Thierry Huillet

Martin Möhle

ASYMPTOTICS OF SYMMETRIC COMPOUND POIS-SON POPULATION MODELS

Keywords: Bell polynomials, coalescent with multiple collisions, compound Poisson model, condensation, Dirichlet model, Kingman coalescent, saddle point method, Wright-Fisher model 2010 Mathematics Subject Classification: Primary 60F05, 60G09, 60J10, 60J80, Secondary 60K35, 92D10, 92D25

Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radius r of convergence. It is shown that the asymptotic behavior of symmetric compound Poisson models is mainly determined by the characteristic value θrφ ′ (r-). If θrφ ′ (r-) ≥ 1, then the model is in the domain of attraction of the Kingman coalescent. If θrφ ′ (r-) < 1, then under mild regularity conditions a condensation phenomenon occurs which forces the model to be in the domain of attraction of a discrete-time Λ-coalescent. The proofs are partly based on the analytic saddle point method. They draw heavily from local limit theorems and from results of S. Janson on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analyzed.

Introduction

Conditional branching process models, introduced by Karlin and McGregor [START_REF] Karlin | Direct product branching processes and related Markov chains[END_REF][START_REF] Karlin | Direct product branching processes and related Markov chains. I. Calculations of rates of approach to homozygosity[END_REF], are population models with fixed population size N ∈ N := {1, 2, . . .} and non-overlapping generations. They are defined in terms of a sequence (ξ n ) n∈N of independent non-negative integer-valued random variables satisfying P(ξ 1 + • • • + ξ N = N ) > 0. If, for i ∈ {1, . . . , N }, µ N,i denotes the number of offspring of the ith individual alive in some fixed generation, then the random variables µ N,1 , . . . , µ N,N have (by definition) joint distribution

P(µ N,1 = j 1 , . . . , µ N,N = j N ) = P(ξ 1 = j 1 ) • • • P(ξ N = j N ) P(ξ 1 + • • • + ξ N = N ) , j 1
, . . . , j N ∈ N 0 := {0, 1, . . .} with j 1 + • • • + j N = N . For convenience we will often drop the index N and simply write µ i instead of µ N,i . For some more information on conditional branching process models we refer the reader to [9, Section 3] and [START_REF] Huillet | Correction on 'Population genetics models with skewed fertilities: a forward and backward analysis[END_REF].

We now turn to the definition of compound Poisson population models. Let θ 1 , θ 2 , . . . be strictly positive real numbers and let φ(z) = ∞ m=1 φ m z m /m!, |z| < r, be a power series with radius r ∈ (0, ∞] of convergence and with non-negative coefficients φ m ≥ 0, m ∈ N. It is also assumed that φ 1 > 0. Compound Poisson models are particular conditional branching process models where each random variable ξ n has probability generating function (pgf)

f n (x) := E(x ξn ) = exp(-θ n (φ(z) -φ(zx))), |x| ≤ 1. (1)
In [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF], z is viewed as a fixed parameter satisfying 0 < z < r. If M n is a random variable having a Poisson distribution with parameter θ n φ(z) and if X 1 , X 2 , . . . are independent random variables and independent of M n each with pgf x → φ(zx)/φ(z), |x| ≤ 1, then Mn j=1 X j has pgf [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF]. This subclass of conditional branching process models is therefore called the compound Poisson class. In order to analyze compound Poisson models it is useful to view z as a variable and to introduce, for θ ∈ [0, ∞), the Taylor expansion exp(θφ(z)) = ∞ k=0 σ k (θ)z k /k!, |z| < r. The coefficients σ k (θ) depend on (φ m ) m∈N and they satisfy the recursion

σ 0 (θ) = 1 and σ k+1 (θ) = θ k l=0 k l φ k-l+1 σ l (θ), k ∈ N 0 , θ ∈ [0, ∞). (2)
The coefficients σ k (θ) are mainly introduced, since, by [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF], the distribution of ξ n , n ∈ N, satisfies

P(ξ n = k) = σ k (θ n ) k! z k exp(-θ n φ(z)), k ∈ N 0 . (3)
Note furthermore that ξ 1 + • • • + ξ n , n ∈ N, has distribution

P(ξ 1 + • • • + ξ n = k) = σ k (Θ n ) k! z k exp(-Θ n φ(z)), k ∈ N 0 ,
where Θ n := θ 1 + • • • + θ n . From (2) it follows by induction on k ∈ N that φ k = lim θ→0 σ k (θ)/θ for all k ∈ N, so the coefficients φ k , k ∈ N, of the power series φ can be recovered from the coefficients σ k (θ), k ∈ N 0 , θ ∈ (0, ∞). From φ 1 > 0 it follows that σ k (θ) is a polynomial in θ of degree k. In the literature (see, for example, [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF] or [START_REF] Charalambides | An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients[END_REF]) the σ k (θ) are called the exponential or Bell polynomials. We have σ 1 (θ) = θφ 1 , σ 2 (θ) = θφ 2 + θ 2 φ 2 1 , σ 3 (θ) = θφ 3 + 3θ 2 φ 1 φ 2 + θ 3 φ 3 1 , σ 4 (θ) = θφ 4 + θ 2 (4φ 1 φ 3 + 3φ 2 2 ) + 6θ 3 φ 2 1 φ 2 + θ 4 φ 4 1 , and so on. The coefficients B kl (φ 1 , φ 2 , . . .), k ∈ N 0 , l ∈ {0, . . . , k}, of the polynomials σ k (θ) = k l=0 B kl (φ 1 , φ 2 , . . .) θ l , k ∈ N 0 , are called the Bell coefficients. In the following, for x ∈ R and k ∈ N 0 , the notation (x) k := x(x -1) • • • (x -k + 1) and [x] k := x(x + 1) • • • (x + k -1) is used for the descending and ascending factorials respectively, with the convention that (x) 0 := 1 and [x] 0 := 1. It is readily checked that ξ n has descending factorial moments 2 and so on. The descending factorial moments therefore satisfy the recursion

E((ξ n ) k ) = f (k) n (1) = z k k l=0 B kl (φ ′ (z), φ ′′ (z), . . .) θ l n , n ∈ N, k ∈ N 0 , i.e. E(ξ n ) = θ n zφ ′ (z), E((ξ n ) 2 ) = θ n z 2 φ ′′ (z) + θ 2 n z 2 (φ ′ (z))
E((ξ n ) k+1 ) = θ n k l=0 k l z k-l+1 φ (k-l+1) (z)E((ξ n ) l ), n ∈ N, k ∈ N 0 .
It is known (see, for example, [9, p. 535]) that µ = (µ 1 , . . . , µ N ) has distribution

P(µ = j) = N ! σ N (Θ N ) N n=1 σ jn (θ n ) j n ! , j = (j 1 , . . . , j N ) ∈ ∆(N ), (4)
where Θ N := θ 1 + • • • + θ N and ∆(N ) denotes the discrete N -simplex consisting of all j = (j 1 , . . . , j N ) ∈ N N 0 satisfying j 1 + • • • + j N = N . Note that the distribution of µ is not necessarily exchangeable. Moreover (see, for example, [10, Eq. ( 4)]), µ has joint factorial moments

E((µ 1 ) k1 • • • (µ N ) k N ) = N ! σ N (Θ N ) j 1 ≥k 1 ,...,j N ≥k N j 1 +•••+j N =N σ j1 (θ 1 ) • • • σ j N (θ N ) (j 1 -k 1 )! • • • (j N -k N )! , (5)
N ∈ N, k 1 , . . . , k N ∈ N 0 , θ 1 , . . . , θ N ∈ (0, ∞). For some compound Poisson models, namely for skewed Wright-Fisher models and for skewed Dirichlet models, the alternative and simpler formula (see, for example, [10, Lemma 2.1] and the remarks thereafter)

E((µ 1 ) k1 • • • (µ N ) k N ) = (N ) k σ k1 (θ 1 ) • • • σ k N (θ N ) σ k (θ 1 + • • • + θ N ) (6)
holds for all k 1 , . . . , k N ∈ N 0 and all θ 1 , . . . , θ N ∈ (0, ∞), where k

:= k 1 + • • • + k N .
For some more information on compound Poisson models we refer the reader to [START_REF] Möhle | Coalescent processes derived from some compound Poisson population models[END_REF].

The article is organized as follows. The main results are presented in the following Section 2. Proofs are provided in Section 3. Examples of compound Poisson models are studied in Section 4. One purpose of this article is to provide proofs of the results stated without proof in [START_REF] Huillet | Correction on 'Population genetics models with skewed fertilities: a forward and backward analysis[END_REF].

Results

The following Proposition 2.1, stated without proof in [10, Proposition 2.2], provides expressions for the transition probabilities P i,j of the backward process X for an arbitrary compound Poisson model. The proof of Proposition 2.1 is provided in Section 3.

Proposition 2.1 If, for each n ∈ N, the random variable ξ n has a pgf of the form (1), then the backward process X of the associated compound Poisson model has transition probabilities

P i,j = 1 N i 1≤n1<•••<nj ≤N k 1 ,...,k j ∈N 0 k:=k 1 +•••+k j ≤N (N ) k k 1 ! • • • k j ! • • ( j i=1 σ ki (θ ni )) σ N -k (Θ N - j i=1 θ ni ) σ N (Θ N ) l 1 ,...,l j ∈N l 1 +•••+l j =i k 1 l 1 • • • k j l j , i, j ∈ S, (7)
with the convention that P i,0 = δ i0 , i ∈ S. Here Θ ), are recursively defined via (2). In particular,

N := θ 1 + • • • + θ N , S := {0, . . . , N }, and the coefficients σ k (θ), k ∈ N 0 , θ ∈ [0, ∞
P i,1 = N n=1 N k=i N -i k -i σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) , i ∈ {1, . . . , N }. ( 8
)
Remark. Proposition 2.1 in particular yields the coalescence probability

c N := P 2,1 = N n=1 N k=2 N -2 k -2 σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) , N ≥ 2, (9)
and the probability

d N := P 3,1 = N n=1 N k=3 N -3 k -3 σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) , N ≥ 3, (10)
that three individuals, picked at random and without replacement from some arbitrary but fixed generation of the population, share a common parent.

In the following we focus on the unbiased (symmetric) case, when all the parameters θ n = θ are equal to some constant θ ∈ (0, ∞). In this case the distribution (4) of µ is exchangeable and ( 7) reduces to

P i,j = N j N i k 1 ,...,k j ∈N 0 k:=k 1 +•••+k j ≤N (N ) k k 1 ! • • • k j ! ( j i=1 σ ki (θ)) σ N -k ((N -j)θ) σ N (N θ) l 1 ,...,l j ∈N l 1 +•••+l j =i k 1 l 1 • • • k j l j , (11)
i, j ∈ S, with the convention that P i,0 = δ i0 , i ∈ S. In particular,

P i,1 = N N k=i N -i k -i σ k (θ) σ N -k ((N -1)θ) σ N (N θ) , i ∈ {1, . . . , N }, (12)
c N = N N k=2 N -2 k -2 σ k (θ) σ N -k ((N -1)θ) σ N (N θ) , N ≥ 2, (13)
and

d N = N N k=3 N -3 k -3 σ k (θ) σ N -k ((N -1)θ) σ N (N θ) , N ≥ 3, (14)
in agreement with (8), [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF], and [START_REF] Huillet | Correction on 'Population genetics models with skewed fertilities: a forward and backward analysis[END_REF].

We are interested in the asymptotic behavior of symmetric compound Poisson models as N → ∞.

The following results show that the characteristic value θrφ ′ (r-) ∈ (0, ∞] is of fundamental interest in this context, where r ∈ (0, ∞] denotes the radius of convergence of φ. Theorem 2.2 below, stated in slightly different form and without proof in [10, Theorem 2.3], clarifies that many symmetric compound Poisson models are in the domain of attraction of the Kingman coalescent.

Theorem 2.2 (supercritical case) Suppose that θrφ ′ (r-) ∈ (1, ∞] such that the equation θzφ ′ (z) = 1 has a unique real solution z(θ) ∈ (0, r). Then, µ N,1 → X in distribution as N → ∞ with convergence E(µ p N,1 ) → E(X p ) as N → ∞, p ∈ (0, ∞)
, of all moments, where X is a non-negative integer-valued random variable with distribution

P(X = k) = σ k (θ) k! (z(θ)) k exp(-θφ(z(θ))), k ∈ N 0 , (15)
and mean E(X) = 1. Moreover, in the sense of [9, Definition 2.1 (a)], the associated symmetric compound Poisson population model is in the domain of attraction of the Kingman coalescent.

The effective population size

N e := 1/c N satisfies N e ∼ ̺N as N → ∞ with ̺ := 1/Var(X) = 1/(1 + θ(z(θ)) 2 φ ′′ (z(θ))) ∈ (0, 1].

Remarks.

1. Two independent proofs of Theorem 2.2 are provided in Section 3. The first proof involves the saddle point method and the second proof is based on the local limit theorem. Both proofs differ significantly from the erroneous proof in [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF]. To the best of the authors knowledge these proofs are the first approaches where saddle point methods or, alternatively, local limit theorems are used to establish convergence to the Kingman coalescent. The distribution of the limiting variable X coincides with the distribution (3) of ξ 1 with the parameter z in (3) replaced by z(θ). Note that X has mean E(X) = θz(θ)φ ′ (z(θ)) = 1. Thus, conditioning ξ 1 on the event that ξ 1 + • • • + ξ N = N and afterwards taking N → ∞, has altogether the effect that the distribution of ξ 1 is 'nearly' recovered. Only the information about the mean of ξ 1 is lost.

2. If φ ′ (r-) = ∞, then for all θ ∈ (0, ∞) the equation θzφ ′ (z) = 1 has a solution z(θ) ∈ (0, r). If φ ′ (r-) < ∞, then a solution z(θ) ∈ (0, r) of the equation θzφ ′ (z) = 1 exists if and only if θrφ ′ (r-) > 1. Examples satisfying φ ′ (r-) < ∞ are provided in Section 4. The solution z(θ) (if it exists) is unique since the map z → zφ ′ (z)
is strictly increasing on (0, r). Closed expressions for the solution z = z(θ) of the equation u(z) := zφ ′ (z) = 1/θ seem to be not available in general. By the inversion formula of Lagrange (see, for example, [5, Section 3.8]), for k ∈ N and x ∈ (0, ∞),

[x k ]u -1 (x) = [z k-1 ](z/u(z)) k k = [z k-1 ](1/φ ′ (z)) k k = B k-1,k (ψ 0 , ψ 1 , . . .),
where ψ n := ψ (n) (0), n ∈ N 0 , with ψ(z) := 1/φ ′ (z). Choosing x := 1/θ and noting that z(θ) = u -1 (x) yields the formal expansion

z(θ) = ∞ k=1 [z k-1 ](ψ(z)) k k θ -k = ∞ k=1 B k-1,k (ψ 0 , ψ 1 , . . .)θ -k .
Note however that, depending on φ and θ, this series may not converge, so we can only speak about a formal series here. Alternatively one may approximate the root z(θ) of the map

z → θzφ ′ (z) -1 numerically. Theorem 2.2 is not applicable if θrφ ′ (r-) ≤ 1. In this case we have φ ′ (r-) < ∞, which implies that φ(r-) < ∞, since zφ ′ (z) = ∞ m=1 φ m z m /(m -1)! ≥ ∞ m=1 φ m z m /m! = φ(z)
for all z ∈ [0, r). Moreover, from φ(r-) < ∞ it follows that r < ∞. Assume from now on that θrφ ′ (r-) ≤ 1 and introduce a non-negative integer-valued random variable X with distribution

P(X = k) = σ k (θ) k! r k exp(-θφ(r-)), k ∈ N 0 . (16)
Note that X has pgf s → E(s X ) = exp(-θ(φ(r-) -φ((rs)-))), |s| ≤ 1, and characteristic function t → E(exp(itX)) = exp(-θ(φ(r-) -φ((re it )-))), t ∈ R. In particular, E(X) = θrφ ′ (r-) ≤ 1.

Moreover, E(X k ) < ∞ if and only if φ (k) (r-) < ∞, k ∈ N. Our second asymptotic result, Theorem 2.3 below, addresses the critical case E(X) = θrφ ′ (r-) = 1. The subcritical case E(X) < 1 is considered at the end of this section. In general c N can tend to zero very slowly. The proof of Theorem 2.3, provided in Section 3, does not provide direct information on the speed of convergence. Under additional assumptions more can be said about the asymptotics of c N . The proof of the following Theorem 2.4 draws solely but heavily from local limit theorems. For examples we refer the reader to Section 4 from Example 4.6 on.

Theorem 2.4 (critical case, speed of convergence) Assume that θrφ ′ (r-) = 1.

a) If the random variable X with distribution ( 16) satisfies E(X 2 ) < ∞ or, equivalently, if φ ′′ (r-) < ∞, then µ N,1 → X in distribution as N → ∞. Moreover, the coalescence probability c N satisfies c N ∼ Var(X)/N as N → ∞, where Var(X) = 1 + θr 2 φ ′′ (r-).

b) If there exists a constant c = c(θ) > 0 such that the distribution (16) of the random variable

X satisfies P(X = k) ∼ ck -3 as k → ∞, then µ N,1 → X in distribution as N → ∞. Moreover, the coalescence probability c N satisfies c N ∼ c 2 log N N , N → ∞. (17)
c) If there exist constants α ∈ (1, 2) and κ = κ(α) ∈ (0, ∞) such that the power series φ of the compound Poisson model satisfies

φ(z) = φ(r-) + φ ′ (r-)(z -r) + κ(r -z) α + O((r -z) 2 ), z → r, (18)
then the distribution (16) of the random variable X satisfies P(X = k) ∼ ck -α-1 as k → ∞ with c := θκr α /Γ(-α). Moreover, µ N,1 → X in distribution as N → ∞ and the coalescence probability c N satisfies

c N ∼ c ∞ 0 x 1-α g(-x) dx g(0) N 2(1-α)/α , N → ∞. ( 19
)
Here g denotes the density of an α-stable random variable with characteristic function t → exp(d(-it) α ), t ∈ R, where d := cΓ(-α) = θκr α .

Remarks.

1. Assume that α ∈ (1, 2). Lemma 5.2, applied with

β := 1 -α ∈ (-1, 0), shows that ∞ 0 x 1-α g(-x) dx = d (1-α)/α Γ(1 -α)/Γ((1 -α)/α) and g(0) = d -1/α |Γ(-1/α)| -1
. Thus, the fraction in [START_REF] Sagitov | Convergence to the coalescent with simultaneous multiple mergers[END_REF] can be expressed in terms of the gamma function via

∞ 0 x 1-α g(-x) dx g(0) = d (2-α)/α Γ(1 -α)|Γ(-1/α)| Γ((1 -α)/α) , α ∈ (1, 2).
2. Theorem 2.4 b) can be viewed as the boundary case α = 2 of Theorem 2.4 c). Note that, under the assumptions of Theorem 2.4 b) or c) respectively,

E(X 2 1 {X≤N } ) N = 1 N N k=1 k 2 P(X = k) ∼ c N N k=1 k 1-α ∼ c 2-α N 1-α for α ∈ (1, 2), c log N N for α = 2.
(20) The naive prospect, based on the equation (N -1)c N = E((µ N,1 ) 2 ), that the coalescence probability c N could be asymptotically equal to (20), is wrong due to Theorem 2.4 b) and c). Theorems 2.2, 2.3, and 2.4 are not applicable if E(X) < 1. In this subcritical case it seems to be not straightforward to derive asymptotic results without further regularity assumptions on the weights w k := σ k (θ)/k!, k ∈ N 0 . Our last asymptotic result below essentially states that a 'condensation phenomenon' occurs which forces the model to be in the domain of attraction of a discrete-time Λ-coalescent with Λ(dt)/t 2 being a Dirac measure (Dirac coalescent). The result in particular shows that, under certain circumstances, Dirac coalescents arise naturally in the limit as the total population size N tends to infinity and indicates that Dirac coalescents are more important in the context of ancestral population genetics as it seems to be at a first glance.

Theorem 2.5 (subcritical case) Fix θ ∈ (0, ∞) and assume that there exist constants c > 0 and β > 2 such that w

k := σ k (θ)/k! ∼ ck -β /r k as k → ∞. If θrφ ′ (r-) < 1, then µ N,1 → X in distribution as N → ∞,
where X is a non-negative integer-valued random variable with distribution [START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF]. Moreover, lim N →∞ c N = u 2 , where u := 1 -E(X) = 1 -θrφ ′ (r-) ∈ (0, 1) and, in the sense of [9, Definition 2.1 (b)], the associated symmetric compound Poisson population model is in the domain of attraction of a discrete-time Λ-coalescent with Λ := u 2 δ u .

Remarks.

1. Note that lim N →∞ P i,j = i j-1 u i-j+1 (1 -u) j-1 for all i, j ∈ N with i > j. 2. The physical image is the one of a very prolific individual giving birth to a fixed fraction u of the total population, the others adjusting their random offspring to ensure a constant population size N in each generation. This 'condensation phenomenon' is reminiscent of the one observed by Eldon and Wakeley [START_REF] Wakeley | Coalescent processes when the distribution of offspring number among individuals is highly skewed[END_REF] (see also [START_REF] Huillet | On the extended Moran model and its relation to coalescents with multiple collisions[END_REF]Proposition 4]) in the context of extended Moran models. However, as observed in [9, Proposition 3.1], compound Poisson models and extended Moran models constitute disjoint reproduction laws.

3. It remains unclear whether Theorem 2.5 holds without any regularity condition on the weight sequence (w k ) k∈N0 . Note that, for arbitrary weights (not necessarily coming from a compound Poisson model), theorems of this form do not hold without any regularity condition on the weight sequence (w k ) k∈N0 . We refer the reader to the remark before Example 19.33 and to the Examples 19.37 and 19.38 of [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF] for analog comments and more details. In our situation however, the weights are coming from a compound Poisson model and have the particular structure ω k = σ k (θ)/k!. Thus, our weights cannot be arbitrarily irregular and hence, there is some chance that Theorem 2.5 could hold without any conditions on (w k ) k∈N0 . We leave this problem open for future work.

Proofs

Proof. (of Proposition 2.1) For j ∈ {1, . . . , N }, pairwise distinct n 1 , . . . , n j ∈ {1, . . . , N }, and

k 1 , . . . , k j ∈ N 0 with k := k 1 + • • • + k j ≤ N we have P(µ n1 = k 1 , . . . , µ nj = k j ) = ( j i=1 P(ξ ni = k i )) P( m∈[N ]\{n1,...,nj } ξ m = N -k) P(ξ 1 + • • • + ξ N = N ) = (N ) k k 1 ! • • • k j ! σ k1 (θ n1 ) • • • σ kj (θ nj ) σ N -k (Θ N - j i=1 θ ni ) σ N (Θ N )
and, therefore, for l 1 , . . . , l j ∈ N 0 ,

E j i=1 µ ni l i = k1,...,kj j i=1 k i l i (N ) k k 1 ! • • • k j ! σ k1 (θ n1 ) • • • σ kj (θ nj )σ N -k (Θ N - j i=1 θ ni ) σ N (Θ N )
,

where the sum k1,...,kj extends over all k 1 , . . . , k j ∈ N 0 satisfying k [START_REF] Flajolet | Analytic Combinatorics[END_REF] follows from [9, Eq. ( 4)]. For j = 1, (7) reduces to [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF]. Alternatively, (8) follows as well via

:= k 1 + • • • + k j ≤ N . Thus (
P i,1 = N n=1 E((µ n ) i ) (N ) i = N n=1 N k=i (k) i (N ) i P(µ n = k) = N n=1 N k=i (k) i (N ) i N k σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) = N n=1 N k=i N -i k -i σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) . 2
Proof. (of Theorem 2.2) Two independent proofs are provided. The first proof exploits the analytic saddle point method. The second proof is based on the standard local limit theorem for sequences of independent and identically distributed random variables with finite and non-vanishing variance. We think that both proofs are worthwhile, since they demonstrate the intertwining of analysis and probability.

Proof 1. Fix k, l ∈ N 0 and θ ∈ (0, ∞). Let us verify that

σ n-k ((n -l)θ) (n -k)! ∼ (a(θ)) n √ 2πn b kl (θ), n → ∞, (21)
where

a(θ) := e θφ(z(θ)) z(θ) and b kl (θ) := (z(θ)) k e -lθφ(z(θ)) 1 + θ(z(θ)) 2 φ ′′ (z(θ)) . ( 22
)
We proceed similarly as in the proof of [4, Theorem 2.1]. However, note that in [START_REF] Charalambides | An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients[END_REF] asymptotic expansions for σ n (θ) are provided whereas we are essentially interested in the asymptotics of σ n (nθ). By Cauchy's integral formula,

σ n (θ)/n! = (2πi) -1 C z -(n+1) e θφ(z) dz, n ∈ N 0 ,
where C is some contour around the origin. Replacing n by n -k and θ by (n -l)θ it follows that

σ n-k ((n -l)θ) (n -k)! = 1 2πi C e (n-l)θφ(z) z n-k+1 dz = 1 2πi C h(z)e ng(z) dz,
where h(z) := z k-1 e -lθφ(z) and g(z) := θφ(z) -log z. Note that g ′ (z) = θφ ′ (z) -1/z and that g ′′ (z) = θφ ′′ (z) + 1/z 2 . In particular, g ′ has a single real zero in the interval (0, r) at the point z(θ) solving the equation θz(θ)φ ′ (z(θ)) = 1. Note that g ′′ (z) > 0 for all z ∈ (0, r). In order to derive the asymptotics of the integral C h(z)e ng(z) dz we use the saddle point method (see, for example, [START_REF] Bruijn | Asymptotic Methods in Analysis[END_REF] or [START_REF] Flajolet | Analytic Combinatorics[END_REF] for general references) and choose the contour C to be the circle around the origin with radius z(θ) such that it passes through the zero z(θ) of g ′ . Note that g ′ (z(θ)) = 0, so g(z(θ)e it ) has Taylor expansion

g(z(θ)e it ) = ∞ j=0 g (j) (z(θ)) j! (z(θ)) j (e it -1) j = g(z(θ)) + g ′′ (z(θ)) 2 (z(θ)) 2 (e it -1) 2 + g ′′′ (z(θ)) 3! (z(θ)) 3 (e it -1) 3 + O(t 4 )
leading to the Taylor expansions Re(g(z(θ

)e it )) = g(z(θ)) -(z(θ)) 2 g ′′ (z(θ))t 2 /2 + O(t 4 ) and Im(g(z(θ)e it )) = O(t 3
). The saddle point method yields the asymptotics

C h(z)e ng(z) dz ∼ i 2π ng ′′ (z(θ)) e ng(z(θ)) h(z(θ)).
Dividing this expression by 2πi and writing z instead of z(θ) for convenience yields

σ n-k ((n -l)θ) (n -k)! ∼ e ng(z) h(z) 2πng ′′ (z) = 1 √ 2πn (a(θ)) n b kl (θ)
with a(θ) := e g(z) = e θφ(z) /z and b kl (θ) := h(z)/ g ′′ (z) = z k-1 e -lθφ(z) / z -2 + θφ ′′ (z) = z k e -lθφ(z) / 1 + θz 2 φ ′′ (z). Thus, (21) is established. Note that for k = l = 0 we have

σ n (nθ) n! = 1 2πi C e ng(z) z dz = 1 2πi π -π e ng(z(θ)e it ) z(θ)e it iz(θ)e it dt = 1 2π π -π
e ng(z(θ)e it ) dt.

Taking the real part yields

σ n (nθ) n! = 1 2π π -π Re(e ng(z(θ)e it ) ) dt ∼ 1 2π π -π e nRe(g(z(θ)e it )) dt, (23)
where the last asymptotics is based on the Laplace method as follows. Choose a sequence (δ n ) n∈N of positive real numbers satisfying nδ 2 n → ∞ and nδ 3 n → 0, for example, δ n := n -α for some fixed α ∈ (1/3, 1/2). Decomposing the first integral in (23) into the two parts

I 1 := δn -δn
Re(e ng(z(θ)e it ) ) dt and I 2 := {δn<|t|≤π} Re(e ng(z(θ)e it ) ) dt we can approximate I 1 and show that I 2 is negligible (in comparison to I 1 ) for large n. Obviously, 1 -x 2 /2 ≤ cos x ≤ 1 for all x ∈ R. Choosing x := nIm(g(z(θ)e it )) and using Im(g(z(θ)e it )) = O(t 3 ) and nt 3 → 0 as n → ∞ uniformly for all |t| ≤ δ n it follows that

lim n→∞ sup |t|≤δn | cos(nIm(g(z(θ)e it ))) -1| = 0.
Thus, as n → ∞, the map t → cos(nIm(g(z(θ)e it ))) converges uniformly on [-δ n , δ n ] to the constant map t → 1, which implies that

I 1 = δn -δn cos(nIm(g(z(θ)e it )
))e nRe(g(z(θ)e it )) dt ∼ δn -δn e nRe(g(z(θ)e it )) dt.

Let us now turn to the second integral I 2 . Define the two functions f : R → R and m : R → R via f (t

) := Re(φ(z(θ)e it )) = ∞ m=1 (φ m /m!)(z(θ)) m cos(mt) and m(t) := Re(g(z(θ)e it )) = Re(θφ(z(θ)e it ) -log(z(θ)e it )) = θf (t) -log(z(θ)
) for all t ∈ R. Note that f is 2π-periodic and that f (t) = f (-t) for all t ∈ R. The same holds for the function m since it is a linear transformation of f . For m ∈ N define a m := φ m (z(θ)) m /m! and choose some q ∈ (1, r/z(θ)). Then, ∞ m=1 a m q m = φ(z(θ)q) < ∞, since z(θ)q < r. By Lemma 5.1 (applied for the function f and with ε := δ n ) it follows that there exists a constant n 0 ∈ N (which may depend on θ and φ) such that sup δn<|t|≤π m(t) ≤ m(δ n ) for all n ∈ N with n > n 0 . Therefore

|I 2 | ≤ {δn<|t|≤π} e nRe(g(z(θ)e it )) dt = {δn<|t|≤π} e nm(t) dt ≤ 2πe nm(δn) . The expansion m(t) = g(z(θ)) -(z(θ)) 2 g ′′ (z(θ))t 2 /2 + O(t 4 ) applied for t = δ n yields e nm(δn) ∼ e ng(z(θ)) e -(z(θ)) 2 g ′′ (z(θ))nδ 2 n /2 . Thus, |I 2 |/I 1 = O( √ n/e cnδ 2 n ) with c := (z(θ)) 2 g ′′ (z(θ))/2 > 0. Since nδ 2 n = n 1-2α → ∞ it follows that |I 2 |/I 1 → 0 as n → ∞. Thus, the integral I 2 is negligible in comparison to I 1 and (23) is established. For k ∈ N 0 and N ∈ N with N ≥ k it follows from (21) that P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) = σ k (θ) k! σ N -k ((N -l)θ) (N -k)! σ N (N θ) N ! ∼ σ k (θ) k! (a(θ)) N b k1 (θ)/ √ 2πN (a(θ)) N b 00 (θ)/ √ 2πN = σ k (θ) k! b k1 (θ) b 00 (θ) = σ k (θ) (z(θ)) k k! e -θφ(z(θ)) .
Thus, µ N,1 → X in distribution as N → ∞, where X has distribution [START_REF] Karlin | Direct product branching processes and related Markov chains. I. Calculations of rates of approach to homozygosity[END_REF]. In the following, for fixed p ∈ (0, ∞), the convergence E(µ p N,1 ) → E(X p ) as N → ∞ of the p-th moments is established. For all N ∈ N and all k, l ∈ {0, . . . , N } we have

σ N -k ((N -l)θ) (N -k)! ≤ σ N -k (N θ) (N -k)! = 1 2πi C z k-1 e N g(z) dz = 1 2πi π -π (z(θ)e it ) k-1 e N g(z(θ)e it ) iz(θ)e it dt = (z(θ)) k 2π π -π
e ikt e N g(z(θ)e it ) dt.

Taking the complex absolute value it follows for all N ∈ N and all k, l ∈ {0, . . . , N } that

σ N -k ((N -l)θ) (N -k)! ≤ (z(θ)) k 2π π -π |e N g(z(θ)e it ) | dt = (z(θ)) k 2π π -π
e N Re(g(z(θ)e it )) dt.

Since, by ( 23), (2π) -1 π -π e N Re(g(z(θ)e it )) dt ∼ σ N (N θ)/N !, it follows that there exists a constant N 0 ∈ N (which may depend on θ and φ but not on k and l) such that

σ N -k ((N -l)θ) (N -k)! ≤ 2(z(θ)) k σ N (N θ) N !
for all N ≥ N 0 and all k, l ∈ {0, . . . , N }. In particular, for all N ≥ N 0 and all k ∈ {0, . . . , N },

P(µ N,1 = k) = σ k (θ) k! σ N -k ((N -1)θ) (N -k)! σ N (N θ) N ! ≤ σ k (θ) k! 2(z(θ)) k = κ(θ)P(X = k),
where κ(θ) := 2e θφ(z(θ)) ∈ (0, ∞). For all p ∈ (0, ∞) the map k → k p P(X = k), k ∈ N 0 , is integrable with respect to the counting measure ε N0 on N 0 , since [START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF] or [16, Theorem 4 (b)]) that the considered symmetric compound Poisson model is in the domain of attraction of the Kingman coalescent. The proof is complete.

k p P(X = k)ε N0 (dk) = ∞ k=0 k p P(X = k) = E(X p ) < ∞. Dominated convergence yields the convergence E(µ p N,1 ) → E(X p ) as N → ∞ of all moments. In particular, (N -1)c N = E((µ N,1 ) 2 ) → E((X) 2 ) > 0 and (N -1)(N -2)d N = E((µ N,1 ) 3 ) → E((X) 3 ) as N → ∞. Thus, d N /c N = O(1/N ) → 0 as N → ∞, which ensures (see
Proof 2. Let X 1 , X 2 , . . . be independent copies of the random variable X with distribution [START_REF] Karlin | Direct product branching processes and related Markov chains. I. Calculations of rates of approach to homozygosity[END_REF].

For N ∈ N define S N := X 1 + • • • + X N . Since 0 < Var(X) < ∞, the local limit theorem lim N →∞ sup k∈Z |d N (k)| = 0 holds, where d N (k) := √ N P(S N = k) -g((k -N )/ √ N
) and g denotes the density of the normal distribution N (0, Var(X)). We have

P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) = P(X = k) P(S N -1 = N -k) P(S N = N ) . Since P(S N = N ) = (g(0) + d N (N ))/ √ N ∼ g(0)/ √ N as N → ∞ and, for arbitrary but fixed k, P(S N -1 = N -k) = (g((1 -k)/ √ N ) + d N -1 (N -k))/ √ N -1 ∼ g(0)/ √ N as N → ∞
thanks to the fact that the convergence in the local limit theorem holds uniformly for all k, it follows that P(S N -1 = N -k)/P(S N = N ) → 1 as N → ∞ and, therefore,

P(µ N,1 = k) → P(X = k) as N → ∞ for all k ∈ N 0 . Thus, µ N,1 → X in distribution as N → ∞. Fix p ∈ (0, ∞). For all N ∈ N, E(µ p N,1 ) = N k=1 k p P(X = k) P(S N -1 = N -k) P(S N = N ) .
Applying the local limit theorem to both probabilities P(S N -1 = N -k) and P(S N = N ) it follows for arbitrary but fixed p ∈ (0, ∞) that

E(µ p N,1 ) ∼ N k=1 k p P(X = k) g((1 -k)/ √ N -1) + d N -1 (N -k) g(0) ≤ N k=1 k p P(X = k) g(0) + d N -1 (N -k) g(0) ∼ N k=1 k p P(X = k) → ∞ k=1 k p P(X = k) = E(X p ),
where we have used that the density g is non-decreasing on (-∞, 0] and that d N -1 (N -k) → 0 as N → ∞ uniformly for all k by the local limit theorem. Similarly,

E(µ p N,1 ) ≥ ⌊N 1/4 ⌋ k=1 k p P(X = k) P(S N -1 = N -k) P(S N = N ) ∼ ⌊N 1/4 ⌋ k=1 k p P(X = k) g((1 -k)/ √ N -1) + d N -1 (N -k) g(0) ≥ ⌊N 1/4 ⌋ k=1 k p P(X = k) g((1 -N 1/4 )/ √ N -1) + d N -1 (N -k) g(0) ∼ ⌊N 1/4 ⌋ k=1 k p P(X = k) → ∞ k=1 k p P(X = k) = E(X p ),
where we used again that g is non-decreasing on (-∞, 0] and that the convergence in the local limit theorem holds uniformly for all k. Thus, the convergence E(µ p N,1 ) → E(X p ) as N → ∞, p ∈ (0, ∞), is established. The rest of the proof is identical to the last four lines of Proof 1.

2

Proof. (of Theorem 2.3) We verify that lim N →∞ c N = 0. Fix ε ∈ (0, 1) and define A i := {µ i ≤ N ε} and B i := {µ i > N ε}, i ∈ {1, . . . , N }. As in the proof of [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF]Theorem 19.2], applied to the weights w k := e -θφ(r-) σ k (θ)/k! and with n := m := N , it follows that there exists a constant

c ε ∈ (-∞, 0) such that P(µ 1 > N ε) ≤ exp(c ε N + o(N )).
In particular, lim N →∞ N P(µ 1 > N ε) = 0 and

c N = 1 (N ) 2 N i=1 E((µ i ) 2 1 Ai ) + 1 (N ) 2 N i=1 E((µ i ) 2 1 Bi ) ≤ N ε (N ) 2 N i=1 E(µ i 1 Ai ) + N (N ) 2 E((µ 1 ) 2 1 B1 ) ≤ N ε (N ) 2 E(µ 1 + • • • + µ N ) + N E(1 B1 ) = N 2 ε (N ) 2 + N P(µ 1 > N ε) → ε, N → ∞.
Thus, lim N →∞ c N = 0, since ε > 0 can be chosen arbitrarily small. In order to determine the asymptotics of the associated compound Poisson population model, let µ (1) ≥ • • • ≥ µ (N ) denote the ranked offspring sizes, i.e. the offspring sizes µ 1 , . . . , µ N , but permutated in non-increasing order. For all ε > 0 we have P(µ (1) > N ε) = P(

N i=1 {µ i > N ε}) ≤ N P(µ 1 > N ε) → 0 as N → ∞. Thus, µ (1) /N → 0 in probability as N → ∞. For arbitrary but fixed dimension d ∈ N it follows that (µ (1) /N, . . . , µ (d) /N ) → (0, . . . , 0) ∈ R d in distribution as N → ∞. Treating the infinite simplex ∆ := {x = (x 1 , x 2 , . . .) : x 1 ≥ x 2 ≥ • • • ≥ 0, ∞ i=1 x i ≤ 1}
as a subset of the metric space R ∞ equipped with the topology of pointwise convergence, this convergence of the finite-dimensional distributions is already equivalent (see, Billingsley [2,p. 19]) to the convergence of the full processes (µ (1) /N, . . . , µ (N ) /N, 0, 0, . . .) → (0, 0, . . .) ∈ ∆ in distribution as N → ∞. Comparing this limit with Sagitov's convergence result [START_REF] Sagitov | Convergence to the coalescent with simultaneous multiple mergers[END_REF]Theorem 2.1] shows that the model is in the domain of attraction of the Kingman coalescent.

2

Proof. (of Theorem 2.4) Let X 1 , X 2 , . . . be independent random variables all with the same distribution (16) of X. Define

S N := X 1 + • • • + X N , N ∈ N.
For all k ∈ N 0 and all N ≥ k,

P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) = P(X = k) P(S N -1 = N -k) P(S N = N ) .
The basic idea of the proof is to apply local limit theorems to the two probabilities P(S N -1 = N -k) and P(S N = N ) in the formula above. It turns out that the details of the proof differ for parts a), b) and c), so we have to treat these parts separately. a) By assumption, φ ′′ (r-) < ∞ or, equivalently, Var(X) < ∞. Following the proof of Theorem 2.2, but with the root z(θ) replaced by r, it follows that µ N,1 → X in distribution as N → ∞, where X has distribution [START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF]. For p ∈ (0, 2] the convergence of moments E(µ p N,1 ) → E(X p ) as N → ∞ is shown in the same way as in Proof 2 of Theorem 2.2 by exploiting the standard local limit theorem for sequences of independent and identically distributed random variables having finite and non-vanishing variance. In particular, (N -

1)c N = E((µ N,1 ) 2 ) → E((X) 2 ) = Var(X) = 1 + θr 2 φ ′′ (r-) as N → ∞. Thus,
d N (k) = √ N log N P(S N = k) -g((k -N )/ √ N log N ), N ∈ N, k ∈ Z, and g(x) := (πc) -1/2 exp(-x 2 /c), x ∈ R,
is the density of the normal distribution N (0, c/2). Applying this local limit theorem to both probabilities P(S N = N ) and P(S N -1 = N -k) it follows in the same way as in Proof 2 of Theorem 2.2 that µ N,1 → X in distribution as N → ∞. Let us now turn to the asymptotics of the coalescence probability c N . We have

(N -1)c N = E((µ N,1 ) 2 ) = N k=2 (k) 2 P(µ N,1 = k) = N k=2 (k) 2 P(X = k) P(S N -1 = N -k) P(S N = N ) (24)
∼ N k=2 (k) 2 P(X = k) g((1 -k)/ √ N log N ) + d N -1 (N -k) g(0) ≥ ⌊ √ N ⌋ k=2 (k) 2 P(X = k) g((1 -k)/ √ N log N ) + d N -1 (N -k) g(0) ≥ ⌊ √ N ⌋ k=2 (k) 2 P(X = k) g(-1/ √ log N ) + d N -1 (N -k) g(0) ∼ ⌊ √ N ⌋ k=2 (k) 2 P(X = k).
Since P(X = k) ∼ ck -3 as k → ∞, the latter sum is asymptotically equal to c

⌊ √ N ⌋ k=2 1/k ∼ c √ N 2 1/x dx ∼ c log √ N = (c/2) log N . Thus, lim inf N →∞ (N c N )/ log N ≥ c/2. It remains to verify that lim sup N →∞ (N c N )/ log N ≤ c/2. Define B N := ⌊ √ N log N ⌋, N ∈ N,
and decompose the sum in (24) into a first part over all k ≤ B N and a second part over all k > B N . The second part is negligible, since, uniformly for all k ∈ {B N + 1, . . . , N },

P(S N -1 = N -k) P(S N = N ) ∼ g((1 -k)/ √ N log N ) + d N -1 (N -k) g(0) ≤ g(- √ log N ) + d N -1 (N -k) g(0) → 0
as N → ∞. Note that we used that the normal density g is non-decreasing on (-∞, 0], that g(-√ log N ) → 0 as N → ∞ and that d N -1 (N -k) → 0 as N → ∞ uniformly for all k. For the first part we use g(x) ≤ g(0), x ∈ R, and obtain

B N k=2 (k) 2 P(X = k) P(S N -1 = N -k) P(S N = N ) ∼ B N k=2 (k) 2 P(X = k) g((1 -k)/ √ N log N ) + d N -1 (N -k) g(0) ≤ B N k=2 (k) 2 P(X = k) g(0) + d N -1 (N -k) g(0) ∼ B N k=2 (k) 2 P(X = k),
again thanks to the fact that the convergence in the local limit theorem holds uniformly for all k. From P(X = k) ∼ ck -3 as k → ∞ it follows that the last sum is asymptotically equal to c

B N k=2 1/k ∼ c log B N ∼ (c/2) log N . Thus, lim sup N →∞ (N c N )/ log N ≤ c/2 and the asymptotics c N ∼ (c/2)(log N )/N is established. c) Let us first verify that P(X = k) ∼ ck -α-1 as k → ∞ with c := θκr α /Γ(-α). It is readily checked that (18) implies that exp(θφ(z)) = A + B(z -r) + C(r -z) α + O((r -z) 2 ), z → r,
with constants A := exp(θφ(r-)), B := θφ ′ (r-)A, and C := θκA. From that representation it follows that

P (X = k) = r k exp(-θφ(r-)) σ k (θ) k! = r k A [z k ] exp(θφ(z)) ∼ r k A C[z k ](r -z) α = r k A C r α r k α k (-1) k ∼ C A r α 1 Γ(-α)k α+1 = c k α+1 , k → ∞.
By the generalized central limit theorem (see also Lemma 5.3 in the appendix), (S N -N )/N 1/α → S in distribution as N → ∞, where S is an α-stable random variable with characteristic function ϕ(t)

:= E(exp(itS)) = exp(cΓ(-α)(-it) α ), t ∈ R. Define the constant C := -cΓ(-α) cos πα 2 ∈ (0, ∞) for convenience. For all p ∈ [0, ∞), ∞ -∞ |t| p |ϕ(t)| dt (25) = ∞ -∞ |t| p exp(cΓ(-α)Re((-it) α )) dt = ∞ -∞ |t| p exp(cΓ(-α)|t| α cos πα 2 ) dt = 2 ∞ 0 t p exp(-Ct α ) dt = 2 αC (p+1)/α ∞ 0 u (p+1)/α-1 exp(-u) du = 2Γ((p + 1)/α) αC (p+1)/α ∈ (0, ∞), (26)
where we have used the substitution u = Ct α (⇒ dt/du = 1/(αC 1/α )u 1/α-1 ). In particular (choose p = 0), ϕ is integrable with respect to the Lebesgue measure on R. Thus, by the Fourier inversion formula, S has density

g(x) = 1 2π ∞ -∞ exp -ixt + cΓ(-α)(-it) α dt = 1 π ∞ 0 exp cΓ(-α)
) := N 1/α P(S N = k) -g((k -N )/N 1/α ), N ∈ N, k ∈ Z. In particular, P(S N = N ) = (g(0) + d N (N ))/N 1/α ∼ g(0)/N 1/α and P(S N -1 = N -k) = g((1 -k)/(N -1) 1/α ) + d N -1 (N -k) (N -1) 1/α
The convergence µ N,1 → X in distribution therefore follows again in the same way as in Proof 2 of Theorem 2.2. We furthermore obtain

N c N ∼ E((µ N,1 ) 2 ) ∼ E(µ 2 N,1 ) = N k=1 k 2 P(X 1 = k) P(S N -1 = N -k) P(S N = N ) ∼ N k=1 k 2 P(X = k) g((1 -k)/(N -1) 1/α ) + d N -1 (N -k) g(0) = T 1 + T 2 ,
where

T 1 := 1 g(0) N k=1 k 2 P(X = k) g((1 -k)/(N -1) 1/α )
and

T 2 := 1 g(0) N k=1 k 2 P(X = k) d N -1 (N -k).
Let us first analyse T 2 . Since [START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF] holds by assumption, it follows that sup k∈Z |d N -1 (Nk)| = O(N 1-2/α ) by Theorem 5.4 (strong local limit theorem) in the appendix. Moreover,

N k=1 k 2 P(X = k) ∼ c N k=1 k 1-α = O(N 2-α ). Thus, T 2 = O(N 1-2/α ) N k=1 k 2 P(X = k) = O(N 1-2/α )O(N 2-α ) = O(N δ ) with δ := 3 -2/α -α.
Let us now turn to T 1 . Since the density g is bounded, any finite number of values of the summation index k in the sum of T 1 do not contribute to the asymptotics of T 1 . It is therefore asymptotically allowed to replace P(X = k) by its asymptotic expression ck -α-1 and we obtain

T 1 ∼ c g(0) N k=1 k 1-α g((1 -k)/(N -1) 1/α ).
For the last sum we have

N k=1 k 1-α g((1 -k)/(N -1) 1/α ) = N k=1 N 1/α (k+1)/N 1/α k/N 1/α ⌊xN 1/α ⌋ 1-α g((1 -⌊xN 1/α ⌋)/(N -1) 1/α ) dx = N 1/α (N +1)/N 1/α 1/N 1/α ⌊xN 1/α ⌋ 1-α g((1 -⌊xN 1/α ⌋)/(N -1) 1/α ) dx ∼ N 1/α ∞ 0 x 1-α N (1-α)/α g(-x) dx = N (2-α)/α ∞ 0 x 1-α g(-x) dx.
Note that the last integral is finite, since

∞ 0 x 1-α g(-x) dx ≤ g 1 0 x 1-α dx+ ∞ 1 g(-x) dx ≤ g /(2 -α) + 1 < ∞. Since δ = 3 -2/α -α < (2 -α)/α, it follows in summary that N c N ∼ T 1 + T 2 ∼ T 1 ,
(θ) = r k σ k (θ), k ∈ N 0 , θ ∈ (0, ∞). Define Φ(z) := Φ θ (z) := exp(θφ(z)) for |z| < 1 and θ ∈ (0, ∞). Note that [z k ]Φ(z) = σ k (θ)/k! for k ∈ N 0 and θ ∈ (0, ∞).
By assumption there exist constants c = c(θ) ∈ (0, ∞) and β = β(θ) > 2 such that σ k (θ)/k! ∼ ck -β as k → ∞. We are hence in the situation to apply Janson's [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF] condensation result (Theorem 19.34) to the weights

w k := σ k (θ)/k!, k ∈ N 0 . Note that the weight sequence (w k ) k∈N0 has pgf ∞ k=0 w k z k = exp(θφ(z)) = Φ(z), |z| < 1. Moreover, in Janson's notation, Ψ(z) := zΦ ′ (z)/Φ(z) = θzφ ′ (z), |z| < 1,
(m, n) = [z m ]Φ nθ (z) = σ m (nθ)/m! satisfies σ N -k ((N -l)θ) (N -k)! ∼ c (λ -ν) β (Φ(1-)) N -l-1 N β-1 (27)
as N → ∞. For all k ∈ N 0 and all N ∈ N with N ≥ k we have

P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) . ( 28
)
Using the asymptotics (27) for l = 0 and for l = 1 it follows that lim

N →∞ P(µ N,1 = k) = σ k (θ) k! 1 Φ(1-) = σ k (θ) k! exp(-θφ(1-)), k ∈ N 0 ,
which is the right hand side of ( 16) for r = 1. Thus, µ N,1 → X in distribution as N → ∞.

As in the proof of Theorem 2.3 let µ (1) ≥ • • • ≥ µ (N ) denote the ranked offspring sizes. Theorem 19.34 (i) of Janson [START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF] implies that (µ (1) /N, . . . , µ (N ) /N, 0, 0, . . .) → (u, 0, 0, . . .)

in distribution as N → ∞, where u := λ -ν = 1 -θφ ′ (1-) ∈ (0, 1).

In the following it is verified that lim N →∞ c N = u 2 . We have µ (1) /N → u in distribution as N → ∞. Since µ (1) /N is bounded for all N (between 0 and 1) it follows that µ (1) /N → u in L 2 . In particular,

E((µ (1) ) 2 )/(N ) 2 → u 2 as N → ∞. Since c N = 1 (N ) 2 N i=1 E((µ (i) ) 2 ) = E((µ (1) ) 2 ) (N ) 2 + 1 (N ) 2 N i=2 E((µ (i) ) 2 )
it hence remains to verify that lim

N →∞ 1 (N ) 2 N i=2 E((µ (i) ) 2 ) = 0. ( 30
)
In order to establish the convergence (30) fix ε ∈ (0, 1) and define A := {µ (2) ≤ N ε} and its complement B := {µ (2) > N ε}. We have

1 (N ) 2 N i=2 E((µ (i) ) 2 ) = 1 (N ) 2 N i=2 E((µ (i) ) 2 1 A ) + 1 (N ) 2 N i=2 E((µ (i) ) 2 1 B ) ≤ N ε (N ) 2 N i=2 E(µ (i) 1 A ) + N (N ) 2 N i=2 E(µ (i) 1 B ) = N ε (N ) 2 E((µ (2) + • • • + µ (N ) )1 A ) + N (N ) 2 E((µ (2) + • • • + µ (N ) )1 B ) ≤ N 2 ε (N ) 2 + N 2 (N ) 2 P(µ (2) > N ε) → ε, as N → ∞, since P(µ (2) > N ε) → 0 as N → ∞.
Since ε > 0 can be chosen arbitrarily small, (30) holds and, hence, the convergence lim N →∞ c N = u 2 ∈ (0, 1) is established. Comparing the limit (29) with Sagitov's convergence result [START_REF] Sagitov | Convergence to the coalescent with simultaneous multiple mergers[END_REF]Theorem 2.2] it follows that the associated compound Poisson model is in the domain of attraction of a discrete-time Ξ-coalescent with Ξ(dx)/ ∞ i=1 x 2 i = δ (u,0,0,...) (dx) being the Dirac measure at (u, 0, 0, . . .) ∈ ∆ := {x = (x 1 , x 2 , . . .) :

x 1 ≥ x 2 ≥ • • • ≥ 0, ∞ i=1 x i ≤ 1}.
Hence, the limiting process is a discrete-time coalescent with multiple collisions (Λ-coalescent) with Λ(dt)/t 2 = δ u (dt) being the Dirac measure at u. 2

Examples

We start with the two most popular examples, the Wright-Fisher model and the Dirichlet model. Note that (6) holds for these two models. These two examples have the advantage that most calculations can be done explicitly. For example, we will verify the asymptotic results stated in Theorem 2.2 directly.

(θ) = θ k , k ∈ N 0 , θ ∈ (0, ∞).
For k ∈ N 0 and θ ∈ (0, ∞) it follows that

P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) = N k 1 N k 1 - 1 N N -k .
Thus, µ N,1 has a binomial distribution with parameters N and 1/N . In particular, for arbitrary but fixed p ∈ N, E((µ N,1 Therefore, ξ 1 has a negative binomial distribution with parameters θ and 1 -z. In particular,

) p ) = (N ) p /N p → 1 = E((X) p ) as N → ∞,
E(ξ 1 ) = θzφ ′ (z) = θz/(1 -z). Moreover, σ k (θ) = [θ] k := θ(θ + 1) • • • (θ + k -1) = Γ(k + θ)/Γ(θ), k ∈ N 0 , θ ∈ (0, ∞).
Thus, for k ∈ N 0 and θ ∈ (0, ∞),

P(µ N,1 = k) = N k [θ] k [(N -1)θ] N -k [N θ] N = [θ] k k! (N ) k Γ(N θ -θ + N -k)Γ(N θ) Γ(N θ -θ)Γ(N θ + N ) .
Since Γ(x + c) ∼ x c Γ(x) as x → ∞ for any c ∈ R, it follows that

P(µ N,1 = k) ∼ [θ] k k! (N ) k (N θ + N ) -θ-k Γ(N θ + N )Γ(N θ) (N θ) -θ Γ(N θ)Γ(N θ + N ) = [θ] k k! (N ) k (N θ + N ) k θ + 1 θ -θ → [θ] k k! 1 θ + 1 k θ θ + 1 θ . Thus µ N,1 → X in distribution as N → ∞
where X has a negative binomial distribution with parameters θ and θ/(θ + 1) ∈ (0, 1). Moreover, for arbitrary but fixed p ∈ N, E((µ N,1 Thus,E(µ p N,1 ) → E(X p ) as N → ∞ for all p ∈ N in agreement with the general results derived in the proof of Theorem 2.2. The solution z(θ) of the equation θz(θ)φ ′ (z(θ)) = 1 is z(θ) = 1/(θ+1). Note that (21) holds with a(θ) := (θ+1) θ+1 /θ θ and b kl (θ) := (θ/(θ + 1)) lθ+1/2 (1/(θ + 1)) k , k, l ∈ N 0 , θ ∈ (0, ∞). The symmetric Dirichlet model has effective population size N e = 1/c N = (N -1)/E((µ N,1 ) 2 ) = (N θ + 1)/(θ + 1) ∼ ̺N with ̺ = θ/(θ + 1) = 1/Var(X), in agreement with Theorem 2.2.

) p ) = (N ) p [θ] p /[N θ] p → [θ] p /θ p = E((X) p ) as N → ∞.
Let us now study examples which do not satisfy [START_REF] Wakeley | Coalescent processes when the distribution of offspring number among individuals is highly skewed[END_REF]. We start with a model which involves the absolute Lah numbers.

Example 4.3 Suppose that φ m = m! for all m ∈ N or, equivalently, that φ(z) = z/(1 -z), |z| < 1. Note that φ ′ (z) = 1/(1 -z) 2 and that φ ′′ (z) = 2/(1 -z) 3 = 2φ ′ (z)/(1 -z), |z| < 1. Then (Comtet [5, Section 3.3, p. 135, Theorem B]), σ n (θ) = n k=1 L(n, k)θ k , n ∈ N, θ ∈ R, where L(n, k) := B nk (1!, 2!, . . .) = n!(n -1)!/(k!(k -1)!(n -k)!), n ∈ N, k ∈ {1, . . . , n}, denote the absolute Lah numbers. The solution z = z(θ) ∈ (0, 1) of the equation 1 = θzφ ′ (z) = θz/(1 -z) 2 is z(θ) = 1 + θ/2 -θ(θ + 4)/2
. By Theorem 2.2, the model is in the domain of attraction of the Kingman coalescent and the effective population size N e satisfies N e ∼ ̺N with

̺ := 1/(1 + θz 2 φ ′′ (z)) = (1 -z)/(1 + z) < 1, since 1 + θz 2 φ ′′ (z) = 1 + θz 2 φ ′ (z) 2 1 -z = 1 + 2z 1 -z = 1 + z 1 -z .
Note that the asymptotics (21) holds with a(θ) = e θφ(z) /z = e θz/(1-z) /z = e 1-z /z and b kl (θ) :=

z k e -lθφ(z) / 1 + θz 2 φ ′′ (z) = z k e -l(1-z) / (1 + z)/(1 -z), k, l ∈ N 0 , θ ∈ (0, ∞).
In the following example, which generalizes Example 4.3, the series φ solves a particular functional equation and is hence (for most parameter constellations) only defined implicitly. Nevertheless an explicit expression for the solution z = z(θ) of the equation θzφ ′ (z) = 1 is derived.

φ m = (m -1)! am m-1 b m-1 , m ∈ N,
where it is assumed that the real parameters a and b are either both negative or b > 0 and a ≥ 1 (such that all the coefficients φ m are non-negative). Note that [1, Eq. ( 43)] the power series φ(z) = ∞ m=1 φ m z m /m! is the solution of the functional equation φ(z) = zf (φ(z)) with f (x) := (1 + bx) a and that φ is related to the generalized binomial series B a (see, for example, [8, p. 200

]) via φ(z) = (B a (bz) -1)/b. It is readily checked that φ has radius of convergence r := lim m→∞ (m + 1)φ m /φ m+1 = (ab) -1 (1 -1/a) a-1 ∈ (0, ∞) and that φ(r-) = 1/(b(a -1)) (= ∞ for a = 1), since, for a = 1, φ(r-) = 1/(b(a -1)
) is the only positive solution of the equation φ(r-) = rf (φ(r-)). Differentiating both sides of the functional equation φ(z) = zf (φ(z)) yields the derivatives

φ ′ (z) = f (φ(z)) 1 -zf ′ (φ(z)) and φ ′′ (z) = 2f ′ (φ(z))φ ′ (z) + zf ′′ (φ(z))(φ ′ (z)) 2 1 -zf ′ (φ(z)) , |z| < r.
(z) = 4z/(1 + √ 1 -4bz) 2 , |z| < 1/(4b). For a = -1 (and b < 0) we have φ m = (2m -2)!/(m -1)!(-b) m-1 , m ∈ N, and φ(z) = ( √ 1 + 4bz -1)/(2b) = 2z/(1 + √ 1 + 4bz), |z| < 1/(-4b).
For general parameters a and b, to the best of the authors knowledge, there seems to be no special function related to the power series φ. Nevertheless, an explicit expression for the solution z = z(θ) of the equation θzφ ′ (z) = 1 is obtained as follows. Let x = x(θ) be the solution in the open interval (0, 1) of the quadratic equation θx = b(1 -x)(a -x), i.e.

x(θ) = θ + b + ab -(θ + b + ab) 2 -4ab 2 2b ∈ (0, 1). ( 31
)
We have 0

< (1 -x)/θ < 1/(b(a -1)) = φ(r-) (= ∞ for a = 1). Thus, z = z(θ) := φ -1 ((1 -x)/θ) ∈ (0, r) is well defined. From (1 -x)/θ = φ(z) = zf (φ(z)) = zf ((1 -x)/θ) it follows that z = 1 -x θ 1 f ( 1-x θ ) = 1 -x θ 1 (1 + b 1-x θ ) a = 1 -x θ 1 (1 + x a-x ) a = 1 -x θ a -x a a = 1 -x θ a -x a a -x a a-1 = x ab a -x a a-1 . (32)
Moreover,

f ′ (φ(z)) = f ′ 1 -x θ = ab 1 + b 1 -x θ a-1 = ab 1 + x a -x a-1 = ab a a -x a-1 = x z
and, hence,

θzφ ′ (z) = θzf (φ(z)) 1 -zf ′ (φ(z)) = θφ(z) 1 -zf ′ (φ(z)) = 1 -x 1 -z x z = 1.
Thus, z = z(θ) satisfies θzφ ′ (z) = 1. Theorem 2.2 is therefore applicable. Straightforward computations show that the limiting random variable X in Theorem 2.2 has variance Var(X) = 1 + θz 2 φ ′′ (z) = (a -x 2 )/(a(1 -x) 2 ). By Theorem 2.2, the model is in the domain of attraction of the Kingman coalescent and the effective population size N e satisfies N e ∼ ̺N as N → ∞ with ̺ = 1/Var(X) = a(1 -x) 2 /(a -x 2 ) ∈ (0, 1). Note that the asymptotics (21) holds, where a(θ) = e θφ(z) /z = e 1-x /z and

b kl (θ) = z k e -lθφ(z) 1 + θz 2 φ ′′ (z) = z k e -l(1-x) a(1 -x) 2 a -x 2
with x = x(θ) and z = z(θ) defined in (31) and (32). At first glance the solution x(θ) of the quadratic equation θx = b(1 -x)(a -x) seems to come 'from nowhere'. In the following an intuitive argument is provided showing how x(θ) comes into play. It is known [1, Eq. ( 51)] that

σ n (θ) = n j=1 B(n, j)θ j with B(n, j) = (n -1)! (j -1)! an n -j b n-j . (33)
The fraction

B(n, n -j + 1)(nθ) n-j+1 B(n, n -j)(nθ) n-j = nθj (n -j)(an -j + 1)b
is equal to 1 for j = j n with

j n := n(θ + b + ab) + b -(n(θ + b + ab) + b) 2 -4b(bn + abn 2 ) 2b ∼ n θ + b + ab -(θ + b + ab) 2 -4ab 2 2b = nx(θ). (34)
Intuitively, when n is large, the contribution to the sum σ n (nθ) = n-1 j=0 B(n, n -j)(nθ) n-j is essentially entirely originated from indices j having the property that j/n belongs to a (small) neighborhood of x(θ). The choice of the neighborhood is rather unimportant. It is hence not surprising that x(θ) plays a crucial role in finding the asymptotics of σ n (nθ). Since the Bell numbers (33) are known explicitly, one may carry out the Laplace method in detail leading to an alternative proof of (21). We leave the details to the reader. For non-integer parameter α ∈ (0, ∞) \ N, the first ⌊α⌋ -1 derivatives of the polylog function φ are finite and φ has asymptotic expansion around z = 1 of the form

φ(z) = ⌊α⌋-1 j=0 φ (j) (1-) j! (z -1) j + Γ(1 -α)(1 -z) α-1 + O((1 -z) ⌊α⌋ ), z → 1, (35)
whereas for integer α ∈ N the expansion has the form (see also [7, p. 411])

φ(z) = α-2 j=0 φ (j) (1-) j! (z -1) j + (-1) α (α -1)! (1 -z) α-1 log(1 -z) + O((1 -z) α-1 ), z → 1. ( 36
)
In the following it is verified that, for all α, θ ∈ (0, ∞), the distribution (16) satisfies

P(X = k) = σ k (θ) k! exp(-θφ(1-)) ∼ θ k α , k → ∞. ( 37
)
Assume first that 2 < α < 3 . Then, by (35

), φ(z) = a + b(z -1) + Γ(1 -α)(1 -z) α-1 + O((1 -z) 2 ) as z → 1 with a := φ(1-) = ζ(α) and b := φ ′ (1-) = ζ(α -1)
, where ζ denotes the Riemann zeta function. Thus,

Φ θ (z) := exp(θφ(z)) = A + B(z -1) + C(1 -z) α-1 + O((1 -z) 2 ), z → 1,
where A := e θa , B := θbA and

C := θΓ(1 -α)A. Using [z k ](1 -z) α-1 = α -1 k (-1) k = Γ(k -α + 1) Γ(1 -α) k! ∼ 1 Γ(1 -α)k α , k → ∞, it follows that P(X = k) = exp(-θφ(1-)) σ k (θ) k! = 1 A [z k ]Φ θ (z) ∼ C A [z k ](1 -z) α-1 ∼ θ k α , k → ∞,
and (37) is established for 2 < α < 3. For arbitrary non-integer α ∈ (0, ∞) (37) follows from (35) similarly as shown above for 2 < α < 3. For integer α ∈ N (37) follows from (36) and from [START_REF] Flajolet | Analytic Combinatorics[END_REF]p. 387,Eq. (24)]. Thus, (37) holds for arbitrary α ∈ (0, ∞).

We now turn to the asymptotic behavior of the model as N → ∞. If α ≤ 2, then φ ′ (1-) = ∞ and Theorem 2.2 is applicable for all θ ∈ (0, ∞), so the associated symmetric compound Poisson model is in the domain of attraction of the Kingman coalescent. There seems to be no closed expression available for the solution z = z(θ) of the equation θzφ ′ (z) = 1. The case α = 2 is a nice exception where it is easily seen that z(θ

) = 1 -e -1/θ , θ ∈ (0, ∞). Assume now that α > 2. Then φ ′ (1-) = ζ(α -1) < ∞.
In this case the equation θzφ ′ (z) = 1 admits a solution z = z(θ) ∈ (0, 1) if and only if θ > θ c with critical value θ c := 1/φ ′ (1-) < 1. Thus, Theorem 2.2 is not applicable for θ ≤ θ c . Note that φ ′′

(1-) = a -1 > 0. Moreover, φ ′ (z) = a -α(1 -z) α-1 , φ ′′ (z) = α(α -1)(1 -z) α-2 , and φ ′ (1-) = a ∈ (1, ∞) and φ ′′ (1-) = ∞. It is readily checked that Φ θ (z) := exp(θφ(z)) = A + B(z -1) + C(1 -z) α + O((1 -z) 2 ), z → 1,
where A := Φ θ (1-) = exp(θ(a -1)), B := Φ ′ θ (1-) = θaA and C := θA. Thus,

P(X = k) := exp(-θφ(1-)) σ k (θ) k! = 1 A [z k ]Φ θ (z) ∼ C A [z k ](1 -z) α ∼ c k α+1
as k → ∞, where c := C/(AΓ(-α)) = θ/Γ(-α). If θa > 1 (supercritical case), then, by Theorem 2.2, the associated symmetric compound Poisson population model is in the domain of attraction of the Kingman coalescent and the coalescence probability c N satisfies c N ∼ (1 + θz 2 φ ′′ (z))/N as N → ∞, where z = z(θ) is the solution of the equation θzφ ′ (z) = 1 in the interval (0, 1). If θa = 1 (critical case) then, by Theorem 2.3, the model is still in the domain of attraction of the Kingman coalescent, and, by Theorem 2.4 c), applied with constants κ = 1 and c = θ/Γ(-α) = 1/(aΓ(-α)), the coalescence probability c N satisfies

c N ∼ 1 aΓ(-α) ∞ 0 x 1-α g(-x) dx g(0) N 2(1-α)/α , N → ∞,
with the density g as defined in Theorem 2.4 c). If 0 < θa < 1 (subcritical case), then, by Theorem 2.5, the model is in the domain of attraction of a discrete-time Λ-coalescent with Λ := u 2 δ u and u := 1 -θφ ′ (1-) = 1 -θa ∈ (0, 1).

The following last example has much in common with the polylog model (Example 4.6).

= log(Z(z)), |z| < 1. Note that σ k (θ)/k! = [z k ] exp(θφ(z)) = [z k ](Z(z)) θ ≥ θ/(k + 1) α for all k ∈ N and all θ ∈ (0, ∞). Therefore φ k = lim θ→0 σ k (θ)/θ ≥ k!/(k + 1) α > 0 for all k ∈ N, so φ is a power series of the form φ(z) = ∞ k=1 φ k z k /k! with strictly positive coefficients φ k > 0, k ∈ N. For α = 1 we have φ(z) = log(-log(1 -z)/z) and for α → 0 we approach the Dirichlet model (Example 4.2) with φ(z) = -log(1 -z), |z| < 1. For 2 < α < 3 we have Z(z) = ζ(α) + ζ(α -1)(z -1) + Γ(1 -α)(1 -z) α-1 + O((1 -z) 2 ), z → 1. Proof. Since f (t) = f (-t) for t ∈ R it suffices to find some ε 0 > 0 such that sup ε<t≤π f (t) ≤ f (ε) for all ε ∈ [0, ε 0 ]. Obviously, f (0) -f (t) = ∞ m=1 a m (1 -cos(mt)) ≥ a 1 ( 1
-cos t) > 0 for t ∈ (0, 2π), so f (t) < f (0) for t ∈ (0, 2π). For arbitrary but fixed t ∈ (-∞, log q), the map (m, k) → a m (-1) k (mt) 2k /(2k)! is integrable with respect to the counting measure ε

N ⊗ ε N0 on N × N 0 , since ∞ m=1 ∞ k=0 a m (-1) k (mt) 2k (2k)! ≤ ∞ m=1 a m ∞ l=0 (mt) l l! = ∞ m=1 a m e mt ≤ ∞ m=1 a m q m < ∞
by assumption. Thus, by Fubini's theorem,

f (t) = ∞ m=1 a m cos(mt) = ∞ m=1 a m ∞ k=0 (-1) k (mt) 2k (2k)! = ∞ k=0 (-1) k t 2k (2k)! ∞ m=1 m 2k a m .
so on the interval (-∞, log q) the function f is a power series in t. Together with f ′ (0) = 0 and f ′′ (0) = -∞ m=1 m 2 a m < 0 it follows that there exists a constant δ ∈ (0, π) such that f is nonincreasing on [0, δ]. The continuous function f takes its supremum on the compact set

[δ, π], so sup t∈[δ,π] f (t) = f (t 0 ) for some t 0 ∈ [δ, π]. Now choose ε 0 ∈ (0, δ) sufficiently small such that f (ε 0 ) > f (t 0 ), which is possible since f is continuous and f (0) > f (t 0 ). Now let ε ∈ [0, ε 0 ] arbitrary but fixed. For t ∈ (ε, δ] we obtain f (t) ≤ f (ε), since f is non-increasing on [ε, δ]. For t ∈ [δ, π] we have f (t) ≤ f (t 0 ) < f (ε 0 ) ≤ f (ε), since f is non-increasing on [ε, ε 0 ]. Thus sup ε<t≤π f (t) ≤ f (ε).2
In the following we provide some results for certain α-stable distributions, which, to the best of the authors knowledge, are not stated or at least quite hard to find in the literature. Our first result concerns half-sided moments. Lemma 5.2 Fix d > 0 and 1 < α < 2. Let S be an α-stable random variable with characteristic function t → exp(d(-it) α ), t ∈ R, and let g denote the density of S. Then, for all β ∈ (-1, 0),

E((-S) β 1 {S≤0} ) = ∞ 0 x β g(-x) dx = d β/α Γ(β) Γ(β/α) .
Moreover, P(S ≤ 0) = 1/α and g(0

) = d -1/α |Γ(-1/α)| -1 .
Proof. Since S has the same distribution as d 1/α S 0 , where S 0 is a random variable with characteristic function t → exp((-it) α ), t ∈ R, we may assume without loss of generality that d = 1. By the Fourier inversion formula,

∞ 0 x β g(-x) dx = ∞ 0 x β 1 2π ∞ -∞ exp(itx + (-it) α ) dt dx = 1 2π ∞ -∞ exp((-it) α ) ∞ 0 x β exp(itx) dx dt = Γ(β + 1) 2π ∞ -∞ exp((-it) α )(-it) -β-1 dt = Γ(β + 1) 2π Γ(-β/α) α 2 sin(-πβ/α) = Γ(β + 1) αΓ(1 + β/α) = Γ(β) Γ(β/α) ,
where the second last equality follows from Euler's reflection formula sin(πz)Γ(z) = π/Γ(1 -z), applied to the point z := -β/α < 1/α < 1. The relation P(S ≤ 0) = 1/α follows by taking the limit β → 0 in the second last fraction above. Similarly,

g(0) = 1 2π ∞ -∞ exp((-it) α ) dt = 1 2π Γ(1/α) α 2 sin(π/α) = 1 αΓ(1 -1/α) = |Γ(-1/α)| -1 . 2
Let X 1 , X 2 , . . . be independent copies of a random variable X having distribution [START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF]. We provide a weak limiting result for S N := X 1 +• • •+X N , N ∈ N, which holds under the asymptotic assumption (18) on the power series φ. The result provides detailed information on the rate of convergence in terms of the characteristic functions. As an application (see Theorem 5.4) a local limit theorem is obtained which provides information on the convergence rate.

Lemma 5.3 Assume that θrφ ′ (r-) = 1 (critical case). If there exist constants α ∈ (1, 2) and κ = κ(α) > 0 such that the power series φ of the considered compound Poisson model satisfies [START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF], then (S N -N )/N 1/α → S in distribution as N → ∞, where S is an α-stable random variable with characteristic function ϕ(t) := exp(θκr α (-it) α ), t ∈ R. Moreover, there exists a constant C > 0 such that the difference of the characteristic functions ϕ

(S N -N )/N 1/α of (S N -N )/N 1/α and S satisfies |ϕ (S N -N )/N 1/α (t) -ϕ(t)| ≤ |ϕ(t)|(e Ct 2 N 1-2/α -1) (38)
for all N ∈ N and all t ∈ R. ). Note that we have used that θrφ ′ (r-) = 1 (critical case). Thus, for all t ∈ R, E(e it(S N -N )/B N ) = e log ϕ(t)+O(t 2 N/B 2 N ) = ϕ(t)e O(t 2 N/B 2 N ) ,

where the constant of the O-term so far does not depend on t and N , which implies that (38) holds.

For arbitrary but fixed t we in particular obtain E(e it(S N -N )/B N ) = ϕ(t)(1 + O(N/B 2 N ) = ϕ(t) + O(N/B 2 N ).

Note that N/B 2 N = N 1-2/α → 0 as N → ∞. By the continuity theorem it follows that (S N -N )/N 1/α → S in distribution, where S has characteristic function ϕ.

2

We are now able to verify a local limit theorem which provides information on the rate of convergence.

Theorem 5.4 (strong local limit theorem) Assume that the conditions of Lemma 5.3 are satisfied. Then, the local limit theorem where we choose k N := B N / √ N = N 1/α-1/2 and where ε ∈ (0, π) is a constant, to be determined. Note that the integrals I j , j ∈ {1, 2, 3, 4}, coincide with those on p. 122 of [START_REF] Ibragimov | Independent and Stationary Sequences of Random Variables[END_REF], except for the crucial modification that the constant A on p. 122 in [START_REF] Ibragimov | Independent and Stationary Sequences of Random Variables[END_REF] is replaced by k N satisfying k N ≥ 1 for all N ∈ N and k N → ∞ as N → ∞. We now turn to the estimation of these integrals and will show that I 1 = O(N 1-2/α ) and that I 2 , I 3 and I 4 are smaller than O(N 1-2/α ). (2) As in the proof of Theorem 4.2.1 of [START_REF] Ibragimov | Independent and Stationary Sequences of Random Variables[END_REF] it follows that, for any δ ∈ (0, α), there exist constants c(δ) ∈ (0, ∞) and ε(δ) > 0 not depending on N and a constant N 0 (δ) ∈ N such that |ϕ X (t/B N )| N ≤ e -c(δ)|t| δ for all |t| ≤ ε(δ)B N and all N ≥ N 0 (δ). Thus, choosing δ := α/2 and using the notation c := c(α/2) and ε := ε(α/2), it follows for all N ≥ N 0 (α/2) that

I 2 ≤ k N ≤|t|≤εB N e -c|t| α/2 dt ≤ 2 ∞ k N e -ct α/2 dt ≤ 2 ∞ k N e -c √ t dt ∼ 4 c k N e -c √ k N ,
a quantity being smaller than O(N β ) for any β < 0, so in particular smaller than O(N 1-2/α ).

(3) From [12, Theorem 1.4.2] it follows that there exists a constant q ∈ (0, 1) such that |ϕ X (u)| ≤ q for all u ∈ [ε, π]. Thus, I 3 = B N ε≤|u|≤π |ϕ X (u)| N du ≤ 2πB N q N , a quantity being smaller than O(N β ) for any β < 0.

(4) In order to estimate the integral I 4 define c := θκr α ∈ (0, ∞) and D := -c cos πα 2 ∈ (0, ∞) for convenience. We have

I 4 =

Theorem 2 . 3 (

 23 critical case, convergence) If θrφ ′ (r-) = 1, then lim N →∞ c N → 0 and, in the sense of [9, Definition 2.1 (a)], the associated symmetric compound Poisson population model is in the domain of attraction of the Kingman coalescent.

  part a) of Theorem 2.4 is established. b) Under the situation b) the local limit theorem lim N →∞ sup k∈N |d N (k)| = 0 holds, where

  and, hence, ν := Ψ(1-) = θφ ′ (1-) < 1 by assumption. Fix k, l ∈ N 0 . Applying[START_REF] Janson | Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation[END_REF] Theorem 19.34 (ii)] (with number of balls m := N -k and number of boxes n := N -l and N → ∞, which implies m/n → 1 =: λ > ν) it follows for arbitrary but fixed k, l ∈ N 0 and θ ∈ (0, ∞) that the partition function Z

Example 4 . 1 (

 41 Wright-Fisher model) For the standard symmetric Wright-Fisher model, φ(z) = z. Therefore, ξ n has a Poisson distribution with parameter θz and σ k

  where X has a Poisson distribution with parameter 1. The solution z(θ) of the equation θz(θ)φ ′ (z(θ)) = 1 is z(θ) = 1/θ. Note that (21) holds with a(θ) := eθ and b kl (θ) := θ -k e -l , k, l ∈ N 0 , θ ∈ (0, ∞). The effective population size N e = 1/c N = N coincides with the actual population size N , a well known result.

Example 4 . 2 (

 42 Dirichlet model) For the symmetric Dirichlet model, φ(z) = -log(1-z), |z| < 1.

Example 4 . 4

 44 Suppose that (see [1, p. 402, Eqs. (50) and (51)])

For a = 1 (

 1 and b > 0) we have φ m = m!b m-1 , m ∈ N, and φ(z) = z/(1 -bz), |z| < 1/b. For a = b = 1 we are back in Example 4.3. For a = 2 we have φ

Example 4 . 5

 45 Suppose that φ m = 1 for all m ∈ N or, equivalently, that φ(z) = e z -1, z ∈ C. Then (see, for example, Comtet[START_REF] Comtet | Advanced Combinatorics[END_REF] Section 3.3, p. 135, Theorem B]) σ n (θ) = n k=1 S(n, k)θ k , n ∈ N, θ ∈ R,where the S(n, k) denote the Stirling numbers of the second kind. In this case the solution z = z(θ) of the equation 1 = θzφ ′ (z) = θze z cannot be expressed in closed form anymore. By Theorem 2.2, µ N,1 → X in distribution, where X has distribution[START_REF] Karlin | Direct product branching processes and related Markov chains. I. Calculations of rates of approach to homozygosity[END_REF], and the model is in the domain of attraction of the Kingman coalescent. The effective population size N e satisfies N e ∼ ̺N with̺ := 1/(1 + θz 2 φ ′′ (z)) = 1/(1 + θz 2 e z ) = 1/(1 + z) < 1.For two further examples where Theorem 2.2 is applicable we refer the reader to[START_REF] Huillet | Correction on 'Population genetics models with skewed fertilities: a forward and backward analysis[END_REF] Examples 1 and 2]. All examples considered so far satisfy φ ′ (r-) = ∞, which implies that for all θ ∈ (0, ∞) the equation θzφ ′ (z) = 1 has a unique real solution z = z(θ) ∈ (0, r). Let us now study examples satisfying φ ′ (r-) < ∞.

Example 4 . 6 (

 46 Polylog model) Fix α ∈ (0, ∞) and suppose that φ is the polylog function, i.e. φ(z) := ∞ m=1 m -α z m , |z| < 1. For α = 1 the polylog model coincides with the Dirichlet model (Example 4.2 with φ(z) = -log(1 -z)). For α → 0 we are back in Example 4.3 whereas for α → ∞ we approach the Wright-Fisher model (Example 4.1).

( 1 -Example 4 . 8 (

 148 ) = ∞ for α ≤ 3 and φ ′′ (1-) = ζ(α -2) -ζ(α -1) < ∞ for α > 3. If θ = θ c (critical case), then by Theorem 2.3, the model is still in the domain of attraction of the Kingman coalescent and, by Theorem 2.4 (part c) applied with α replaced by α -1, so with as k → ∞. If θ > 1/φ ′ (1-) = 2 (supercritical case), then, by Theorem 2.2, the associated compound Poisson model is in the domain of attraction of the Kingman coalescent and the coalescence probability c N satisfies lim N →∞ N c N = 1 + θz 2 φ ′′ (z), where z is the solution of the equation θzφ ′ (z) = 1 in the interval (0, 1). If θ = 2 (critical case), then, by Theorem 2.3, the compound Poisson model is in the domain of attraction of the Kingman coalescent and, by (17), applied with constant c := θ = 2, the coalescence probability satisfies c N ∼ (log N )/N as N → ∞.If θ ∈ (0, 2) (subcritical case), then, by Theorem 2.5, the model is in the domain of attraction of a discrete-time Λ-coalescent with Λ(du) := u 2 δ u and u := 1 -θφ ′ (1-) = 1 -θ/2 ∈ (0, 1).We provide another example satisfying φ ′ (r-) < ∞ and φ ′′ (r-) = ∞. Lamperti model) Fix α ∈ (1, 2) and a ∈ (α, ∞) and assume that φ(z) = az -1 + (1 -z) α . Note that φ 1 = a -α > 0 and φ m = (-1) m (α) m > 0 for m ∈ {2, 3, . . .}. The series φ has radius of convergence r = 1 with φ

Example 4 . 9 (

 49 exp polylog model) Fix α ∈ (0, ∞), let Z(z) := ∞ k=0 z k /(k + 1) α be the polylog function divided by z and define φ(z) :

Proof. (of Lemma 5 . 3 )

 53 Let t ∈ R. By[START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF], the characteristic function ϕ X of the random variable X with distribution[START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF] satisfies log ϕ X (t) θ = φ(re it ) -φ(r-) = φ ′ (r-)(re it -r) + κ(r -re it ) α + O((r -re it ) 2 ) = rφ ′ (r-)(e it -1) + κr α (1 -e it ) α + O(r 2 (1 -e it ) 2 ) = rφ ′ (r-)(it + O(t 2 )) + κr α (-it + O(t 2 )) + O(t 2 ) = rφ ′ (r-)it + κr α (-it) α + O(t 2 ), t → 0.For convenience, define B N := N 1/α for all N ∈ N. The characteristic function of (S N -N )/B N satisfies E(e it(S N -N )/B N ) = e -itN/B N (ϕ X (t/B N )) N and, therefore,log E(e it(S N -N )/B N ) = -itN/B N + N log ϕ X (t/B N ) = -itN/B N + N θ rφ ′ (r-)it/B N + κr α (-it/B N ) α + O((t/B N ) 2 ) = -itN/B N 1 -θrφ ′ (r-) + θκr α (-it) α + O(t 2 N/B 2 N ) = θκr α (-it) α + O(t 2 N/B 2 N ) = log ϕ(t) + O(t 2 N/B 2 N

≤ I 1 + I 2 + I 3 + I 4

 1234 sup k∈Z |d N (k)| = O(N 1-2/α ) holds, where d N (k) := |N 1/α P(S N = k)-g((k -N )/N 1/α )| and g denotes the density of the limiting α-stable random variable S defined in Lemma 5.3. Proof. (of Theorem 5.4) We modify the proof of the local limit theorem for lattice distributions of Ibragimov and Linnik [12, Theorem 4.2.1]. By the Fourier inversion formula, S has density g(x) = (2π) -1 ∞ -∞ e -itx ϕ(t) dt, x ∈ R.As in the proof of Lemma 5.3 define B N := N 1/α , N ∈ N. The discrete inversion formula ensures that the integer valued random variableS N := X 1 +• • •+X N has distribution P (S N = k) e -itk/B N (ϕ X (t/B N )) N dt e -it(x+N/B N ) (ϕ X (t/B N )) N dt, k ∈ Z,where we have used the substitution u = t/B N and where we use the notation x := (k -N )/B N . Thus,d N (k) = |B N P(S N = k) -g(x)| = 1 2π πB N -πB N e -itx e -itN/B N (ϕ X (t/B N )) N -∞ -∞ e -itx ϕ(t) dt ≤ πB N -πB N |e -itN/B N (ϕ X (t/B N )) N -ϕ(t)| dt + |t|>πB N |ϕ(t)| dt |e -itN/B N (ϕ X (t/B N )) N -ϕ(t)| dt, I 2 := k N ≤|t|≤εB N |ϕ X (t/B N )| N dt,I 3 := εB N ≤|t|≤πB N |ϕ X (t/B N )| N dt, and I 4 := |t|>k N |ϕ(t)| dt,

( 1 ) 2 Ne 2 Nt 2

 1222 By (38) there exists a constant C > 0 (not depending on t and N ) such that|e -itN/B N (ϕ X (t/B N )) N -ϕ(t)| ≤ |ϕ(t)|(e Ct 2 N/B 2 N -1), N ∈ N, t ∈ R.Applying the inequality ex -1 ≤ xe x , x ∈ R, to the point x := Ct 2 N/B 2 N yields |e -itN/B N (ϕ X (t/B N )) N -ϕ(t)| ≤ |ϕ(t)|Ct 2 N B Ct 2 N/B 2 N , N ∈ N, t ∈ R.For all |t| ≤ k N we have Ct 2 N/B 2 N ≤ C and it follows that|e -itN/B N (ϕ X (t/B N )) N -ϕ(t)| ≤ Ce C N B |ϕ(t)|, N ∈ N, |t| ≤ k N .Since, by (26),∞ -∞ t 2 |ϕ(t)| dt < ∞, it follows for all N ∈ N that I 1 ≤ Ce C (N/B 2 N ) ∞ -∞ t 2 |ϕ(t)| dt = O(N 1-2/α ) as required.

2 ∞k

 2 |t|>k N e cRe((-it) α ) dt = |t|>k N e c|t| α cos πα 2 dt = 2 ∞ k N e -Dt α dt.Noting that k N ≥ 1 for all N ∈ N and that α > 1 it follows that I 4 ≤ N e -Dt dt = (2/D)e -Dk N , again being smaller than O(N β ) for any β < 0. 2

  t α cos πα Note that g is bounded, i.e. g := sup x∈R |g(x)| < ∞. Moreover (see, for example, Ibragimov and Linnik [12, Theorem 4.2.1]) the local limit theorem lim N →∞ sup k∈Z |d N (k)| = 0 holds, where d N (k

	2	cos xt + cΓ(-α)t α sin πα 2	dt,	x ∈ R.

  and[START_REF] Sagitov | Convergence to the coalescent with simultaneous multiple mergers[END_REF] follows immediately. The proof of part c) is complete.

2

Proof. (of Theorem 2.5) Without loss of generality assume that φ has radius of convergence r = 1. Otherwise consider the new power series φ * defined via φ * (z) := φ(rz), |z| < 1, and note that the coefficients σ * k (θ) of the new power series φ * and the coefficients σ k (θ) of the original power series φ are related via σ * k

Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise,

AvenueAdolphe Chauvin, 95302 Cergy-Pontoise, France, E-mail: thierry.huillet@u-cergy.fr 2 Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany, E-mail: martin.moehle@uni-tuebingen.de

constants κ := Γ(1 -α) and c := θ c , the coalescence probability c N satisfies

where, in the first case α ∈ (3, ∞),

and, in the third case α ∈ (2, 3), g denotes the density of an (α -1)-stable random variable with characteristic function

Finally, if 0 < θ < θ c (subcritical case), then, by Theorem 2.5, the model is in the domain of attraction of a discrete-time Λ-coalescent with Λ := u 2 δ u and u

The following example is a slight modification of the polylog model. In comparison to the polylog model it has the advantage that for integer parameter α explicit expressions for the power series φ are available.

It does not come as a surprise that this model behaves very similar as the polylog model. We therefore leave the details to the reader and provide more information exemplary only for α = 3, in which case the explicit formula

is available. We have

and, for θ ∈ (0, ∞),

where

Taking the θ-power yields

as z → 1. Using standard singularity analysis (see Flajolet and Sedgewick [7, p. 385, Theorem VI.2] it follows that

as k → ∞. The same asymptotic result for P(X = k) is checked similarly for arbitrary α ∈ (0, ∞). Let us now discuss the asymptotic behavior of the model. If α ≤ 2, then φ ′ (1-) = ∞, so Theorem 2.2 is applicable for all θ ∈ (0, ∞), which implies that the associated compound Poisson model is in the domain of attraction of the Kingman coalescent. Assume now that α > 2. Then φ

In this case the equation θzφ ′ (z) = 1 admits a solution z = z(θ) ∈ (0, 1) if and only if θ > θ c with critical value θ c := 1/φ ′ (1-) ∈ (0, ∞). Thus, Theorem 2.2 is not applicable for θ ≤ θ c . If θ = θ c (critical case), then, by Theorem 2.3, the model is still in the domain of attraction of the Kingman coalescent and, by Theorem 2.4, part c) applied with α replaced by α -1, so with constants κ

where, in the first case α ∈ (2, 3), g denotes the density of an (α -1)-stable random variable with characteristic function t → exp(θ c Γ(1 -α)(-it) α-1 ), t ∈ R, and in the third case α ∈ (3, ∞),

Appendix

The following simple but useful lemma is needed in the proof of Theorem 2.2.

Lemma 5.1 Let a 1 , a 2 , . . . ∈ [0, ∞) such that ∞ m=1 a m q m < ∞ for some q ∈ (1, ∞). Define f (t) := ∞ m=1 a m cos(mt), t ∈ R. If a 1 > 0 then there exists ε 0 > 0 such that sup ε<|t|≤π f (t) ≤ f (ε) for all ε ∈ [0, ε 0 ].


Jean-Christophe Bach
email: jeanchristophe.bach@inria.fr

Pierre-Etienne Moreau
email: pierre-etienne.moreau@loria.fr

Marc Pantel
email: marc.pantel@enseeiht.fr

Tom-based tools to transform EMF models in avionics context

Model Driven Engineering (MDE) is now widely used in many industrial contexts (such as AeroSpace) which require a high level of system safety. Model-checking is one of the formal techniques which are used to assess a system compliance to its requirements. It relies on verification dedicated languages to model the system under verification and the expected properties. In order to ease the use of these tools, model transformations are provided that translate the end user provided system model to the formal languages than can be verified. In order to rely on these activities for system certification, the correctness of these transformation steps must be assessed (qualification of the development and verification tools). One of the goal of our work is to provide tools to implement the transformation steps between end user source languages used for the system development and target languages used for formal verification. This paper present the Tom rule-based approach used in a research project involving industrial partners: Airbus and Ellidiss.

Introduction and related work

New safety critical system development chains are based on Domain Specific Modeling Languages (DSML) and on qualified model transformations (that preserve required properties) between these DSMLs. The quarteFt5 project aims to develop technologies to make this approach easier in the field of real-time embedded systems. The associated case study relies on Fiacre6 , which is an intermediate language used for formal verification of real-time aspects in the TOPCASED 7project. Starting from an avionic model expressed in AADL 8 , the goal is to extract a behavioral model and its intended properties which can be verified by a model-checker such as TINA 9 or CADP 10 . To make this transformation simpler, several intermediate languages have been introduced: basic and parametrized Fiacre; and a real-time extension to Fiacre (RT-Fiacre). The following parts show how a rule based language, implemented in the Tom framework [START_REF] Moreau | A pattern matching compiler for multiple target languages[END_REF][START_REF] Balland | Tom: piggybacking rewriting on java[END_REF], can be used to express the AADL2Fiacre transformation in an expressive and executable way.

In MDE , many domain specific languages (DSLs) have been developed to manipulate and transform models, trying to implement the Query View Transformation QVT standard defined by the Object Management Group (OMG). QVT proposes two main approaches: relational and operational. The latter requires to describe explicitly the control as part of the transformation whereas it is handled implicitly by the execution engine for relational one. The operational approach expresses a transformation as a sequence of elementary steps that builds the resulting model step by step from the source model. It can be implemented using DSL such as Kermeta [START_REF] Jézéquel | Model driven language engineering with kermeta[END_REF], QVT-Operational, or GPL (General Purpose Languages) such as Java using reflexive libraries or generated code such as Eclipse Modeling Framework (EMF) [START_REF] Steinberg | EMF: Eclipse Modeling Framework 2.0. 2nd edn[END_REF]. The relational approach defines a model transformation as the relations that must exist between the source and the target models at the end of the transformation. Some transformations are not directly executable. When they can be executed, they are usually translated into an operational transformation.

A third approach [START_REF] Sendall | Model transformation: the heart and soul of modeldriven software development[END_REF] can be considered: the introduction of intermediate representations such as XML or any textual concrete syntax. The model transformation can then be described using a language such as XSLT or any concrete language transformation toolset like ASF+SDF 11 , Rascal [START_REF] Klint | Rascal: A domain specific language for source code analysis and manipulation[END_REF], Spoofax(based on Stratego/XT) [START_REF] Hemel | Code generation by model transformation: a case study in transformation modularity[END_REF][START_REF] Kats | The spoofax language workbench: rules for declarative specification of languages and ides[END_REF], etc.

Implementations of transformations may be written using GPL or DSL. Both of these approaches have advantages and drawbacks: GPL are usually well equipped with multiple libraries and external tools and are well known from the common software engineer. However some specific tasks may be difficult to implement whereas DSL may be more precise and efficient. Integration and maintainability has also to be taken in account when choosing a language to transform models. DSLs are usually not known by most software engineers and a significant cost must be spent in their learning and the development of associated tools.

ATL [START_REF] Jouault | Atl: A model transformation tool[END_REF] does not rely on the QVT concrete syntax but implements most of the operational and part of the relational approaches from QVT. It fills a part of the gap (use of both relational and operational approches), but is still a full new language that must be learnt from scratch.

This contribution proposes to bridge previous approaches by embedding a DSL into a GPL, to take advantages both of GPLs tooling and DSL expressivity. As the Tom language is compiled into Java relying on model management libraries like EMF, the end user can benefit from both classical programming, and model transformation technologies (operational and relational), without major efficiency penalty and without the additional cost of learning a completely new language. Therefore Java developers have only to learn few new constructs to use within a Java program to easily implement a model transformation, contrary to the use of a dedicated transformations language such as Kermeta which are more complete for this task but also much more complex. By remaining in the Java world, developers can benefit from a third advantage of our approach: keep using existing Java tooling.

The rest of the paper is organized as follows. Section 2 presents the Tombased tools we developed to transform EMF models. Section 3 introduces the transformations chain and the industrial use case AADL2Fiacre, then it explains the use of Tom in this context. Section 4 concludes and presents current and future work.

2 Tom-based tools to transform EMF models Tom is a language designed to extend general-purpose programming languages (Java, C, C#, Python and Caml) by providing term rewriting, pattern matching and strategic programming facilities. Tom is well-suited for implementing programs that manipulate tree structures such as AST (Abstract Syntax Tree) or XML documents. Only the main constructs of the language that are used in the case study will be presented, but the interested reader may refer to the Tom reference manual12 .

Short introduction to Tom

Pattern-matching. The main Tom feature is pattern-matching using the %match construct. It is a generalization of switch-case construct which can be applied to any data-structure, and not only to atomic types like switch-case. It is composed of a set of rules where the left-hand side is a pattern (having a tree structure), and the right-hand side is an action (a block of host and Tom code).

Backquote construct. The backquote (character ') is the Tom construct which allows to build a new algebraic term or to retrieve the value of a variable instantiated by pattern matching.

Listing 1.1 illustrates the use of %match and ': at line 1, a TypeDeclaration is created with '. It has two parameters: rec is the value of the type, and "int" is the name of the type. Then, at line 2, this TypeDeclaration is given as the parameter (called subject) of the %match construct. Line 3 shows how to write a transformation rule: on the left-hand side, the patterns checks that the subject corresponds to a TypeDeclaration. In this example, the variable name is instantiated by the first argument ("int" in this case). The second argument of the pattern is a nested pattern which recursively checks that the second subterm of typeDecl is rooted by RecordType_fiacre, etc.. It is also possible to consider disjunction (||) and conjunction (&&) of patterns. The right-hand side of the rule is the action (-> {...}), composed of a block of Tom+Java code. In this example, the variable name is printed. Strategic-programming. Strategic programming is a way to increase the control over the application of rewriting rules. Strategies -implemented by the %strategy construct -allow to dissociate the treatment (rewriting rules) and the control (tree traversal). Then complex strategies can be obtained by composition of elementary transformation strategies and combinators such as Sequence, Repeat, TopDown, and the recursion for instance. For the interested reader, more detail can be found in [START_REF] Balland | Rewriting strategies in java[END_REF]. Listing 1.2 shows the use of %strategy construct and how a strategy can be composed and called (line 8). DataTrans() is an elementary strategy. TopDown(DataTrans()) is also a strategy. visit is a method which apply a strategy on a given term (subject in this example). Algebraic views (mappings). In order to be able to use pattern-matching feature, one has to establish an anchor between the implementation data-structure (i.e. the Java classes) and the Tom algebraic signature. This is done by a mechanism called mapping. It is composed of two constructs: %typeterm and %op. As shown in Listing 1.3, %typeterm associates the algebraic sort (DataImplementation) to the implementation data-type (org.osate.aadl2.DataImplementation) with the construct implement. The equals construct defines an equality predicate. The %op construct illustrated in Listing 1.4 specifies how to view a Java object as a Tom constructor. The predicate is_fsym(t) should be true when the Java object t can be seen as the considered constructor (DataImplementation in this example). get_slot specifies how to retrieve a given field of the datastructure. make defines how to build an instance of the constructor and is used by ' construct to build terms. This example relies on the EMF reflexive framework. It could also rely on the generated classes. However, these ones are also implemented using the reflexive framework. This work thus also follows that scheme. %oplist and %oparray are variant of %op for list operators (see Listing 1.5) used for associative and associative/commutative operators. Their use is similar to %op, but they define additional constructs to build a list and to retrieve a given element in a list, and the size of a list. Tom-EMF is a tool whose goal is to offer a support to manipulate EMF datastructures with Tom. It is composed of two main elements: EcoreContain-mentIntrospector which is used for using strategies on EMF elements, and a mappings generator. This latter loads and inspects a Java meta-model implementation generated by Eclipse from an ECore file. It retrieves all the EClassifier (EClass and EDataType) and EStructuralFeatures in order to generate the corresponding %typeterm and %op constructs. These mappings will be the view for EMF elements in Tom. For instance, the mappings presented in Listing 1.3 and Listing 1.4 have been automatically generated by Tom-EMF from the aadl2fiacre.jar archive, which corresponds to java classes which are automatically generated by EMF from an ECore meta-model. If a reference shows that an element may have many instances (which is represented by a collection), the Tom-EMF tool generates list operators such as RecordTypeLabelEList illustrated in Listing 1.5.

%transformation: an easy way to express a model transformation in Tom+Java

To write a model transformation, one can use full Java or Tom+Java. In the case of Tom+Java, every elementary transformation composing the whole transformation can be encoded by a strategy. This approach is appropriate for simple transformations, but in some cases, an elementary transformation is dependent on elements created by another one. For instance, let us consider two transformations transA and transB, which respectively transform elements of sorts A and B. The result of transA may need to be connected to some elements produced by transB. A way to express these connexions is thus needed. This problem can be addressed in two ways:

1. a first solution consists in finding a general strategy (usually a recursive one) which inspect the model and transforms the elements in an order such that all the links can be built during the transformation itself. Unfortunately such a strategy does not always exist. 2. a second approach consists in not considering any particular order for the transformation steps. In that case, an additional mechanism has to be provided in order to denote and to build a link to an element which results from a transformation. This operation is called link resolution in the following. This approach is derived from the one available in ATL and the relational part of QVT.

A method using the second solution to implement a models transformation with Tom+Java is explained in [START_REF] Bach | Model transformations with tom[END_REF]. It constitutes the core contribution of the new higher-level Tom %transformation construct. It addresses the problem and hides a part of the work from the developer. %transformation is a construct composed of a set of definitions (corresponding to Tom strategies) which may have one or many rules (corresponding to rules of rewriting strategies). An example is given in the next section (see Listing 1.9). The grammar of %transformation construct is given by Listing 1.6 in Appendix A.

As the execution order of transformation steps is not considered, some definition may need to reference an element created in another transformation step definition in order to be able to assemble the elements resulting from the various steps. It also may create an element which has to be referenced in another transformation step. That is the reason why we introduced two new Tom constructs to be able to perform the link resolution: %tracelink and %resolve. They build a link model during the transformation steps to keep references between the source and the target. Then, it is used to resolve the links once all target elements have been created. These constructs ensure a form of traceability of the transformation, which is a mandatory step to be able to verify it.

%tracelink whose syntax is given by Listing 1.7 in Appendix A allows to save in the link model an element which will be necessary in another step. It takes three parameters: the name and the type the element has, and the term itself. In the link model, a field of the defined type having the given name will be created. The object will be instantiated with the object represented by the backquote term.

%resolve construct whose grammar is shown in Listing 1.8 (in Appendix A) can be considered as the corresponding construct of %tracelink. It creates a virtual element the developer can manipulate in a definition as if it were the one he needs (whether it has effectively already been created or not). This kind of elements is removed at the end of the transformation during the link resolution. The first two parameters of a %resolve construct are the name and the type of a source element. The last two parameters are the name and the type of a resulting element obtained by the transformation of the source element in another definition.

An industrial use case in avionics: AADL2Fiacre

This case study consists in transforming an AADL model into a Fiacre model in order to verify this one with TINA. This concrete use case is being implemented by the Ellidiss Software13 company in Adele14 to address the needs of Airbus15 in the quarteFt project.

Important note: due to industrial constraints, all source code and documents produced during quarteFt project are still unavailable for public use. All materials will be released publicly after the end of the project. [START_REF] Feiler | The sae architecture analysis & design language (aadl) a standard for engineering performance critical systems[END_REF] is an architecture description language initially developed for avionics domain. It is designed for modeling the software and hardware architecture of embedded real-time systems, more particularly in the context of automotive engineering, aeronautics and aerospace. The purpose of AADL language is to describe an architecture to be able to analyze the system, to verify properties, to generate code and documentation, etc.. Components may be defined to model a part of the system. Each component may contain one or many sub-components, and it may be connected to other components to build a more complex system by simulating a network. In the context of quarteFt, AADL is the starting point of the transformations chain.

AADL. AADL

Fiacre. Fiacre16 is a high level description language designed for representing both the behavioral and timing aspects of embedded and distributed systems for formal verification and simulation purposes. In the quarteFt project, the last transformation step is the compilation of Fiacre descriptions into Time Transition Systems (TTS) for TINA.

For a detailed explanation of Fiacre language, the reader is invited to refer to [START_REF] Berthomieu | Fiacre: an Intermediate Language for Model Verification in the Topcased Environment[END_REF] where the use case is a first proposal of AADL2Fiacre transformation written in Kermeta. Although ATL and Kermeta have been experienced and were useful tools to precise a part of the project, major issues concerning performance and integration constraints led to consider Tom as the transformation language used in quarteFt. Moreover, the declarative way to write a transformation in Tom is well adapted to correctness proof construction.

AADL2Fiacre. Because of the complexity of the whole transformation and the lack of place, this work focuses on a part of the step which transforms AADL model to RT-Fiacre (Real-Time Fiacre) model. This transformation can be divided into six elementary transformations: AADL.Data, AADL.Thread, AADL.-ThreadImpl, AADL.BehaviorAnnex, AADL.System and AADL.Process translations.

Due to space constraints and because this transformation is still under development, the six elementary transformations will not be detailled. Snippets of code will be extracted from the shortest and easiest one: the translation of AADL Data into RT-Fiacre Record.

To begin this transformation, one has to generate the Tom mappings to have an algebraic representation of models. Starting from source meta-model aadl2.ecore17 and target meta-model fiacre.ecore18 , one generates with EMF the corresponding Java code which will be used during the transformation. Then, one uses Tom-EMF tool to generate AADL and Fiacre mappings. This transformation step consists in generating a RT-Fiacre type-record (Re-cordType_fiacre) for each Data component (implemented by DataImplementation. Each data-field declared in the subcomponents part corresponds to a field of the RT-Fiacre record.

By using Tom, this elementary transformation can be implemented by a single strategy composed with a TopDown traversal strategy or being a definition of a %transformation construct as shown in Listing 1.9 (Appendix B).

As a %transformation is compiled like a composition of strategies, the application and use of AADL2Fiacre transformation is like every other strategy as illustrated in Listing 1.10 (Appendix B).

A special strategy tom_StratResolve_AADL2Fiacre() is generated when needed (in case of use of %tracelink and %resolve constructs). It replaces all resolve objects ("virtual" ones created by the %resolve construct) by the corresponding and therefore it resolves all links. It allows to the developer not to take care of the application order of transformation steps.

In a pure Java approach, Tom facilities such as pattern-matching should be replaced by lots of hand-written tests and iterators (for list-matching). Strategies would be replaced by hand-written traversal of the source model, using loops and recursivity. In addition, the developer would have to be very careful on the instructions sequence in order not to use not yet created objects.

Conclusions, future work

This paper presented tools designed to transform more easily EMF models in a general purpose language such as Java. These tools are based on the Tom language which relies on strong formal foundations (rewriting calculus); has been developed since year 2000; and has been used in many research and industrial project. As the Tom language is composed of a limited set of constructs that developers include in host languages programs, it is easy to learn and its use allows developers to be quickly efficient. One the one hand its DSL approach is useful to express a model transformation in an easier way than in pure Java. On the other hand, its integration in a GPL allows to take advantage of existing tooling and libraries. To complete this goal, Eclipse plugins for Tom and Tom-EMF are also being developed.

A real industrial case study which is currently being developed in an avionicsrelated project was also described. Our goal is to provide an efficient tool usable in an state-of-the-art industrial context.

Tom-EMF and extensions of the Tom language are currently experiencing a testing period. The first version of %transformation construct will be available in the Tom 2.10 release. These tools should be improved in many ways. For the moment, Tom models transformations tools are exclusively based on EMF. The support of multimodeling frameworks such as GMS19 is currently being implemented. Another focus is the extension of %transformation construct to support multiple inputs. With this feature, the pattern-matching possibilities will be extended to be able to match several patterns at the same time in an elementary transformation. The extension and the generalization of the EMF-specialized introspector to be able to easily follow other links than containment associations in a model is also studied. As the verification of models transformations is one of our core concerns, the traceability of a model transformation is being investigated. The start point of this part consists in working on the improvement of the currently generated link meta-model; and then to apply constraints (as in OCL) on this link meta-model to ensure that properties are verified.

Finally, this work represents a first step towards the increase safety in software development chains through the design of new tools and the verification of models transformations.

Listing 1 . 1 :

 11 TypeDeclaration typeDecl = 'TypeDeclaration("int",rec); %match(typeDecl) { TypeDeclaration(name,RecordType_fiacre(RecordTypeLabelEList())) -> { System.out.println (' name); Example of %match and backquote constructs

%strategy

  DataTrans() extends Identity() { visit DataImplementation { dt@DataImplementation[name=name] -> { <Tom+Java code block> } } } public static void main(String[] args) { ... 'TopDown(DataTrans()).visit(subject); ... } Listing 1.2: Example of %strategy construct

  %typeterm DataImplementation { implement { org.osate.aadl2.DataImplementation } equals(t1,t2) { t1.equals(t2) } } Listing 1.3: Example of %typeterm construct

  %typeterm RecordTypeLabelEList { implement { org.eclipse.emf.common.util.EList<fiacre.RecordTypeLabel> } equals(t1,t2) { ... } } %oparray RecordTypeLabelEList RecordTypeLabelEList ( RecordTypeLabel * ) { is_fsym(t) { ... } make_empty(n) { / * allocate a list of size n * / } make_append(e,l) { / * add an element e to the end of the list l * / } get_element(l,n) { / * get the n-th element of the list l * / } get_size(l) { / * returns the size of the list l * / } } Listing 1.5: Example of variadic operator mapping 2.2 Tom-EMF: from EMF representation to Tom representation

The quarteFt project (http://quarteft.loria.fr/) is funded by FNRAE (http: //www.fnrae.org/), a French research foundation for aeronautics and space.

http://projects.laas.fr/fiacre/

http://www.topcased.org/

Architecture Analysis & Design Language, formerly Avionics Architecture Description Language, http://www.aadl.info/

http://www.laas.fr/tina/

http://cadp.inria.fr/

http://www.meta-environment.org/Meta-Environment/ASF%2BSDF/

http://tom.loria.fr/

http://www.ellidiss.com/

TOPCASED AADL editor : http://gforge.enseeiht.fr/projects/adele/

http://www.airbus.com

Fiacre stands for "Format Intermédiaire pour les Architectures de Composants Répartis Embarqués", french for "Intermediate Format for the Embedded Distributed Component Architectures".

Can be found in Osate2 plugin from Ellidiss update site: http://aadl.ellidiss.fr

An ECore metamodel has been defined to integrate Fiacre in the TOPCASED environment, and therefore can be found in the TOPCASED bundle.

Ada modeling framework used in the P project: http://www.open-do.org/ projects/p/

Many elements are note defined in this syntax block in order to have a readable %transformation syntax block. TransformationName, FileName, SubjectName, AlgebraicType and DefinitionName are Identifier with self-explanatory names. Strategy represents a Tom strategy such as 'TopDown(MyStrategy()). Pattern is a pattern as defined in the Tom grammar on the official Tom website. BlockList is a block composed of both host and Tom code, as defined in the Tom grammar.