is a biphasic in situ generation of dimethyldioxirane by oxone (KHSO 5 ) over acetone.

The alkoxyamine of styrene has not been successfully obtained using the Miura's procedure. [39] The metal complex synthesis has been carried out according to ref. [35] by using MBPh 4 (M = Li + , Na + , K + ). In every case, a good yield has been reached with the exception of the potassium complex. The ESR experiments are summarized in Table 1. It gives the respective hyperfine coupling constants and the g-factors for

3, 3Li and 3Na in several solvents.

Insert Table 1 All the spectra have been simulated with an EPR software from Duling et al. [40,41] It was impossible for 3 to find a correlation between the E(T)N of the solvent and the a N or the g-factor. The spectra of 3Li and 3Na (Figure 1) show triplets of quadruplets with a a N ≈ 15.8 G and a Metal ≈ 2.5 G. Our goal is to obtain the highest value for a N ; it implies that the B mesomeric form (Scheme 4) is favoured.

Considering that the lithium and sodium salts are innocuous, their ecological impact is low. The oversized potassium does not enter into the crown ; therefore only the spectrum of 3 could be observed (Figure 1).

Insert Figure 1 Insert Scheme 4

The UV spectra of 3 in styrene have been recorded for several concentrations. 3

shows two absorption peaks, viz. at 313 nm (ε = 46 mol -1 .dm 3 .cm -1 ) and at 417 nm (ε = 4 mol -1 .dm 3 .cm -1 ).

Polymerization

Three polymerization processes for 3 have been studied : in bulk at 90 °C (Bulk 90 in Figure 2-4 and Table 2), in bulk at 120 °C (Bulk 120) and in miniemulsion at 90 °C (Miniemulsion). The polymerization of 3Na was only performed in bulk at 90 °C (Bulk Na). After removing the inhibitor from the monomer by a distillation, the BPO initiator was added with an initial concentration around 45 mM. In all the experiments, a Nitroxide to Initiator ratio of 1.1 was used. Table 2 summarizes our results. They are compared to data obtained by Georges et al. [5] for TEMPO at 120 °C (TEMPO 120) and to our experiments for TEMPO at 90 °C (TEMPO 90).

Insert Table 2 Insert Figure 2 Figure 2 shows the evolution of the conversion versus time. Disregarding the reaction conditions, the rate of 3 is faster than TEMPO at 120 °C. The reaction rate follows the order Bulk 120 > Miniemulsion > Bulk 90. In the case of TEMPO at 120 °C, an inhibition time of 10 hours was observed before normal polymerization occurred. The conversion at 90 °C with the same nitroxide was only 4 % after 51 hours

Bulk 90

The variation of M n versus Conversion (Figure 3) is not entirely linear ; the presence of thermal radicals certainly accounts for this curvature. The experimental M n reproduced in Table 2 are significantly higher than the theoretical values showing that the number of chains is lower than what is expected.

Insert Figure 3 The Insert Figure 4 However, the increasing molecular weights and the relatively low polydispersities suggest that this process is a living polymerization.

Bulk 120

At this temperature, the BPO decomposition is swift. As expected, a faster polymerization rate is observed at the beginning of the reaction (Figure 2). Later on, it is probably controlled by thermal initiation or will depend on the equilibrium constant [42] ; this explains why the straight line does not pass through zero in the ln([M] 0 /[M]) versus Time graphic (Figure 4).

In this case, experimentally measured Mn (Mn Exp.) is lower than simulated Mn (Mn calc.) at the end of the reaction (Table 2). At lower conversions, the nitroxide deactivation causes a higher molecular weight than what is theoretically expected.

However, at the end, this effect is compensated by a thermal initiation and a chain transfer to the monomer ; therefore lower molecular weights are obtained. This behaviour also accounts for the very high polydispersity.

Miniemulsion

When the miniemulsion results are compared to Bulk 90, a higher polymerization rate, higher molecular weights and higher polydispersities are observed. This increased rate may be the result of a compartmentalisation in the emulsion polymerization, leading to a less bimolecular termination. The higher molecular weights and polydispersities suggest more nitroxides side reactions or a nitroxides partitioning to the aqueous phase. Furthermore the latter may also explain the higher polymerization rate due to a higher fraction of propagating chains inside the particles.

Bulk Na

The general behaviour of 3Na is close to TEMPO. When using 3Na as the counter radical, the reaction has an inhibition period of 7.5 hours and a slower rate than 3.

A 75% conversion is the maximum yield reached by 3Na.

Conclusion

By analyzing Figure 3, it can be concluded that there is some control with 3 and 3Na because M n linearly increases with conversion and because the polydispersities are not very high. Ultimately, the results should be better if more ideal experimental conditions were met, as for instance with another initiator or a different initiator/nitroxide ratio. These enhancements are still under evaluation in our laboratory. kd is s o s .

Captions

(1)

Reversible chain transfer

  Gel permeation chromatography (GPC) experiments were achieved in THF solution (from Biosolve, stabilized with BHT) at 22 °C with a flow rate of 1.0 cm 3 .min -1 by using a WATERS Model 510 pump, a Model 410 refractive index detector (at 40 °C) and a Model 486 UV detector (at 254 nm). 50 µl injections were performed by means of a WATERS Model WISP 712 autoinjector in a low molecular weight set of columns i.e. a PLgel guard (5 µm particles) 50*7.5 mm precolumn followed by two successive PLgel columns of 50 nm (5 µm particles) and 10 nm (5 µm particles). Polystyrene samples from Polymer Laboratories (M = 580 to M = 7.1*10 6 ) were the calibration standards. Data acquisition and processing were performed by means of the WATERS Millennium32 (v3.05) software. The conversion was measured by gravimetry. The ESR spectra were recorded at room temperature on a Bruker ESP 300E spectrometer fitted with a Xband resonator (9.41 GHz), a Bruker ER035M NMR gaussmeter and an HP 8535B microwave frequency counter. The signal was detected at a 100 kHz magnetic field modulation. The UV spectra were acquired on an HP 8453 UV-Visible spectrophotometer with an HP 89090A heating unit. Polymerization BULK In 50 ml flasks, styrene, nitroxide and BPO were mixed as described hereunder. The mixture was thoroughly degassed for 30 minutes by argon bubbling and then dipped into an oil bath heated at the appropriate temperature. Aliquots were picked at various time intervals and stored over hydroquinone. They were characterised by gel permeation chromatography. The conversion was measured by gravimetry. Bulk 90: m BPO = 0.256 g, m 3 = 0.280 g, m STY = 20.005 g. Bulk 120: m BPO = 0.260 g, m 3 = 0.281 g, m STY = 20.015 g.

  homogenous emulsion was obtained at room temperature by means of a sonifier (Dr. Hielscher UP 400S) operated for 5 minutes at 50 % duty, on power 5 and on pulsed mode. Miniemulsion: m BPO = 0.123 g, m 3 = 0.136 g, m STY = 10.040 g, m hexadecane = 0.505 g, m Dowfax 8390 = 0.315 g, m H2O = 40.058 g.

  polymerization rate depends on the instantaneous concentration of BPO and should slowly decrease over time. When analyzing the ln([M] 0 /[M]) versus Time plot (Figure 4), the linear variation for neat styrene in bulk cannot be reproduced here ; thermally generated radicals control the polymerization rate. A ln([M] 0 /[M]) variation yields a straight line when the radical concentration stays equal throughout the reaction. It seems not to be the case here by considering that the half-time of the BPO initiator (approximately 1.5 hour) is short as compared to the reaction time.

Figure 1 .

 1 Figure 1. ESR spectra of 3 (a, experimental) and 3Na (b, experimental ; c,

Figure 2 .Figure 3 .

 23 Figure 2. Conversion vs. Time for 3, 3Na and TEMPO

  Figure 2

  Figure 4

  Scheme 1

Table 1

 1

	. Features of nitroxides 3, 3Li, 3Na and 3K At Room Temperature. ESR
	conditions: 0.5 G for modulation amplitude, time constant of 163.84 ms, scan
	time of 84 s and 2 W for power. The receptor gain was adjusted to obtain a good
	signal to noise ratio. [Nitroxide] 0 ≈ 300 mM.

Table 2 .

 2 Different Free radical Polymerization Process of Styrene carried out in

	Presence of 3, 3Na and TEMPO using BPO as initiator and a Nitroxide to BPO
	ratio of 1.1.

Table 1 :

 1 Features of nitroxides 3, 3Li, 3Na and 3K At Room Temperature.

	ESR conditions: 0.5 G for modulation amplitude, time constant of 163.84 ms,
	scan time of 84 s and 2 W for power. The receptor gain was adjusted to obtain a
	good signal to noise ratio. [Nitroxide] 0 ≈ 300 mM.
	Compound	Solvent (E(T)N)	a N /G	a Metal /G	g-factor
	3	Toluene (0.099)	14.97		2.0056 3
		Dichloromethane (0.31)	15.30		2.0056 1
		Ethanol (0.65)	15.61		2.0055 3
		MMA	15.00		2.0056 3
		STY	15.04		2.0056 3
	3Li	Dichloromethane	15.81	2.41	2.0055 1
	3Na	Dichloromethane	15.86	2.58	2.0055 0
	3K	Dichloromethane	15.33		Not measured

Table 2 :

 2 Different Free radical Polymerization Process of Styrene carried out in Presence of 3, 3Na and TEMPO using BPO as initiator and a Nitroxide to BPO ratio of 1.1. I = M w /M n b f (= M n calc./M n exp.) is the efficiency of the initiating step

	Process	Time /H Conversion /% M n Exp. M n Calc.	I a	f b
	Bulk 90	0.50	2	1009	313	1.18	0.31
		1.50	22	7564	4293	1.86	0.57
		2.58	40	12950	7916	1.63	0.61
		3.50	49	13972	9761	1.77	0.70
		6.50	62	18605	12339	1.65	0.66
		25.25	95	25392	18950	1.63	0.75
	Bulk 120	0.50	36	12493	7194	1.61	0.58
		1.50	49	15017	9844	1.76	0.66
		2.58	63	16616	12691	2.06	0.76
		3.50	72	17884	14343	2.36	0.80
		6.50	98	17642	19665	3.51	1.11
	Mini-	0.50	16	9130	3236	2.18	0.35
	emulsion	1.58	38	17389	7517	2.26	0.43
		3.67	62	25883	12311	2.38	0.48
		6.08	74	29632	14809	2.86	0.50
	Bulk Na	17.25	36	9367	7273	1.18	0.78
		24.25	56	12248	11124	1.21	0.91
		41.50	64	15916	12789	1.26	0.80
		48.75	68	16551	13622	1.29	0.82
		65.75	72	15954	14499	1.42	0.91
	TEMPO	0.50	0.2	443	36	1.00	0.08
	90	25.25	1.3	670	237	1.18	0.35
		30.47	1.9	762	346	1.23	0.45
		51.60	4.3	1176	785	1.31	0.67

a

Acknowledgement

The authors would like to thank R. Bussels and Dr. A. Mercier for their great help and also Dr. S. van Es for the helpful discussions.

Scheme 4


Gilles Olive

Marcel H P Van Genderen

A 1 H, 13 C, 31 P and 15 N NMR study of (pyrrolidine-2,2-diyl)bisphosphonic acid, tetraalkyl(pyrrolidine-2,2-diyl)bisphosphonates and acyclic tetraethyl bisphosphonates

Keywords: NMR, 1 H NMR, 13 C NMR, 31 P NMR, 15 N NMR, tetraalkyl bisphosphonates, bisphosphonic acid

A multinuclear NMR study ( 1 H, 13 C, 31 P, 15 N) was performed on a series of new cyclic pyrrolidine bisphosphonates and acyclic bisphosphonates. Values are reported and discussed for the chemical shifts and coupling constants of the various nuclei.

INTRODUCTION

Gem-bisphosphonates or gem-bisphosphonic acids can complex well with calcium and magnesium. [START_REF] Ebrahimpour | Determination of solubility and calcium ion stability constants of a phosphonoalkylphosphinate (PAP) and bisphosphonates (BPs) such as EHDP, Risedronate, Alendronate, 3-pic AMBP, and 3-pic AMPMP[END_REF] As a result their use in various areas, for instance in the medical field [START_REF] Ohanessian | [END_REF] , as plant growth regulators 3 and in the nuclear industry [START_REF] Ohanessian | [END_REF] , has been known for some time. We report here a multinuclear NMR study, in particular nitrogen-15, of new tetraalkyl bisphosphonates in a cyclic series and acyclic forms. It is interesting that these compounds can be labeled with 15 N for a direct study in biological material, which has been a considerable problem for several decades. 4

EXPERIMENTAL

The synthesis of compounds 1 to 6 (Figure 1) has been described 5,6 and the synthesis and full characterization of 7 and 8 will be published in a forthcoming paper. [START_REF] Ebrahimpour | Determination of solubility and calcium ion stability constants of a phosphonoalkylphosphinate (PAP) and bisphosphonates (BPs) such as EHDP, Risedronate, Alendronate, 3-pic AMBP, and 3-pic AMPMP[END_REF] H, 13 C and 31 P NMR spectra were measured at 25 ºC on Bruker AC-100, AC-200 and AMX-400 spectrometers. [START_REF] Ebrahimpour | Determination of solubility and calcium ion stability constants of a phosphonoalkylphosphinate (PAP) and bisphosphonates (BPs) such as EHDP, Risedronate, Alendronate, 3-pic AMBP, and 3-pic AMPMP[END_REF] H spectra were recorded at 100, 200 and 400 MHz. 13 C spectra were recorded at 25.18, 50.32 and 100.65 MHz. 31 P spectra were recorded at 40.54 MHz. 15 N NMR spectra were measured at 25 ºC on a Varian Unity Inova 500 spectrometer at 50.65 MHz. In all cases, a field-frequency lock on the solvent 2 H signal was used. Chemical shifts are reported in ppm downfield from the standards, which were: TMS (δ = 0) or residual HDO in aqueous NaOD (δ = 4.81) for 1 H spectra, TMS (δ = 0) for 13 C spectra, external 85% H 3 PO 4 (δ = 0) for 31 P spectra, and external CD 3 NO 2 (δ = 0) for 15 N spectra. Samples were prepared in C 6 D 6 , CDCl 3 or aqueous NaOD (50 mg Na/ml D 2 O), in concentrations of 50-100 mg/ml for 1 H, 13 C and 31 P spectra in 5-mm tubes, and in a 80:20 (v/v) ratio of compound and solvent for natural abundance 15 N measurements in 10-mm tubes. Acquisition parameters for 1 H spectra were: pulse width 2-6 µs (45° pulse), 16 transients, 4 s acquisition time, 2 s relaxation delay, spectral width 10 ppm (1-4 kHz), 8-32 K data points, 0.24 Hz resolution. [START_REF] Ebrahimpour | Determination of solubility and calcium ion stability constants of a phosphonoalkylphosphinate (PAP) and bisphosphonates (BPs) such as EHDP, Risedronate, Alendronate, 3-pic AMBP, and 3-pic AMPMP[END_REF] H spectra were obtained with line broadening (lb = 0.1). Acquisition parameters for 13 C spectra were: pulse width 3 µs (45° pulse), 100 transients, 0.7-1.4 s acquisition time, 5 s relaxation delay, spectral width 6-24 kHz, 16-32 K data points, 0.74-1.45 Hz resolution. 13 C spectra were obtained with line broadening (lb = 1) and 1 H broadband decoupling.

Acquisition parameters for 31 P spectra were: pulse width 5 µs (90° pulse), 80 transients, 5 s acquisition time, 3 s relaxation delay, spectral width 16 kHz, 16 K data points, 2 Hz resolution. 31 P spectra were obtained with 1 H broadband decoupling. Acquisition parameters for 15 N spectra were: pulse width 10 µs (90° pulse: 20.5 µs), 400 transients, 1.9 s acquisition time, 10 s relaxation delay, spectral width 8 kHz, 30000 data points, 0.5 Hz resolution. 15 N spectra were obtained with line broadening (lb = 1) and inverse-gated 1 H broadband decoupling.

RESULTS AND DISCUSSION

For all bisphosphonates, except 1, the 15 N spectrum (Table 1) shows a triplet, indicating that the two phosphorus nuclei are equivalent. This means that the ring pseudorotation is very fast for 2 to 5, as previously reported for the nitroxide 7 (confirmed by the fact that we obtained triplets in 13 C NMR). The two-bond coupling constants between phosphorus and nitrogen are in the range 2-9 Hz, which agrees with the few instances of 2 J NP reported in the literature. 8,[START_REF] Berger | NMR Spektroskopie von Nichtmetallen -Band 2 : 15 N-NMR Spektroskopie[END_REF] It is evident that the linear compounds 6, 7 and 8 have smaller coupling constants (6.3, 6.1 and 5.2 Hz respectively) than the cyclic structures.

Except for 1 (due to a different solvent) and for the picrate of 2, the 15 N chemical shifts of the cyclic forms 2-4 are all similar (max. change 0.5 ppm) and no correlation with the structure was found. Cyclic 5 has a different chemical shift because it is a tertiary amine (the donating methyl group induces an upfield shift).

All 15 N chemical shifts (Table 1) for amines 2-8 are very different from the starting materials (respectively -264.3 ppm for the pyrrolidine-2-one, -271.9 ppm for N-methyl pyrrolidone, -262.1 ppm for the precursor of 6, -242.0 ppm for the precursor of 7, and -244.5 ppm for the precursor of 8).

It is also not possible to make a correlation between the structure and the chemical shift in the 31 P spectra (Table 1). In all cases we see only one line in the 31 P spectra. Hence the two phosphorus nuclei are equivalent and the ring pseudorotation is very fast for 1-5. For 1 H (Table 2) and 13 C NMR (Table 3) there are some cases (for example for 3) where the methyl groups are not equivalent, presumably free rotation around the P-C bond is prevented. [START_REF] Olive | Synthèse de Nouvelles Nitrones du Type Pyrroline-N-Oxyde et leur Utilisation en Spin-Trapping (98AIX30024)[END_REF] The analysis of the 13 C spectra (Table 3) for the α and β-carbons of the phosphonate ester groups, that is an AA'X system, has been described in reference 5.

(

(6) (7) (7) (8) f CH 3 CH 2 C N (CH 3 -CH 2 -O) 2 (O)P (CH 3 -CH 2 -O) 2 (O)P CH 2 H 6 a b c d e f (7) (1) (2) (3) 5) (5) (5) (6) (6) (6)

Figure 1 .

 1 Figure 1. Structures and numbering of the compounds studied.Letters refer to protons and numbers to carbons

Table 1 .

 1 31 P and 15 N NMR spectral data for bisphosphonates 1-8: δ( 31 P) and

			δ( 15 N) (ppm)
	(Parentheses indicate coupling constants in Hz)
			31 P			15 N
		δ	Solvent	Ref	δ	Solvent
	1	15.9 Na/D 2 O d	5	-312.8 Na/D 2 O d
	2	22.5	CDCl 3	5	-340.7 (8.7) a	C 6 D 6
					-12.2 b
	Picrate of 2	16.2	CDCl 3	5	-15.4 c -322.1	CDCl 3
					(2.2)
	3	21.2	CDCl 3	5	-340.3 (9.3)	C 6 D 6
	4	22.7	CDCl 3	6	-340.5 (8.5)	C 6 D 6
	5	21.8	CDCl 3	6	-344.8 (8.3)	C 6 D 6
	6	21.5	CDCl 3	6	-340.4 (6.3)	C 6 D 6
	7	22.5	CDCl 3		-324.3 (6.1)	C 6 D 6
	8	23.3	CDCl 3		-330.8 (5.2)	C 6 D 6
	a -341.6 (8.6) in CDCl 3 , b nitro ortho, c nitro para, d 50 mg Na/ml D 2 O

Table 2 .

 2 [START_REF] Ebrahimpour | Determination of solubility and calcium ion stability constants of a phosphonoalkylphosphinate (PAP) and bisphosphonates (BPs) such as EHDP, Risedronate, Alendronate, 3-pic AMBP, and 3-pic AMPMP[END_REF] H NMR spectral data for bisphosphonates 1-8: δ( 1 H) (ppm) (Parentheses indicate the pattern of coupling and the coupling constants in Hz)

		a	b	c	d	e	f	g	Solvent Frequency Ref
	1	3.21 (t, 7.0)	1.85 (quint., 7.0)	2.17 (sept., 7.1 e ; 14.5 f )					Na/D 2 O c 200	5
	2	2.88 (t, 6.5)	1.69 (quint., 6.8)	2.42 (tt, 7.2 e ; 17.7 f )	4.17 (m)	1.11 (t, 7.1 e ) (t, 7.1 e ) 1.10			C 6 D 6 400	5
	Picrate	3.64	2.15-2.67	4.24	1.36 (t, 7.1 e )	8.86 a	7.99 b	CDCl 3
	of 2	(t, 6.9)	(m)		(m)	1.33	(s)	(s)	100
						(t, 7.1 e )

Table 3 .

 3 13 C NMR spectral data for bisphosphonates 1-8: δ( 13 C) (ppm) (Parentheses indicate coupling constants with 31 P in Hz) quaternary aromatic, b meta aromatics, c ortho aromatics, d para aromatic, e coupling can not be resolved because they are to small, f 2 J CP (+ 4 J CP' ) h , g 3 J CP h , h see ref 5, i 50 mg Na/ml D 2 O

	C 6 D 6
	400
	)

a

a aromatics, b -NH 2 + , c 50 mg Na/ml D 2 O, d the two protons are not equivalent, e H-H coupling, f P-H coupling