The common ancestor type distribution of a Λ-Wright-Fisher process with selection and mutation

Using graphical methods based on a ‘lookdown’ and pruned version of the ancestral selection graph, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from [17], now including the case of heavy-tailed offspring, directed by a reproduction measure Λ. The representation is in terms of the equilibrium tail probabilities of the line-counting process L of the graph. We identify a strong pathwise Siegmund dual of L, and characterise the equilibrium tail probabilities of L in terms of hitting probabilities of the dual process.


Introduction
We consider a Wright-Fisher process with two-way mutation and selection. This is a classical model of mathematical population genetics, which describes the evolution, forward in time, of the type composition of a population with two types. Individuals reproduce and change type, and the reproduction rate depends on the type (the beneficial type reproduces faster than the less favourable one).
In a previous paper [17], we have presented a graphical construction, termed the pruned lookdown ancestral selection graph (p-LD-ASG), which allows to identify the common ancestor of a population in the distant past, and to represent its type distribution. This construction keeps track of the collection of all potential ancestral lines of an individual. As the name suggests, the p-LD-ASG combines elements of the ancestral selection graph (ASG) of Krone and Neuhauser [16] and the lookdown construction of Donnelly and Kurtz [6], which here leads to a hierarchy that encodes who is the true ancestor once the types have been assigned to the lines. In addition, a pruning procedure is applied to reduce the graph.
A key quantity is the process L, which counts the number of potential ancestors at any given time. The ancestral type distribution is expressed in terms of the stationary distribution of L together with the overall type distribution in the past population. The two distributions may be substantially different. This mirrors the fact that the true ancestor is an individual that is successful in the long run; thus, its type distribution is biased towards the favourable type. Explicitly, the ancestral type distribution is represented as a series in terms of the frequency of the beneficial type in the past, where the coefficients are the tail probabilities of the stationary distribution of L and are known in terms of a recursion.
The results obtained so far referred to Wright-Fisher processes. These arise as scaling limits of processes in which an individual that reproduces has a single offspring that replaces a randomly chosen individual (thus keeping population size constant); in the ancestral process, this corresponds to a coalescence event of a pair of individuals. Here we will consider a natural generalisation, the so-called Λ-Wright-Fisher processes. These include reproduction events where a fraction z > 0 of the population is replaced by the offspring of a single individual; this leads to multiple merger events in the ancestral process.
The Λ-Wright-Fisher processes belong to the larger class of Λ-Fleming-Viot processes (which also include multi-(and infinite-)type generalisations). These, together with their ancestral processes, the so-called Λ-coalescents, have become objects of intensive research in the past two decades. Although less is known for the case with selection, progress has been made in this direction as well (see for example [1,5,6,7,10]).
Besides deriving our main result on the common ancestor type distribution of a Λ-Wright-Fisher process (stated in Sec. 2), the purpose of our paper is twofold: First, we will extend the p-LD-ASG to include multiple-merger events; this will lead to the p-LD-Λ-ASG. Second, in the footsteps of Clifford and Sudbury [3], we will construct a Siegmund dual of the linecounting process L of the p-LD-Λ-ASG. In line with a classical relation between entrance laws of a monotone process and exit laws of its Siegmund dual (discovered by Cox and Rösler [4]), the tail probabilities of L at equilibrium correspond to hitting probabilities of the Siegmund dual. This Siegmund dual is a new element of the analysis: In [17], the recursions for the tail probabilities were obtained from the generator of L, in a somewhat technical manner. The duality provides a more conceptual approach, which is interesting in its own right, and yields the recursion in an elegant way, even in the more involved case including multiple mergers. It will also turn out that the Siegmund dual of L is a natural generalisation (to the case with selection) of the so-called fixation line (or fixation curve), introduced by Pfaffelhuber and Wakolbinger [19] for Kingman coalescents and investigated by Hénard [13] for Λ-coalescents.
The paper is organised as follows. In Section 2, we recapitulate the Λ-Wright-Fisher model with mutation and selection, and the corresponding ancestral process, the Λ-ASG; we also provide a preview of our main results. In Section 3, we extend the p-LD-ASG to the case with multiple mergers. Section 4 is devoted to the Siegmund dual. The dynamics of this dual process is identified via a pathwise construction and thus yields a strong duality. Once the dual is identified, it leads to the tail probabilities of L with little effort.

Model and main result
We will consider a population consisting of individuals each of which is either of deleterious type (denoted by 1) or of beneficial type (denoted by 0). The population evolves according to random reproduction, two-way mutation, and fertility selection (that is, the beneficial type reproduces at a higher rate), with constant population size over the generations. The parameters of the model are • the reproduction measure Λ, which is a probability measure on [0, 1], and whose meaning will be explained along with that of the generator G X below Eq. (2.1), • the selective advantage σ (a non-negative constant that quantifies the reproductive advantage of the beneficial type and is scaled with population size), • the mutation rates θν 0 and θν 1 , where θ, ν 0 , and ν 1 are non-negative constants with ν 0 + ν 1 = 1. Thus, ν i , i ∈ {0, 1}, is the probability that the type is i after a mutation event; note that this includes silent events, where the type remains unchanged.
We will work in a scaling limit in which the population size is infinite and time is scaled such that the rate at which a fixed pair of individuals takes part in a reproduction event is 1. The process X := (X t ) t∈R describing the type-0 frequency in the population then has the generator (cf. [7,10]) (2.1) The first and second terms of this generator describe the neutral part of the reproduction. In the case Λ = δ 0 (to which we refer as the Kingman case), the first term vanishes and X is a Wright-Fisher diffusion with selection and mutation. Concerning the part of Λ concentrated on (0, 1], the measure dt Λ(dz)/z 2 figures as intensity measure of a Poisson process, where a point (t, z), t ∈ R, z ∈ (0, 1], means that at time t a fraction z of the total population is replaced by the offspring of a randomly chosen individual. Consequently, if the fraction of type-0 individuals is x at time t−, then at time t the frequency of type-0 individuals in the population is x + z(1 − x) with probability x and x(1 − z) with probability 1 − x. The third term of generator (2.1) describes the systematic (logistic) increase of the frequency x due to selection, and the type flow due to mutation.
In the absence of both selection and mutation (i.e. when σ = θ = 0), the moment dual of the Λ-Wright-Fisher process is the line-(or block-)counting process of the Λ-coalescent. The latter was introduced independently by Pitman [20], Sagitov [21], and Donnelly and Kurtz [18], see [2] for an introductory review.
The rate at which any given tuple of j out of b blocks merges into one is Thus the transition rate of the line-counting process from state b to state c < b is given by b b−c+1 λ b,b−c+1 . Note that Λ = δ 0 corresponds to Kingman's coalescent; here, λ b,j = δ 2,j for all b ≥ 2. The measure Λ is said to have the property CDI if the Λ-coalescent comes down from infinity, i.e. ∞ is an entrance boundary for its line counting process.
When selection is present (i.e. σ > 0), an additional component of the dynamics of the genealogy must be taken into account. In this case, in addition to the (multiple) coalescences just described, the lines (or blocks) may also undergo a binary branching at rate σ per line. The resulting branching-coalescing system of lines is a straightforward generalisation of the ancestral selection graph (ASG) of Krone and Neuhauser [16] to the multiple-merger case; we will call it the Λ-ASG. The Λ-ASG belonging to a sample of n individuals taken from the population at time t = 0 describes all potential ancestors of this sample at times t < 0. Throughout we use the variables t and r for forward and backward time, respectively.
We denote the line-counting process of the Λ-ASG by K = (K r ) r≥0 . The generator of K is The process K is the moment dual of the Λ-Wright-Fisher process with selection coefficient σ and mutation rate θ = 0, see e.g. [7,Thm 4.1]. Throughout we will work under the Remark 2.2. a) It is proved in [12] that σ * = lim k→∞ , where T 1 is the first time at which the line counting process of the Λ-coalescent hits 1. In particular, if the measure Λ has the property CDI, then σ * = ∞ and hence Assumption 2.1 is satisfied for all σ ≥ 0. b) Under Assumption 2.1 the process K has the property E k [T 1 ] < ∞ for all k ∈ N, where now T 1 is the first time at which K hits 1. Indeed, the proof of [9, Lemma 2.4], though stated there only in the case when ∞ k=2 1/δ(k) = ∞ (with δ(k) defined in [9], (2.1)), works literally also for the case ∞ k=2 1/δ(k) < ∞. To see this, note that in this proof the assumption that ∞ k=2 1/δ(k) is infinite is only required to guarantee the non-explosiveness of K. The latter property, however, is automatic because K is dominated by a pure birth process with birth rate bσ, b ∈ N, which is non-explosive.
In view of Remark 2.2b), under Assumption 2.1 the process K has a unique equilibrium distribution and a corresponding time-stationary version indexed by r ∈ R. Similarly, there exists a time-stationary version of the Λ-ASG, which we call the equilibrium Λ-ASG, and which will be a principal object in our analysis.
Mutations can be superimposed as independent point processes on the lines of the Λ-ASG: On each line, independent Poisson point processes of mutations to type 0 come at rate θν 0 and to type 1 at rate θν 1 .
For t < t and for a given frequency x of type-0 individuals in the population at time t, the Λ-ASG may be used to determine the types in a sample taken at time t, together with its ancestry between times t and t, by the following generalisation of the procedure in [16]. Each line of the Λ-ASG at time t is assigned type 0 with probability x and type 1 with probability 1 − x, in an iid fashion. Let the types then evolve forward in time along the lines: after each beneficial or deleterious mutation, the line takes type 0 or 1, respectively. At each neutral reproduction event (which is a coalescence event backward in time), the descendant line inherits the type of the parent. This is also true for the (potential) selective reproduction events (the branching events backward in time), but here one first has to decide which of the two lines is parental. The rule is that the incoming branch (the line that issues the potential reproduction event) is parental if it is of type 0; otherwise, the continuing branch (the target line on which the potential offspring is placed) is parental. When all selective events have been resolved this way, the lines that are not parental are removed, and one is left with the true genealogy.
For a given assignment of types to the lines of the stationary Λ-ASG at time 0, there is exactly one true ancestral line between the times t = 0 and t = ∞. The line of this ancestor throughout time is called the immortal line or line of the common ancestor. Our main result is a characterisation of its type distribution at time 0, conditional on the type frequency in the population at that time. For the following definition, let I t be the type of the immortal line in the stationary Λ-ASG at time t. Definition 2.3 (Common ancestor type distribution). In the regime of Assumption 2.1, and for x ∈ [0, 1], let h(x) := P(I 0 = 0 | X 0 = x) be the probability that the immortal line in a stationary Λ-ASG with two-way mutations carries type 0 at time 0, given the type-0 frequency in the population at time 0 is x.
is also the limiting probability (as t → ∞) that the ancestor at the past time −t of the population at time 0 is of the beneficial type, given that the frequency of the beneficial type at time −t was x.
Theorem 2.4. The probability h(x) has the series representation

4)
where the coefficients a n in (2.4) are monotone decreasing, and the unique solution to the system of equations  (a n − a c−1 ) + (σ + θ)a n = σa n−1 + θν 1 a n+1 , n ≥ 1, with the convention In the Kingman case the system of equations (2.5) simplifies to 1 n n + 1 2 + σ + θ a n = 1 n and we immediately obtain In the Kingman case, the coefficients in (2.4) satisfy the recursion with a 0 = 1 and lim n→∞ a n = 0.
Recursion (2.8) appears in [8] in connection with a time-stationary Wright-Fisher diffusion (with selection and mutation). 1 In [23], the representation (2.4) together with (2.8) was derived by analytic methods. In [17], again for the Kingman case, we gave a new, more probabilistic proof, interpreting the coefficients a n as equilibrium tail probabilities of the line-counting process of the pruned lookdown ASG (see Sec. 3). In the present paper we give a twofold extension: (i) we include the case of multiple mergers, and (ii) we use a strong Siegmund duality (and thus a fully probabilistic method) in order to derive the recursion (2.5).
An analogue of the quantity h(x) can also be defined for a Moran model with finite population size N : for k ∈ {0, 1, . . . , N }, let h N k be the probability that the individual whose offspring will take over the whole population at some later time is of type 0 at time 0, given the number of type-0 individuals in the population at time 0 is k. In [15] it is shown (for the Kingman case) that h N k converges to h(x) as N → ∞ and k/N → x. Here, we work in the infinite-population limit right away, in order to carve out some important features of the underlying mathematical structure.
In order to determine h(x), it turns out to be helpful to remove certain lines from the ASG. Indeed, as the mutations convey information on the types of lines, some lines in the Λ-ASG can be deleted from the set of potential ancestors. This way, the Λ-ASG may be pruned, according to the ideas presented in [17] for the Kingman case. In the following section we review the p-LD-ASG [17] and extend it to the p-LD-Λ-ASG.

The pruned lookdown-Λ-ancestral selection graph
At each time r, the pruned lookdown Λ-ASG G consists of a finite number L r of lines, numbered by the integers 1, . . . , L r , to which we refer as levels. The evolution of the lines as r increases is determined by a point configuration on R × P(N) ∪ R × N × { * , ×, •} , where P(N) is the set of subsets of N. Each of the points (r, τ ) stands for a transition element τ occurring at time r, that is, a merger, a selective branching, a deleterious mutation, or a beneficial mutation at time r. One of the lines, called the immune line, is distinguished; its level at time r is denoted by M r . The role of this line will become clear from Proposition 3.1. The reason why we call it immune is because it is not killed by mutations.
Let us now detail the transition elements and their effects on G (see Figs. 1 and 2): • A merger at time r is a pair (r, η), where η is a subset of N. If |{1, . . . , L r− } ∩ η| ≤ 1, then G is not affected. If, however, {1, . . . , L r− } ∩ η = {j 1 , . . . , j κ } with j 1 < · · · < j κ and κ ≥ 2, then the lines at levels j 2 , . . . , j κ merge into the line at level j 1 . The remaining lines in G are relocated to 'fill the gaps' while retaining their original order; this renders L r = L r− − κ + 1. The immune line simply follows the line on level M r− . Proof. In the absence of multiple mergers (i.e. if all mergers have exactly two elements), this is Theorem 4 in [17]. In its proof, the induction step for binary mergers directly carries over to multiple mergers.
The transition elements arrive via independent Poisson processes: For each i ∈ N, the 'stars', 'crosses', and 'circles' at level i come as Poisson processes with intensities σ, θν 1 and θν 0 , respectively. For each 2-element subset η of N, the 'η-mergers' come as a Poisson process with intensity Λ({0}). In addition, we have a Poisson process with intensity measure 1 {z>0} 1 z 2 Λ(dz) dr, where each z generates a random subset H(z) := {i : V i = 1} ⊂ N, with (V i ) i∈N being a Bernoulli(z)-sequence, and the point (r, z) gives rise to the merger (r, H(z)). All these Poisson processes are independent. The points (r, τ ) constitute a Poisson configuration Ψ, whose intensity measure we denote by µ ⊗ ρ, where µ is Lebesgue measure on R. With the transition rules described above, this induces Markovian jump rates upon L r and (L r , M r ). With the help of (2.2), it is easily checked that the generator G L of L is  given by . (3.1) Due to Assumption 2.1 and Remark 2.2b), and because L is stochastically dominated by K, the process L obeys Thus L has a time-stationary version L (which is L ≡ 1 if σ = 0), and likewise the pruned lookdown Λ-ASG has an equilibrium version as well. We now set L eq := L 0 and denote the tail probabilities of L eq by α n := P(L eq > n), n ∈ N 0 . Because of (3.2), for almost all realisations of L, there exists an r 0 < 0 such that L r0 = 1. Hence, arguing as in [17, proof of Theorem 5], we conclude from Proposition 3.1 the following Corollary 3.2. Given the frequency of the beneficial type at time 0 is x, the probability that the immortal line in the equilibrium p-LD-Λ-ASG at time 0 is of beneficial type is In order to further evaluate the representation (3.3), we need information about the equilibrium tail probabilities α n . This is achieved in the following sections via a process D which is a Siegmund dual for L.

Tail probabilities and hitting probabilities
It is clear that L is stochastically monotone, that is, P n (L r ≥ i) ≥ P m (L r ≥ i) for n ≥ m and for all i ∈ S (where the subscript refers to the initial value of the process). It is well known [22] that such a process has a Siegmund dual, that is, there exists a process D such that Lemma 4.1. The tail probabilities of the stationary distribution of L are hitting probabilities of the dual process D. To be specific, Proof. This is a special case of [4, Thm. 1] for entrance and exit laws. In our case the entrance law is the equilibrium distribution of L, the exit law is a harmonic function (in terms of hitting probabilities), and the proof reduces to the following elementary argument. Namely, evaluating the duality condition (4.1) for = 1 and d = n + 1, n ≥ 0, gives Taking the limit u → ∞, the left-hand side converges to P(L eq > n) = α n by positive recurrence and irreducibility. Setting = d = 1 in (4.1), we see that 1 is an absorbing state for D. Hence we have for the right-hand side of (4.3) lim u→∞ P n+1 (D u = 1) = P n+1 (∃t ≥ 0 : D t = 1) ∀n ≥ 0, and the lemma is proven.
Next we want to show that the (shifted) hitting probabilities α n = P n+1 (∃t ≥ 0 : D t = 1), n ≥ 0, (4.4) satisfy the system of equations (2.5). More precisely, (2.5) will emerge as a first-step decomposition of the hitting probabilities. For this purpose, we first have to identify the jump rates of D. This can be done via a generator approach that translates the jump rates of the process L (which appear in (3.1)) into their dual jump rates, see, for instance, formula (12) in [3] or in [22]. For the jump rates coming from the mergers this is somewhat technical, see the calculations in the appendix in [13]. Inspired by [3] we will therefore take a 'strong pathwise approach' that consists in decomposing the dynamics of L into so-called flights, which can be 'dualised' one by one. While Clifford and Sudbury, starting from the generator of a monotone process, in [3, Thm 1] construct a special Poisson process of flights for which they form the duals ([3, Thm 2]), in our situation the Poisson process of flights is naturally given (being induced by the transition elements for G defined in Sec. 3, see Sec. 4.3 below). Consequently, we will show in Proposition 4.3 that the approach of [3, Thm 2] works also when starting from a more general Poisson process of flights.

Flights and their duals
In [3], Clifford and Sudbury introduced a graphical representation that allows to construct a monotone homogeneous Markov process L together with its Siegmund dual D on one and the same probability space. The method requires that the state space S of the processes L and D be (totally) ordered. We restrict ourselves to the case S := N ∪ {∞}, which is the relevant one in our context (and which is prominent in [3] as well).
The basic building blocks of Clifford and Sudbury's construction are so-called flights. A flight f is a mapping from S into itself that is order-preserving, so f (k) ≤ f ( ) for all k < with k, ∈ S; let us add that each flight leaves state ∞ invariant, so f (∞) = ∞. By the construction described below, a flight f that appears at time r will induce the transition to L r = f ( ), given L r− = . This way, transitions from different initial states will be coupled on the same probability space. A flight f is graphically represented as a set of simultaneous arrows pointing from to f ( ), for all ∈ S, so that the process simply follows the arrows. Examples are shown in Fig. 3.
We denote the set of all flights by F, and consider a Poisson process Φ on R × F whose intensity measure is of the form µ ⊗ γ, where µ is again Lebesgue measure on R, and the measure γ has the property Property (4.5) implies that with probability 1, for all ∈ N and r ∈ R, among all the points (s, f ) in Φ with s > r and f ( ) = , there is one whose s is minimal. We denote this time by v(r, ). For r ∈ R and ∈ N, we define inductively a sequence (s 0 , 0 ), (s 1 , 1 ), . . . with r =: s 0 < s 1 < · · · , =: 0 , 1 , 2 , . . . ∈ S, by setting (Note this procedure will terminate if i = ∞ for some i ∈ N.) With the notation just introduced, Φ induces a semi-group (a flow) of mappings, indexed by r < s ∈ R, and defined by for ∈ N, with F r,s (∞) := ∞.
Assuming property (4.5), we say that Φ represents the process L if for all s > 0 the distribution of F 0,s ( ) is a version of the conditional distribution of L s given {L 0 = }, ∈ N. Equivalently, for all r ∈ R and u > 0, P (L u ∈ (.)) = P(F r,r+u ( ) ∈ (.)). (4.7) We now describe, in the footsteps of Clifford and Sudbury [3], the construction of a strong pathwise Siegmund dual D, based on the same realisation of the flights as for the original process L. For examples, see Fig. 3, and note that we insinuate r = −t.
we define F in terms of Φ in the same way as F was defined in terms of Φ by (4.6).
It is clear that f is order preserving. Since f is monotone increasing by assumption, (4.8) is equivalent to with the convention max(∅) = 0. Note further that (4.9) is implied by (4.5) together with The following proposition is an adaptation of [3, Theorem 2] to our setting. Compare also [14,Section 4.1]. Proposition 4.3. Assume (4.5) and (4.11), and assume that ∞ is unattainable for the process L represented by the Poisson process Φ with intensity measure µ ⊗ γ. Then the following strong pathwise duality relation is valid: For all s > 0; , d ∈ N, almost surely.  (4.5) and the assumption that ∞ is unattainable, Y has a.s only finitely many jumps; let us denote the jump times by −r 1 , . . . , −r n . We write J for the union of {r 1 , . . . , r n } and the set of jump times of Y . Because of (4.9), J has a smallest element, a second-smallest element, and so on. We denote these elements by u 1 < u 2 < . . ., and show that Proceeding by induction, for (4.13) it is sufficient to show for all flights f , and j, k ∈ N. Let f ∈ F. On the one hand, f (j) ≥ k yields where we have used order preservation of f and f as well as (4.8). On the other hand, f (j) < k is equivalent to f (j) + 1 ≤ k. By order preservation and (4.10), this entails We have thus shown (4.14), and hence also (4.13). If (u i ) has no accumulation point, then it has a maximal element, say u m . Choosing i = m in the r.h.s. of (4.13) yields (4.12) (since u m = s with probability 1). If (u i ) has an accumulation point, say τ , then, because of (4.9), we have lim t↑τ Y t = lim i→∞ Y ui = ∞. Because Y remains bounded by assumption, this together with (4.13) enforces that Y 0 < Y 0 . This means that the l.h.s. of (4.12) takes the value 0. However, this is the case also for the r.h.s of (4.12), since ∞ = Y τ = Y s > .
In view of (4.7) we immediately obtain the following compare also Fig. 3. The flights are indeed order preserving. The structure of f η , f i, * , and f i,• is clearly inherited from that of the corresponding transition elements. The flights f i,× ( ) forget about the position (but not about the existence) of the immune line within the p-LD-Λ-ASG. Indeed, recall that the downward jump rate of L due to deleterious mutations is ( − 1)θν 1 ; this reflects the fact that crosses arrive at rate θν 1 per line, but are ignored on the immune line, no matter where it is located. This is taken into account in the definition of the flight f i,× by setting f ,× ( ) = .
Following Definition 4.2, we can now consider a process D represented by Φ. According to Corollary 4.4, L and D then obey the duality relation (4.1). It remains to read off the jump rates of D from the intensities of the (dual) flights.
Lemma 4.5. The generator G D of the process D is given by where we again use the convention (2.6).
Proof. We claim that the flights that are dual to those in (4.15) are of the form  Fig. 3).
Let us now consider the contribution of the various types of flights to G D . For c = d ∈ N we have to compute γ({f : f (d) = c}). It is clear that the contributions from γ * , γ × and γ • yield the last 3 summands in (4.16). For the contribution coming from γ m , we have for d < c < ∞  The contribution from the Kingman mergers to the right-hand side of of (4.19) is Λ({0}) c−1 2 if c = d + 1, and 0 otherwise. For z > 0, the probability that a z-merger does not affect level c but does affect c − d + 1 out of the levels 1, . . . , c − 1 is c−1 c−d+1 z c−d+1 (1 − z) d−1 . Integrating this with respect to 1 z 2 Λ(dz) and adding the Kingman component shows that the right-hand side of (4.19) equals c−1 c−d+1 λ c,c−d+1 . These are the jump rates from d to c < ∞ that appear in the first sum on the r.h.s. of (4.16). It remains to take into account the jump rate of D from d to ∞. For this we note that f N ( ) = 1, = 1, 2, . . ., and consequently f (d) = 1 if d = 1 and f (d) = ∞ if d ≥ 2. These flights appear at rate Λ({1}), and thus for d ≥ 2 add the term (g(∞) − g(d))Λ({1}) to the generator. Remark 4.6. In the case without selection and mutation (that is, σ = θ = 0), our process D shifted by one, that is, D − 1, is equal to the so-called fixation line in [13]. In this case one has no pruning, and the line counting process K has generator (2.3) (with σ = 0). The (Siegmund) duality between K and D is stated in [13,Lemma 2.4]. See also [11,Thm 2.3] for a corresponding statement on the still more general class of exchangeable coalescents.