Coalescents with Simultaneous Multiple Collisions

We study a family of coalescent processes that undergo ``simultaneous multiple collisions,'' meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. This family of processes, which we obtain from simple assumptions about the rates of different types of mergers, essentially coincides with a family of processes that Mohle and Sagitov obtain as a limit of scaled ancestral processes in a population model with exchangeable family sizes. We characterize the possible merger rates in terms of a single measure, show how these coalescents can be constructed from a Poisson process, and discuss some basic properties of these processes. This work generalizes some work of Pitman, who provides similar analysis for a family of coalescent processes in which many clusters can coalesce into a single cluster, but almost surely no two such mergers occur simultaneously.


Introduction
In this paper, we study a family of coalescent processes that undergo "simultaneous multiple collisions," meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. These processes were previously introduced in [13] by Möhle and Sagitov, who obtained them by taking limits of scaled ancestral processes in a population model with exchangeable family sizes. Here we take a different approach to characterizing these processes. The approach is similar to that used by Pitman in [16] for "coalescents with multiple collisions," also called Λ-coalescents, in which many clusters of particles can merge at one time into a single cluster but almost surely no two such mergers occur simultaneously. The family of processes studied here includes the Λ-coalescents as a special case, and we generalize several facts about Λ-coalescents.
Let P n denote the set of partitions of {1, . . . , n}, and let P ∞ denote the set of partitions of N = {1, 2, . . .}. Given m < n ≤ ∞ and π ∈ P n , let R m π be the partition in P m obtained by restricting π to {1, . . . , m}. That is, if 1 ≤ i < j ≤ m, then i and j are in the same block of the partition R m π if and only if they are in the same block of π. Following [16], we identify each π ∈ P ∞ with the sequence (R 1 π, R 2 π, . . .) ∈ P 1 × P 2 × . . .. Each P n is given the discrete topology and P ∞ is given the topology that it inherits from the product P 1 × P 2 × . . ., so P ∞ is compact and metrizable. We equip P ∞ with the Borel σ-field associated with this topology. We call a P n -valued process (Π n (t)) t≥0 a coalescent if it has right-continuous step function paths and if Π n (s) is a refinement of Π n (t) for all s < t. We call a P ∞ -valued process (Π ∞ (t)) t≥0 a coalescent if it has càdlàg paths and if Π ∞ (s) is a refinement of Π ∞ (t) for all s < t. Equivalently, (Π ∞ (t)) t≥0 is a coalescent if and only if for each n, the process (R n Π ∞ (t)) t≥0 is a coalescent.
In [16], Pitman studies "coalescents with multiple collisions," which are P ∞ -valued coalescents (Π ∞ (t)) t≥0 with the property that for each n ∈ N , the process (R n Π ∞ (t)) t≥0 is a P n -valued Markov chain such that when R n Π ∞ (t) has b blocks, each possible merger of k blocks into a single block is occurring at some fixed rate λ b,k that does not depend on n, and no other transitions are possible. It is shown in [16] that given a collection of rates {λ b,k : 2 ≤ k ≤ b < ∞}, such a process exists if and only if the consistency condition λ b,k = λ b+1,k + λ b+1,k+1 (1) holds for all 2 ≤ k ≤ b. Theorem 1 of [16] shows that (1) holds if and only if whenever 2 ≤ k ≤ b, we have for some finite measure Λ on [0, 1]. The process is then called the Λ-coalescent. When Λ is a unit mass at zero, we obtain Kingman's coalescent, a process introduced in [10] in which only two blocks can merge at a time and each pair of blocks is merging at rate 1. The case in which Λ is the uniform distribution on [0, 1] was studied by Bolthausen and Sznitman in [3].
In [18], Sagitov obtains all of the Λ-coalescents, up to a time-scaling constant, as limits of ancestral processes in a haploid population model with an exchangeable distribution of family sizes in which there are N individuals in each generation. The ancestral processes are P n -valued processes obtained by sampling n out of N individuals from the current generation and tracing their ancestors backwards in time. A simpler formulation of this convergence result is presented in [12], and similar results for a diploid population model are given in [14].
An important property of the Λ-coalescent is that the rate at which blocks are merging does not depend on the size of the blocks or on which integers are in the blocks. It is therefore natural to pursue a generalization to a larger class of processes that still have this property but that may undergo "simultaneous multiple collisions." The possibility of such a generalization is mentioned in section 3.3 of [16]. We define a (b; k 1 , . . . , k r ; s)-collision to be a merger of b blocks into r + s blocks in which s blocks remain unchanged and the other r blocks contain k 1 , . . . , k r ≥ 2 of the original blocks. Thus, b = r j=1 k j + s. The order of k 1 , . . . , k r does not matter; for example, any (5; 3, 2; 0)-collision is also a (5; 2, 3; 0)-collision. It is easily checked (see equation (11) of [15]) that if r ≥ 1, k 1 , . . . , k r ≥ 2, s ≥ 0, b = r j=1 k j + s, and l j is the number of k 1 , . . . , k r that equal j, then the number of possible (b; k 1 , . . . , k r ; s)-collisions is b!
We define a coalescent with simultaneous multiple collisions to be a P ∞ -valued coalescent process (Π ∞ (t)) t≥0 with the property that for each n ∈ N , the process (R n Π ∞ (t)) t≥0 is a P n -valued Markov chain such that when R n Π ∞ (t) has b blocks, each possible (b; k 1 , . . . , k r ; s)-collision is occurring at some fixed rate λ b;k 1 ,...,kr;s .
In [13], Möhle and Sagitov generalize the proofs in [12] and obtain coalescents with simultaneous multiple collisions as limits of ancestral processes in a haploid population model. We now describe their model and their results. Assume there are N individuals in each generation. For all a ≥ 0, let ν N,N ), which we denote by µ N , is assumed to be exchangeable and to be the same for all a. Therefore, we will suppress the superscript in the notation when we are concerned only about the distributions of the family sizes. Möhle and Sagitov consider a random sample of n ≤ N distinct individuals from the 0th generation. They define the Markov chain (Ψ n,N (a)) ∞ a=0 , where Ψ n,N (a) is the random partition of {1, . . . , n} such that i and j are in the same block if and only if the ith and jth individuals in the sample have a common ancestor in the ath generation backwards in time. Let (m) k = m(m − 1) . . . (m − k + 1), and let (m) 0 = 1. Let c N be the probability that two individuals chosen randomly from some generation have the same ancestor in the previous generation. ¿From equation (5) of [13], we have The following result is part of Theorem 2.1 in [13]. where x i ) s (7) and for 1 ≤ j ≤ m, when we set i 0 = −1 and i j+1 = s + 1.
Möhle and Sagitov point out that when s = 0, equation (6) gives so the moments of the measures F r are collision rates. Also, equation (9) above and equations (16) and (19) of [13] imply that for all r ≥ 1, k 1 , . . . , k r ≥ 2, and b = r j=1 k j . Note that some condition like the existence of the limits in (5) is needed to relate the distributions µ N for different values of N . The condition lim N →∞ c N = 0 ensures that the limit obtained is a continuous-time process. If instead lim N →∞ c N = c > 0, then Theorem 2.1 of [13] states that the limit is a discrete-time Markov chain.
In section 4, we will prove Lemma 18, which shows that this condition implies the existence of a P ∞ -valued coalescent (Π ∞ (t)) t≥0 with the property that (R n Π ∞ (t)) t≥0 and (Ψ n,∞ (t)) t≥0 have the same distribution for all n ∈ N . Thus, coalescents with simultaneous multiple collisions can be derived from the ancestral processes studied in [13].
However, Möhle and Sagitov leave open the question of whether every possible coalescent with simultaneous multiple collisions can be obtained as a limit of ancestral processes in their population model. They also do not discuss the questions of which sequences of measures (F r ) ∞ r=1 satisfying conditions A1, A2, and A3 of Proposition 1 are associated with coalescent processes in the manner described above, and whether there is a natural probabilistic interpretation of the measures F r , aside from the interpretation of their moments as collision rates.
The primary goals of this paper are to answer these questions, and to establish an alternative characterization of coalescents with simultaneous multiple collisions based on a single measure Ξ on the infinite simplex We will show that, up to a scaling constant, all coalescents with simultaneous multiple collisions can be obtained as limits of ancestral process as described above, and therefore these coalescents can be characterized either by a single measure Ξ or by a sequence of measures (F r ) ∞ r=1 . One advantage to the characterization based on Ξ is that every finite measure on the infinite simplex is associated with a coalescent process. However, for a sequence of measures (F r ) ∞ r=1 satisfying conditions A1, A2, and A3 of Proposition 1 to be associated with a coalescent process, we will show that it must satisfy an additional consistency condition that does not appear to be easy to check.
The rest of this paper is organized as follows. In section 2, we summarize the results that establish the two characterizations of coalescents with simultaneous multiple collisions. We also state results that give interpretations of the characterizing measures. In section 3, we give a Poisson process construction of these coalescents. The Poisson process construction is an important tool for studying the coalescents and is used in most of the proofs in the paper. In section 4 we prove the results stated in section 2. In section 5, we build on work done for the Λ-coalescent to establish some further properties of coalescents with simultaneous multiple collisions. We establish there some regularity properties of the coalescents and derive a condition for a coalescent with simultaneous multiple collisions to be a jump-hold Markov process with bounded transition rates. We also present some results related to the question of whether the coalescents "come down from infinity," meaning that only finitely many blocks remain at any time t > 0 even if the coalescent is started with infinitely many blocks at time zero. Finally in section 6, we discuss the discrete-time analogs of these processes, which can also arise as limits of ancestral processes in the population models studied in [13].

Summary of results characterizing the coalescents
In this section, we summarize the results needed to establish the two characterizations of the coalescents with simultaneous multiple collisions, one involving a single measure Ξ and the other involving a sequence of measures (F r ) ∞ r=1 . The proofs of all propositions and theorems in this section are given in section 4.
We first state the main theorem of this paper, which characterizes coalescents with simultaneous multiple collisions in terms of a measure Ξ on the infinite simplex ∆. In the statement of this result, and throughout the rest of the paper, we refer to the point (0, 0, . . .) ∈ ∆ as "zero," and we denote a generic point in ∆ by x = (x 1 , x 2 , . . .).

Definition 3
We call a coalescent process satisfying B2 whose collision rates are given by (11) for a particular finite measure Ξ on ∆ a Ξ-coalescent. We call a Ξ-coalescent satisfying B1 the standard Ξ-coalescent.
Suppose π ∈ P ∞ and B 1 , B 2 , . . . are the blocks of π. If (Π ∞ (t)) t≥0 is a standard Ξ-coalescent, then we can define a Ξ-coalescent (Π π ∞ (t)) t≥0 satisfying Π π ∞ (0) = π by defining i ∈ B k and j ∈ B l to be in the same block of Π π ∞ (t) if and only if k and l are in the same block of Π ∞ (t). Since any Ξ-coalescent can thus be derived easily from the standard Ξ-coalescent, we will restrict our attention to the standard Ξ-coalescent whenever it is simpler to do so.
For any finite measure Ξ on ∆, a collection of nonnegative collision rates can be defined by (11), so Theorem 2 implies that a standard Ξ-coalescent exists. The following proposition states that the collision rates of a coalescent with simultaneous multiple collisions uniquely determine the associated measure Ξ. Thus, there is a one-to-one correspondence between finite measures Ξ on ∆ and coalescent processes satisfying conditions B1 and B2 of Theorem 2.

Proposition 4 Let Ξ and Ξ be finite measures on the infinite simplex
Observe that we can easily recover the Λ-coalescent as a special case of the Ξ-coalescent. Suppose Ξ is concentrated on the subset of ∆ consisting of the sequences (x 1 , x 2 , . . .) such that x i = 0 for all i ≥ 2. Then, λ b;k 1 ,...,kr;s = 0 unless r = 1. When r = 1, the expression inside the double summation in the numerator in the integrand of (11) is zero unless l = 0, so This result agrees with the formula for λ b,k 1 given in (2) when Λ is the projection of Ξ onto the first coordinate. Note that when Ξ is a unit mass at zero, the Ξ-coalescent is therefore Kingman's coalescent.
We next work towards giving an interpretation of the measure Ξ. Equation (11) implies that If Ξ(∆) = 0, then all the collision rates are zero. Otherwise (12) implies that Ξ = λ 2;2;0 G, where G is a probability measure defined by G(S) = Ξ(S)/Ξ(∆) for all measurable subsets S of ∆. From (11), we see that if Ξ is multiplied by a constant, then all of the collision rates are multiplied by the same constant. Therefore, unless Ξ = 0, any Ξ-coalescent can be obtained from a G-coalescent, where G is a probability measure, by rescaling time by a constant factor. To interpret G, we first give the following definition. The following proposition, which is the natural analog of Theorem 4 of [16], gives an interpretation of G. We write #π for the number of blocks in a partition π.
Note that essentially the same result would hold if we defined T to be the time at which two arbitrary fixed integers merged, but we state the result in terms of the collision time of 1 and 2 to simplify notation.
We now turn to the question of whether all coalescents with simultaneous multiple collisions can arise as limits of ancestral processes in a population model of the type discussed in [13]. Since Ψ n,∞ (0), as defined in Proposition 1, equals the partition of {1, . . . , n} into singletons, only standard Ξ-coalescents can arise in this way. Also, it follows from (9) and condition A3 of Proposition 1 that for coalescents obtained from ancestral processes as described in the introduction, we have λ 2;2;0 = F 1 (∆ 1 ) = 1.
It then follows from (12) that Ξ is a probability measure. However, since λ 2;2;0 is just a timescaling factor and the case Ξ = 0 is trivial, Proposition 7 below shows that the family of continuous-time processes that can be obtained from the ancestral processes studied in [13] should be regarded as essentially the same as the family of standard Ξ-coalescents.

Proposition 7
Let Ξ be a probability measure on ∆. Then there exists a sequence (µ N ) ∞ N =1 such that each µ N is a probability distribution on {0, 1, 2, . . .} N that is exchangeable with respect to the N coordinates with the property that if for all N , µ N is the distribution of family sizes in the population model described in the introduction, then for all n, the processes (Ψ n, We can use Proposition 1 and Proposition 7 to characterize the coalescents with simultaneous multiple collisions by a sequence of measures (F r ) ∞ r=1 . We state this result precisely below. Note that the result can be stated without referring to the population model that originally motivated Möhle and Sagitov to study the measures F r .
Conditions A1 and A2 come directly from Proposition 1. The reason for replacing A3 with A3 is to obtain coalescents for which λ 2;2;0 = 1. Condition A4, which is clearly necessary for (F r ) ∞ r=1 to be associated with a coalescent process, can be viewed as a consistency condition on the measures F r . The following example shows that A4 does not always hold.
We next prove a result which interprets the measures F r as distributions of limiting relative frequencies of blocks of random partitions. This result parallels Proposition 6, which provides a similar interpretation of Ξ. We could prove essentially the same result replacing the integers 1, . . . , 2r with any distinct integers i 1 , . . . , i 2r .
We have established in Theorem 2 and Proposition 4 a one-to-one correspondence between finite measures Ξ on the infinite simplex ∆ and coalescent processes satisfying B1 and B2. Proposition 8 shows that these coalescent processes are also in one-to-one correspondence with the sequences of measures (F r ) ∞ r=1 satisfying A1, A2, A3 , and A4. These results, of course, yield a natural one-to-one correspondence between finite measures Ξ on ∆ and sequences of measures (F r ) ∞ r=1 satisfying A1, A2, A3 , and A4. The last result of this section shows how to calculate (F r ) ∞ r=1 given the measure Ξ. We are unable to give a simple formula for calculating Ξ directly from (F r ) ∞ r=1 .

Proposition 11
Let Ξ = Ξ 0 + aδ 0 be a finite measure on the infinite simplex ∆, where Ξ 0 has no atom at zero and δ 0 is a unit mass at zero. Let (F r ) ∞ r=1 be the unique sequence of measures satisfying conditions A1, A2, and A3 of Proposition 8 such that the collision rates of a standard Ξ-coalescent are given by (6). Let S be a measurable subset of ∆ r . Then,

The Poisson process construction
In [16], Pitman gives a Poisson process construction of the Λ-coalescent when Λ has no atom at zero. Here we generalize this idea to obtain a Poisson process construction of the Ξ-coalescent started from any π ∈ P ∞ for all finite measures Ξ on the infinite simplex ∆. Our construction does permit Ξ to have an atom at zero. This construction is a useful tool for studying the Ξ-coalescent, in part because making computations using (11) can be tedious.
We will first define a σ-finite measure L on Z ∞ , which is a Polish space when equipped with the product topology. We will then use a Poisson point process (e(t)) t≥0 with characteristic measure L, as defined in appendix B, to construct the Ξ-coalescent. For each x = (x 1 , x 2 , . . .) ∈ ∆, define a probability measure P x on Z ∞ to be the distribution of a sequence ξ = (ξ i ) ∞ i=1 of independent Z-valued random variables such that for all i we have P (ξ i = j) = x j for all j ∈ N and Then define a measure L on Z ∞ by for all product measurable A ⊂ Z ∞ . To show that L is σ-finite, and to establish some facts that will be useful later, we define for all b ≥ 2 and for all k = l. Note that if k = l then P x ({ξ : ξ k = ξ l = j}) = x 2 j for all j ∈ N , which means Define Note that L(A c ∞ ) = 0. Thus, the union of the sets in the countable collection consisting of A c ∞ and A b for all b ≥ 2 equals Z ∞ , and L assigns finite measure to each set in this collection. Hence, L is σ-finite, so we can define a Poisson point process (e(t)) t≥0 with characteristic measure L.
We now use this Poisson point process to construct a Ξ-coalescent starting from an arbitrary π ∈ P ∞ . First, we define for each n a P n -valued coalescent Π π n = (Π π n (t)) t≥0 as follows. Let T 0,n = 0 and for k ≥ 1, define T k,n = inf{t > T k−1,n : e(t) ∈ A n }. Since L(A n ) < ∞ by (18), it follows from part (b) of Lemma 41 in appendix B that lim k→∞ T k,n = ∞ a.s. Therefore, by the argument used to prove part (c) of Lemma 41 in appendix B, we have e(T k,n ) ∈ A n for all k almost surely. We will define Π π n to have right-continuous step function paths with jumps only possible at the times T k,n for k ≥ 1. Therefore, it suffices to specify Π π n (T k,n ) for all k ≥ 0. Define Π π n (0) = R n π. For k ≥ 1, if Π π n (T k−1,n ) consists of the blocks B 1 , . . . , B b , where the blocks are ordered by their smallest elements, then Π π n (T k,n ) is defined to be the partition of {1, . . . , n}, each of whose blocks is a union of some of the blocks B 1 , . . . , B b , such that B i and B j are in the same block of Π π n (T k,n ) if and only if e(T k,n ) i = e(T k,n ) j , where e(T k,n ) i and e(T k,n ) j denote the ith and jth coordinates respectively of e(T k,n ).
Suppose m < n. We claim that (R m Π π n (t)) t≥0 = (Π π m (t)) t≥0 . If ξ ∈ A m then ξ ∈ A n , so the processes (R m Π π n (t)) t≥0 and (Π π m (t)) t≥0 can only jump at times T k,n for some k ≥ 1. Thus, to prove the claim, it suffices to show that R m Π π n (T k,n ) = Π π m (T k,n ) for all k ≥ 0. We use induction on k. Note first that R m Π π n (T 0,n ) = R m Π π n (0) = R m π = Π π m (0) = Π π m (T 0,n ). Suppose k ≥ 1 and R m Π π n (T k−1,n ) = Π π m (T k−1,n ). Let B 1 , . . . , B b be the blocks of Π π m (T k−1,n ) and let B 1 , . . . , B d be the blocks of Π π n (T k−1,n ), where blocks are ordered by their smallest elements. Fix i, j ∈ {1, . . . , m} and define h(i) and h(j) such that i ∈ B h(i) and j ∈ B h(j) . Since B i = {1, . . . , m} ∩ B i for i = 1, . . . , b, we have i ∈ B h(i) and j ∈ B h(j) . Therefore i and j are in the same block of Π π m (T k,n ) if and only if e(T k,n ) h(i) = e(T k,n ) h(j) . Likewise, i and j are in the same block of Π π n (T k,n ), and thus the same block of R m Π π n (T k,n ), if and only if e(T k,n ) h(i) = e(T k,n ) h(j) . We conclude that Π π m (T k,n ) = R m Π π n (T k,n ), so by induction, (R m Π π n (t)) t≥0 = (Π π m (t)) t≥0 . Now define a P ∞ -valued process Π π ∞ = (Π π ∞ (t)) t≥0 such that i and j are in the same block of Π π ∞ (t) if and only if i and j are in the same block of Π π n (t) for n ≥ max{i, j}. Then, Π π n = R n Π π ∞ . The proposition below establishes that Π π ∞ is a Ξ-coalescent started from π.
Definition 13 Let Ξ be a finite measure on the infinite simplex ∆. Suppose (e(t)) t≥0 is a Poisson point process with characteristic measure L, where L is defined in terms of Ξ by (14). If Π ∞ is defined from (e(t)) t≥0 as described above, then we say Π ∞ is a Ξ-coalescent derived from (e(t)) t≥0 .

Remark 14
Note that by taking Ξ = δ 0 , we can obtain a Poisson process construction of Kingman's coalescent. In this case, (14) reduces to The coalescent derived from a Poisson point process (e(t)) t≥0 with the characteristic measure L defined in (22) has the property that if the blocks are ordered by their smallest elements, then the ith and jth blocks merge at the times t for which e(t) = z ij .
Then, Π ∞ (T A −) and e(T A ) are independent.
Proof. Let (e (t)) t≥0 be defined such that e (t) = δ if e(t) ∈ A and e (t) = e(t) otherwise. By parts (d) and (e) of Lemma 41 in appendix B, we have that e(T A ), T A , and (e (t)) t≥0 are mutually independent. Since Π ∞ (T A −) is a function of (e (t)) t≥0 and T A , it follows that

Preliminary Lemmas
In this subsection, we give some preliminary lemmas that will be useful for some of the proofs of results in section 2. We begin with the following result, which can be proved by a straightforward application of the Daniell-Kolmogorov Theorem.

Lemma 16
Suppose, for each n, that Θ n is a random partition of {1, . . . , n}. Suppose R m Θ n and Θ m have the same distribution for all m < n. Then, there exists on some probability space a random partition Θ ∞ of N such that R n Θ ∞ has the same distribution as Θ n for all n.
The lemma below will enable us to construct P ∞ -valued coalescents from consistently-defined P n -valued coalescents. It is proved by an application of the Daniell-Kolmogorov Theorem as on p.40 of [11].

Lemma 17
Suppose, for each n, that (Π n (t)) t≥0 is a P n -valued coalescent. Suppose, for all m < n, that the processes (R m Π n (t)) t≥0 and (Π m (t)) t≥0 have the same law. Then, there exists on some probability space a P ∞ -valued coalescent (Π ∞ (t)) t≥0 such that (R n Π ∞ (t)) t≥0 has the same law as (Π n (t)) t≥0 for all n.
The next lemma gives the consistency condition that an array of nonnegative real numbers k j + s} must satisfy to be the array of collision rates for a coalescent with simultaneous multiple collisions. This condition is the analog of condition (1) for the Λ-coalescent.
Proof. Continuous-time Markov chains on a finite state space can be constructed with arbitrary nonnegative transition rates. Thus, we can define for each n a Markov chain Π n = (Π n (t)) t≥0 with state space P n and right-continuous paths such that Π n (0) is the partition of {1, . . . , n} into singletons and, when Π n (t) has b blocks, each (b; k 1 , . . . , k r ; s)-collision is occurring at rate λ b;k 1 ,...,kr;s .
Define for each n a process Θ n = (Θ n (t)) t≥0 by Θ n (t) = R n Π n+1 (t). Suppose Θ n and Π n have the same law. Then R m Π n and Π m have the same law for all m < n. By Lemma 17, there exists a coalescent process (Π ∞ (t)) t≥0 such that (R n Π ∞ (t)) t≥0 has the same law as (Π n (t)) t≥0 for all n. The process (Π ∞ (t)) t≥0 satisfies conditions B1 and B2 of Theorem 2. Conversely, suppose Π ∞ = (Π ∞ (t)) t≥0 satisfies B1 and B2 of Theorem 2. Then, Π n has the same law as R n Π ∞ and Θ n has the same law as R n (R n+1 Π ∞ ) = R n Π ∞ , so Θ n and Π n have the same law. Thus, we must show that Θ n and Π n have the same law for all n if and only if (23) holds.
Next, suppose Θ n undergoes a (b; k 1 , . . . , k r ; s)-collision at time t ≤ U . Then, Π n+1 could undergo any of r + s + 1 possible collisions at time t, as can be seen by considering the following three cases: Case 1: The block {n + 1} could remain a singleton at time t, in which case Π n+1 undergoes a (b + 1; k 1 , . . . , k r ; s + 1)-collision at time t.
Case 2: The block {n + 1} could join one of the s blocks of Θ n (t) that consists of a single block of Θ n (t−), in which case Π n+1 undergoes a (b; k 1 , . . . , k r , 2; s − 1)-collision at time t.
Case 3: The block {n + 1} could join one of the r blocks of Θ n (t) consisting of two or more blocks of Θ n (t−). Then, Π n+1 undergoes a (b + 1; Thus, before time U , the rate of any (b; k 1 , . . . , k r ; s)-collision for the process Θ n is the same as the sum of the rates at which Π n+1 is undergoing one of the r + s + 1 collisions described above. The definition of Π n+1 implies that this rate equals the right-hand side of (23).
It follows from the results proved in the last two paragraphs that if (23) holds for all r ≥ 1, k 1 , . . . , k r ≥ 2, s ≥ 0, and b = r j=1 k j + s, then Θ n and Π n have the same law for all n. Conversely, suppose Θ n and Π n have the same law. Then the initial rate at which Π n is undergoing an (n; k 1 , . . . , k r ; s)-collision is λ n;k 1 ,...,kr;s by definition, and arguments in the previous paragraph imply that the initial rate at which Θ n is undergoing an (n; k 1 , . . . , k r ; s)-collision is given by the right-hand side of (23) with b = n. Hence, (23) holds when b = n. Thus, if Θ n and Π n have the same law for all n, then (23) holds for all r ≥ 1, k 1 , . . . , k r ≥ 2, s ≥ 0, and b = r j=1 k j +s.

Proof of Theorem 2
In this subsection, we prove Theorem 2. Recall that in section 3, we constructed from a Poisson point process a standard Ξ-coalescent for an arbitrary finite measure Ξ on the infinite simplex ∆. This construction proves the "if" part of Theorem 2. However, Lemma 18 provides a way of proving the "if" part of Theorem 2 more directly by checking the consistency of the transition rates defined by (11). We provide this alternative proof below.
The proof of the "only if" part of Theorem 2 relies heavily on exchangeability arguments. Some well-known results that we will apply are reviewed in appendix A. It will be convenient to make the following additional definition. Note that any coalescent process satisfying conditions B1 and B2 of Theorem 2 for some collection of rates {λ b;k 1 ,...,kr;s : Proof. We may assume, without loss of generality, that a 1 ≥ a 2 ≥ . . .. Suppose there exists an i such that b i = 0 and a i > 0. Let m = min{i : b i = 0}, and let s = max{r ≥ 0 : a m+r = a m }.
We have We have a k m |b m | > Ba k−1 m+s+1 for sufficiently large k because a m+s+1 < a m and |b m | > 0, which contradicts the fact that the expression on the right-hand side of (25) is zero for all k ∈ N .
Proof of the "only if " part of Theorem 2. Suppose we have a collection of nonnegative real numbers {λ b;k 1 ,...,kr;s : r ≥ 1, k 1 , . . . , k r ≥ 2, s ≥ 0, b = r j=1 k j + s} such that there exists a coalescent process Π ∞ = (Π ∞ (t)) t≥0 satisfying conditions B1 and B2 of Theorem 2 with collision rates λ b;k 1 ,...,kr;s . We wish to show that there is a finite measure Ξ on the infinite simplex ∆ such that all of the collision rates are given in terms of Ξ by (11).
Let T = inf{t : 1 and 2 are in the same block of Π ∞ (t)}. We may assume that T < ∞ a.s. because if the rate at which the blocks containing 1 and 2 are merging is zero, then all collision rates are zero and (11) holds with Ξ = 0. For n ≥ 2, let E n be the event that 1, 2, . . . , n are in distinct blocks of Π ∞ (T −). Since Π ∞ is an exchangeable process, the probability that no pair of integers in the set {1, 2, . . . , n} merges before 1 and 2 merge is at least 2/n(n − 1). Therefore, P (E n ) > 0. Let Γ n be a random partition of {1, . . . , n} whose distribution is the same as the conditional distribution of Π n (T ) given E n . We claim that there exists a random partition Θ ∞ of N such that Θ n = R n Θ ∞ has the same distribution as Γ n for all n. By Lemma 16, it suffices to show that R m Γ n has the same distribution as Γ m for all m < n.
Fix m < n. Let θ be a partition of {1, 2, . . . , m} in which 1 and 2 are in the same block. Let s be the number of singletons in θ, and let k 1 , . . . , k r be the sizes of the larger blocks of θ. It follows from condition B2 that if Π m (t) consists of m singletons, then the merger of the m singletons into the blocks of the partition θ is occurring at rate λ m;k 1 ,...,kr;s . Also, the total rate of all mergers involving the blocks {1} and {2} is λ 2;2;0 . Thus, P (Γ m = θ) = λ m;k 1 ,...,kr;s /λ 2;2;0 . If Π n (t) consists of n singletons, then condition B2 implies that the total rate of all mergers of {1, 2, . . . , n} whose restriction to {1, 2, . . . , m} is the merger of m singletons into the blocks of θ is also λ m;k 1 ,...,kr;s . Therefore, we have P (R m Γ n = θ) = λ m;k 1 ,...,kr;s /λ 2;2;0 . Thus, R m Γ n has the same distribution as Γ m .
Let Θ ∞ be the restriction of Θ ∞ to {3, 4, . . .}. Since Π ∞ is an exchangeable process, Θ ∞ is an exchangeable random partition of {3, 4, . . .}. It follows from Lemma 40 in appendix A that each block of Θ ∞ has a limiting relative frequency. Let P 1 ≥ P 2 ≥ . . . be the ranked sequence of limiting relative frequencies of the distinct blocks of Θ ∞ , where P n = 0 if Θ ∞ has fewer than n blocks with nonzero limiting relative frequencies. Note that the blocks of Θ ∞ also have limiting relative frequencies, and (P j ) ∞ j=1 is the ranked sequence of limiting relative frequencies of distinct blocks of Θ ∞ .
We now label the blocks of Θ ∞ having nonzero limiting relative frequencies by B 1 , B 2 , . . ., where B j has limiting relative frequency P j on {P j > 0} and blocks with the same limiting relative frequency are ordered at random, independently of Θ ∞ . On {P j = 0}, the block B j is undefined. Define a sequence of random variables (Z m ) ∞ m=1 such that Z m = i on {m ∈ B i } and Z m = 0 when the block of Θ ∞ containing m has a limiting relative frequency of zero. Let F denote the σ-field generated by ( We make the following claim regarding the distribution of the sequence (Z m ) ∞ m=1 : Before proving the claim, we show how we can use the claim to complete the proof of the "only if" part of Theorem 2.
Let θ be a partition of {1, 2, . . . , b} into s singletons and larger blocks D 1 , . . . , D r of sizes k 1 , . . . , k r respectively. Assume that 1 and 2 are in the block D 1 . Then, as we showed in the third paragraph of this proof, By Lemma 40, almost surely every block of the exchangeable random partition Θ ∞ having a limiting relative frequency of zero is a singleton. Therefore, if i, j ≥ 3, then almost surely i and j are in the same block of Θ ∞ if and only if Z i = Z j = 0. On {P 1 > 0}, the claim implies that Z 1 = Z 2 > 0 a.s., so for all positive integers i and j, we have that almost surely i and j are in the same block of Θ ∞ if and only if Z i = Z j = 0. Therefore, on {P 1 > 0}, the event that Θ b = θ is the same, up to a null set, as the event that there exist l ∈ {0, . . . , s} and distinct positive integers i 1 , . . . , i r+l such that the following hold: Note that l is the number of singletons in Θ b that are in blocks of Θ ∞ having nonzero limiting relative frequencies. For now, fix l ∈ {0, . . . , s} and fix distinct positive integers i 1 , . . . , i r+l . The claim implies that for j ∈ {2, . . . , r}, we have Since 1 and 2 are in D 1 , it follows from the claim that and On {P 1 = 0}, the claim implies that Z m = 0 almost surely for all m ≥ 3, so Θ ∞ almost surely consists of all singletons. The exchangeability of Π ∞ implies that the probability, conditional on the event that Θ ∞ consists of all singletons, that 1 and 2 are in the same block as q is the same for all q ≥ 3, and therefore must be zero. Therefore, on (31) almost surely.
Using (33) combined with the fact that the random variables Z 1 , Z 3 , Z 4 , . . . are conditionally independent given F, we obtain almost surely for all k ≥ 4. Likewise, almost surely we have, for all k ≥ 4, ¿From (35), (36) and (37), we obtain, for all k ≥ 4, the equation On the set {P i = P l }, we have Q i = Q l a.s. because, in the proof of the "only if" part of Theorem 2, the blocks B 1 , B 2 , . . . having the same limiting relative frequency were labeled in random order, independently of Θ ∞ . Therefore, It follows from (38) and Lemma 20 that

Proofs of Propositions 4 and 6
We can establish Propositions 4 and 6 by a straightforward application of the Poisson process construction in section 3. We prove Proposition 6 first, as it is used in the proof of Proposition 4.
Proof of Proposition 6. Write G = G 0 + αδ 0 , where G 0 has no atom at zero. We may assume without loss of generality that Π ∞ is derived from the Poisson point process (e(t)) t≥0 with characteristic measure L given by (14) with G 0 and α in place of Ξ 0 and a. The construction of Π ∞ implies that T = inf{t : e(t) ∈ A 1,2 }, where A 1,2 is defined by (16). LetΘ be the random partition of {3, 4, . . .} such that i and j are in the same block ofΘ if and only if e(T ) i = e(T ) j . It follows from the definition of (e(t)) t≥0 thatΘ is an exchangeable random partition. Also, we see from the construction of Π ∞ that Θ =Θ on {#Π ∞ (T −) = ∞} and Θ is the restriction of (17), Lemma 15 implies that Π ∞ (T −) is independent of e(T ) and thus is independent ofΘ. Therefore, to show that Θ satisfies the conclusion of Proposition 6, it suffices to show that the ranked sequence of limiting relative frequencies of the blocks ofΘ has distribution G. By (17), (14), Thus, N (e(T )) has distribution G. Since N (e(T )) is the ranked sequence of limiting relative frequencies of blocks ofΘ, the proposition follows.
Proof of Proposition 4. Suppose (Π ∞ (t)) t≥0 is both a standard Ξ-coalescent and a standard Ξ -coalescent. Then, the collision rates associated with the Ξ-coalescent and the Ξ -coalescent must be the same. By (12), we have Ξ(∆) = Ξ (∆). Therefore, it suffices to establish the proposition when Ξ and Ξ , are probability measures.

Proof of Proposition 7
In this subsection, we consider a population model of the type discussed in the introduction, and we adopt the notation used in the introduction. We begin with the following lemma.

Lemma 22
Proof of Proposition 7. We can write Ξ = Ξ 0 + aδ 0 , where Ξ 0 has no atom at zero. Let Π ∞ = (Π ∞ (t)) t≥0 be a standard Ξ-coalescent, and denote the collision rates of Π  (ν 1,N , . . . , ν N,N ). We claim that for all n, the processes (Ψ n,N ( t/c N )) t≥0 derived from the distributions µ N as described in the introduction converge in the Skorohod topology to (R n Π ∞ (t)) t≥0 as N → ∞. This claim will establish the proposition. By Lemma 21, it suffices to show that lim N →∞ c N = 0 and that for all r ≥ 1, k 1 , . . . , k r ≥ 2 and b = r j=1 k j , we have We may assume without loss of generality that Π ∞ is derived from a Poisson point process (e(t)) t≥0 with characteristic measure L, where L is defined from Ξ by (14). Define A N by (15). Recall from (18) where the sum is over distinct positive integers i 1 , . . . , i r . We claim that To see (47) N (ζ N ), . . . , h ir ,N (ζ N ) are nonzero, the probability that these are the first r elements, in order, of the multiset M after a random ordering is 1/(N ) r . Thus, Equation (47) follows by taking expectations of both side of (48).

Proofs of propositions related to
In this subsection, we prove Propositions 8, 10, and 11, all of which relate to the characterization of coalescents with simultaneous multiple collisions by a sequence of measures (F r ) ∞ r=1 . We will use the following lemma. When (F r ) ∞ r=1 is associated with a population model as described in Proposition 1, then this result is exactly Lemma 3.4 of [13]. However, the proof in [13] uses the properties of the population model. Since we wish to apply the result without knowing, a priori, whether the sequence (F r ) ∞ r=1 is associated with a population model, we will prove the result for all (F r ) ∞ r=1 satisfying conditions A1, A2, and A3 of Proposition 8.
By Theorem 2, Π ∞ must be a standard Ξ-coalescent for some finite measure Ξ. Therefore, we may assume that Π ∞ is derived from a Poisson point process (e(t)) t≥0 with intensity measure L, where L is defined in terms of Ξ by (14). We have T r = inf{t : e(t) ∈ A 2r }, where A 2r is defined by (15). Define Then let Θ r be a random partition whose distribution is the same as the conditional distribution ofΘ r given E r . Since Π ∞ is an exchangeable process (see Definition 19), the restriction of Θ r to {2r + 1, 2r + 2, . . .} is exchangeable. Let f j,r be the limiting relative frequency of the block of Θ r containing 2j − 1 and 2j, which exists by Lemma 40 of appendix A. Let Q r be the distribution of (f 1,r , . . . , f r,r ).
We first show that Q r satisfies condition (b) of Proposition 10. Suppose that we have P (#Π ∞ (T r −) = ∞) > 0. By Lemma 15, Π ∞ (T r −) and e(T r ) are independent since L(A 2r ) < ∞. Therefore, the conditional distribution ofΘ r given E r equals the conditional distribution of Θ r given E r ∩ {#Π ∞ (T r −) = ∞}. The conditional distribution of the limiting relative frequencies of the first r blocks ofΘ r given E r equals Q r by definition. Since Θ r =Θ r on {#Π ∞ (T r −) = ∞}, the conditional distribution of the limiting relative frequencies of the first r blocks ofΘ r given E r ∩ {#Π ∞ (T r −) = ∞} equals the conditional distribution of (f 1,r , . . . , f r,r ) given E r ∩ {#Π ∞ (T r −) = ∞}, where f j,r is the limiting relative frequency of the block of Θ r containing 2j − 1 and 2j. Hence, conditional on E r ∩ {#Π ∞ (T r −) = ∞}, the distribution of (f 1,r , . . . , f r,r ) equals Q r , which is condition (b).
Next, we show that Q r satisfies condition (c). Suppose P (#Π ∞ (T r −) = n) > 0. The conditional distribution of Θ r given E r ∩ {#Π ∞ (T r −) = n} equals the conditional distribution of the restriction ofΘ r to {1, . . . , n} given E r ∩ {#Π ∞ (T r −) = n}. By the independence of Π ∞ (T r −) and e(T r ), this equals the conditional distribution of the restriction ofΘ r to {1, . . . , n} given E r , which equals the distribution of Θ r restricted to {1, . . . , n}. Since Q r was defined to be the distribution of f 1,r , . . . , f r,r , it follows that Q r satisfies (c).
where D r is defined by (55). Then, E r is the event that 1, . . . , 2r are in distinct blocks of Π ∞ (U r −), and T r = U r on E r . Since D r ⊂ A 2r , where A 2r is defined by (15), we have L(D r ) < ∞. Therefore, Π ∞ (U r −) and e(U r ) are independent by Lemma 15. Using this independence for the fourth equality and part (d) of Lemma 41 in appendix B for the fifth equality, we have Define a sequence of random variables (Z i ) ∞ i=2r+1 such that Z i = j if i is in the same block of Θ r as 2j and Z i = 0 if j is not in the same block of Θ r as any of 1, . . . , 2r. Since the restriction of Θ r to {2r + 1, 2r + 2, . . .} is an exchangeable random partition, (Z i ) ∞ i=2r+1 is an exchangeable random sequence. By Lemma 40 of appendix A, (Z i ) ∞ i=2r+1 is has a limiting empirical measure µ given by µ({j}) = f j for j = 1, . . . , r and µ({0}) = 1 − r j=1 f j . Let F be the σ-field generated by µ. By Lemma 38, the random variables Z 2r+1 , Z 2r+2 , . . . are conditionally independent given F, and the conditional distribution of each Z i given F is µ. Therefore, Taking expectations of both sides of (59), we get which completes the proof.

Remark 24
Recall that to prove Proposition 8, we used Proposition 1 to establish the fact that if there is a coalescent process satisfying conditions B1 and B2 of Theorem 2, then the collision rates must be determined by (6) for a unique sequence of measures (F r ) ∞ r=1 satisfying conditions A1, A2, and A3 of Proposition 8. Thus, the proof of Proposition 8 made use of the population model introduced in [13]. In the proof of Proposition 10, we started with a coalescent process Π ∞ satisfying B1 and B2. Without using the assumption that the collision rates were derived from (F r ) ∞ r=1 by (6), we then defined a sequence of measures (F r ) ∞ r=1 satisfying A1, A2, and A3 such that the collision rates of Π ∞ were given by (6), with F m in place of F m on the right-hand side. The uniqueness of this sequence of measures, which is part of Proposition 1, follows from (9) and the fact that measures on ∆ r are uniquely determined by their moments. Consequently, we now have an alternative proof of Proposition 8 that does not refer to the population model.

Proof of Proposition 11.
Let Π ∞ be a standard Ξ-coalescent, and denote the collision rates by λ b;k 1 ,...,kr;s . Fix r ∈ N . Define E r as in Proposition 10. First, suppose P (E r ) = 0. Then λ 2r;2,...,2;0 = 0 and F r (∆ r ) = 0 by Proposition 10. We see from (11) that when S = ∆ r , the right-hand side of (13) equals λ 2r;2,...,2;0 , which is zero. Also, the right-hand side of (13) is nonnegative and takes on its maximum value when S = ∆ r , so it is zero for all S. Thus, (13) holds when P (E r ) = 0. Now, suppose P (E r ) > 0. Assume that Π ∞ is derived from a Poisson point process (e(t)) t≥0 whose characteristic measure L is defined by (14). Define T r ,Θ r , Θ r , and f 1,r , . . . , f r,r as in the proof of Proposition 10, and let Q r be the distribution of f 1,r , . . . , f r,r . By the proof of Proposition 10, we have F r (S) = λ 2r;2,...,2;0 Q r (S) for all measurable subsets S of ∆ r . For ξ ∈ Z ∞ and k ∈ Z, define N k (ξ) as in (40), provided this limit exists. Define D r as in (55) and U r as in (57). For each measurable subset S of ∆ r , define On E r , if f j,r denotes the limiting relative frequency of the block ofΘ r containing 2j − 1 and 2j, then (f 1,r , . . . , f r,r ) ∈ S if and only if e(T r ) ∈ A S,r . Since Θ r is defined to have the conditional distribution ofΘ r given E r , we have Q r (S) = P ((f 1,r , . . . , f r,r ) ∈ S) = P ((f 1,r , . . . , f r,r ) ∈ S|E r ) = P (e(T r ) ∈ A S,r |E r ).
Note that T r = U r on E r . By Lemma 15, Π ∞ (U r −) and e(U r ) are independent. Also, recall from the proof of Proposition 10 that E r is the event that 1, . . . , 2r are in distinct blocks of Π ∞ (U r −). Using these facts and part (d) of Lemma 41 of appendix B, we get Q r (S) = P (e(U r ) ∈ A S,r |E r ) = P (e(U r ) ∈ A S,r ) = L(A S,r ) L(D r ) .

Further properties of the Ξ-coalescent
In this section, we establish some properties of the Ξ-coalescent, most of which are straightforward extensions of properties that have been proved in [16] or [19] for the Λ-coalescent.

Regularity Properties of the Ξ-coalescent
In this subsection, we prove some regularity properties of the Ξ-coalescent. The analogous results for the Λ-coalescent are given as part of Theorem 1 in [16]. A consequence of the Feller property proved below is that the Ξ-coalescent satisfies the strong Markov property, a fact we will use later.
Recall from the introduction that we can identify P ∞ with the product space P 1 × P 2 × . . .. Since each P n is finite, the associated product topology is compact and metrizable and has a countable basis. Also, one can check that this topology is induced by the metric d(σ, τ ) = 2 −n , where n = inf{m ≥ 1 : is a semigroup. Let > 0, and fix m such that f m − f < /3. Let n be an integer such that f m ∈ A n . There exists a constant λ n < ∞ such that the total rate of all collisions for the restriction to {1, . . . , n} of any Ξ-coalescent is at most λ n . Thus, if f m > 0, we can choose δ > 0 such that P (Π α n (t) = Π α n (0)) < /(6 f m ) for all t < δ and α ∈ P n . Then, for all t < δ and σ ∈ P ∞ , we have Hence, Therefore, lim t↓0 P t f − f = 0. It follows (see Definition 2.1 in chapter III of [17]) that (P t , t ≥ 0) is a Feller semigroup on C(P ∞ ). Therefore (see Theorem 1.5 and Proposition 2.2 in chapter III of [17]), there exists a P ∞ -valued Feller process starting from the partition of N into singletons with (P t , t ≥ 0) as its transition semigroup. Since P t f (σ) = P n t f (n) (R n σ) for all f ∈ A n and σ ∈ P ∞ , this Feller process has the property that its restriction to {1, . . . , n} has the same law as Π n . It follows that (P t , t ≥ 0) is the transition semigroup for the Ξ-coalescent, so the collection of laws (P Ξ,π , π ∈ P ∞ ) defines a Feller process, as claimed.
We now work towards proving that the law P Ξ,π depends continuously on the measures Ξ and π. To formulate this result precisely, we need to define a topology on ∆. We will use the weakest topology making all of the coordinate functions x → x i continuous; this topology is discussed in section 3 of [9]. With this topology, ∆ is compact and metrizable, and a sequence (x (n) ) ∞ Proof. For all x ∈ ∆, we have Thus, g k 1 ,...,kr (x) ≤ 1 for all x ∈ ∆, which means g k 1 ,...,kr is bounded.
Next, we show that g k 1 ,...,kr is continuous. For n = 1, 2, . . . , ∞, define for all x ∈ ∆. Thus, the sequence of functions (f It remains only to show that g k 1 ,...,kr is continuous at zero. If r = 1 and k 1 = 2, then g k 1 ,...,kr (x) = 1 for all x ∈ ∆, so g k 1 ,...,kr is continuous at zero. Let (x (n) ) ∞ n=1 be a sequence in ∆ converging to zero. If r ≥ 2, then (61) implies Given > 0, choose N > 2/ , and choose M such that for n > M, we have x Thus, lim n→∞ g k 1 ,...,kr (x (n) ) = 0. If r = 1 and k 1 ≥ 3, then which approaches 0 as n → ∞. Hence g k 1 ,...,kr is continuous at zero. Proof. Let (Ξ m ) ∞ m=1 be a sequence of finite measures on ∆ converging weakly to Ξ, and let (π m ) ∞ m=1 be a sequence in P ∞ converging to π. Since ∆ is compact and metrizable, the space of finite measures on ∆ with the topology of weak convergence is metrizable, as shown in section 5 of chapter VIII of [4]. Also, P ∞ is metrizable. Therefore, it suffices to show that (P Ξm,πm ) ∞ m=1 converges weakly to P Ξ,π .
Then, ρ is a metric on P k ∞ which induces the product topology. Thus, all open subsets of P k ∞ are unions of sets of the form where n = sup{j ∈ N : 2 −j ≥ }. Thus, every open subset of P k ∞ can be written in the form where the n j are integers and σ j ∈ P k n j for all j ∈ N . If M < ∞, then there exists a subset S of The sets in (67) increase to the set in (66) as M → ∞. Therefore, to show (65), it suffices to show that for all N < ∞ and all S ⊂ P k N , we have Equation (68) follows from (63) and the fact that for sufficiently large m, we have the equality Π m N (0) = R N π m = R N π = Π N (0). Hence, (P Ξm,πm ) ∞ m=1 converges weakly to P Ξ,π , which completes the proof.

Some Formulas
For b ≥ 2, let λ b denote the total rate of all collisions when the Ξ-coalescent has b blocks. Let N (b; k 1 , . . . , k r ; s) be the number of possible (b; k 1 , . . . , k r ; s)-collisions, which was given in equation (3) of the introduction. We have where s = b − r j=1 k j and the inner sum is over multisets {k 1 , . . . , k r } because we do not have a separate term for each possible ordering of k 1 , . . . , k r . We can obtain another formula for λ b using the Poisson process construction of section 3. Define A b as in (15). Then, we have Also, z ij ∈ A b if and only if i ≤ b, j ≤ b, and i = j, so from (14), we get Note that for the Λ-coalescent, when Ξ is concentrated on {x ∈ ∆ : x i = 0 for all i ≥ 2}, only the l = 0 and l = 1 terms in the integrand of (70) are nonzero. Therefore, (70) reduces to which agrees with equation (6) of [16] because lim Equation (15) of [13] gives another formula for λ b in terms of the sequence of measures (F r ) ∞ r=1 associated with Ξ as in Proposition 11.
Let γ b denote the total rate at which the number of blocks is decreasing when the coalescent has b blocks. Each (b, k 1 , . . . , k r , s)-collision decreases the number of blocks by b − r − s, so for b ≥ 2 we have We record one simple lemma regarding the γ b , which is proved in [19] for the Λ-coalescent by a direct calculation.
Proof. Let Π ∞ be a standard Ξ-coalescent, and let Π n = R n Π ∞ for all n ∈ N . Fix m < n.
Then γ n is the initial rate at which the number of blocks of Π n is decreasing, and γ m is the initial rate at which the number of blocks of Π m is decreasing. For all t ∈ R, we have That is, whenever Π m undergoes a collision, Π n undergoes a collision at the same time, and the collision reduces the number of blocks of Π n by at least as much as it reduces the number of blocks of Π m . It follows that the rate at which the number of blocks of Π n is decreasing is always greater than or equal to the rate at which the number of blocks of Π m is decreasing. Hence γ n ≥ γ m , which proves the lemma.

Jump-hold coalescents
Pitman has shown in subsection 2.1 of [16] that a standard Λ-coalescent is a Markov process of jump-hold type with bounded transition rates and step function paths if and only if His proof is based on the observation that a standard Λ-coalescent is a Markov process of jumphold type with bounded transition rates and step function paths if and only if the sequence (λ b ) ∞ b=2 is bounded. He then uses an explicit formula for λ b to show that (λ b ) ∞ b=2 is bounded if and only if (72) holds. Here, we obtain the following result for the Ξ-coalescent using the Poisson process construction.
Proposition 29 Let Ξ be a finite measure on the infinite simplex ∆. The standard Ξ-coalescent is a jump-hold Markov process with bounded transition rates and step function paths if and only if Ξ has no atom at zero and Proof. Write Ξ = Ξ 0 + aδ 0 , where Ξ 0 has no atom at zero. As for the standard Λ-coalescent, the standard Ξ-coalescent is a jump-hold Markov process with bounded transition rates and step-function paths if and only if (λ b ) ∞ b=2 is bounded. As observed in subsection 5.2, we have , where L is defined by (14) and A b is defined by (15). The sets A b increase to the set A ∞ defined in (19). Therefore, lim b→∞ L( which is finite if and only if Ξ has no atom at zero and (73) holds.

Proper Frequencies
Here, the results for the standard Λ-coalescent discussed in [16] carry over to the standard Ξ-coalescent without much difficulty. If Π ∞ is a standard Ξ-coalescent, then since Π ∞ is an exchangeable coalescent (see Definition 19), Π ∞ (t) is an exchangeable random partition of N for all t > 0. Let B 1 (t), B 2 (t), . . . be the blocks of Π ∞ (t), ordered by their smallest elements, where B j (t) = ∅ if Π ∞ (t) has fewer than j blocks. By Lemma 40 in appendix A, almost surely exists for all j ∈ N , and we denote this limit by f j (t). We say that Π ∞ (t) has proper frequencies if ∞ j=1 f j (t) = 1 a.s. The following result is the analog of Lemma 25 of [16].
Proposition 30 Let Ξ = Ξ 0 + aδ 0 be a finite measure on the infinite simplex ∆, where Ξ 0 has no atom at zero. Let Π ∞ be a standard Ξ-coalescent, and fix t > 0. Then Π ∞ (t) has proper frequencies if and only if a > 0 or Proof. As noted in the proof of Lemma 25 of [16], Lemma 40 of appendix A implies that Π ∞ (t) has proper frequencies if and only if the singleton set {1} is almost surely not a block of Π ∞ (t). Assume that Π ∞ is derived from the Poisson point process (e(t)) t≥0 . Let which is infinite if and only if a > 0 or (75) holds.

Coming down from infinity
Let Π ∞ be a standard Ξ-coalescent. By definition, #Π ∞ (0) = ∞. We say the Ξ-coalescent comes down from infinity if #Π ∞ (t) < ∞ a.s. for all t > 0. We say the Ξ-coalescent stays infinite if #Π ∞ (t) = ∞ a.s. for all t > 0. The problem of whether the Λ-coalescent comes down from infinity has been studied thoroughly. Results of Bolthausen and Sznitman in [3] imply that the Λ-coalescent stays infinite when Λ is the uniform distribution on [0, 1], and Sagitov shows in [18] that if Λ(dx) = (1−α)x −α dx for 0 < α < 1, then the Λ-coalescent comes down from infinity. Pitman shows in [16] that the Λ-coalescent comes down from infinity if Λ has an atom at zero and stays infinite if Pitman also shows in Proposition 23 of [16] that if Λ has no atom at 1, then the Λ-coalescent either comes down from infinity or stays infinite. It is then shown in [19] that the Λ-coalescent comes down from infinity if and only if ∞ b=2 γ −1 b < ∞. This result does not fully generalize to the Ξ-coalescent, but we provide some partial results in this subsection. The problem of finding a necessary and sufficient condition for a Ξ-coalescent to come down from infinity remains open.
Since Ξ 1 has no atom at zero, it follows from (14) that We consider the following three cases:  We have now reduced the problem to the case when Ξ 1 = 0. We begin our analysis of this case with the following straightforward generalization of Proposition 23 of [16].
Suppose p = 0. We wish to show that this implies P (T = ∞) = 1. It suffices to show that P (0 < T < ∞) = 0. Note that T is a stopping time with respect to the completed natural filtration of Π ∞ (see Theorem 2.17 in chapter III of [17]). Therefore, if let T m be the time at which the smallest integer that is not in any of the blocks containing one of the integers n 1 , . . . , n b at time T m−1 merges with a block containing one of n 1 , . . . , n b . Note that T 0 < T 1 < . . . < T a.s. However, for any fixed integers s 1 , . . . , s b , the rate at which any particular block is colliding with one of the blocks containing s 1 , . . . , s b is at most λ b+1 < ∞. Since only countably many sets {s 1 , . . . , s b } are possible, it follows that T m ↑ ∞ a.s. as m → ∞. Thus, which completes the proof.
Then the Ξ-coalescent comes down from infinity.

Proof. By Lemma 31, it suffices to show that E[T
b=2 is increasing by Lemma 28, we can show that E[T ∞ ] < ∞ by the same argument used to prove Lemma 6 of [19].
For the Ξ-coalescent, we are only able to establish a converse to Proposition 32 when an additional condition is satisfied.
Proof. Let Π ∞ be a standard Ξ-coalescent derived from a Poisson point process (e(t)) t≥0 with characteristic measure L defined by (14). Let Π n = R n Π ∞ . Let T n = inf{t : #Π n (t) = 1}. As observed for the Λ-coalescent in equation (31) of [16], we have By the argument used to prove Proposition 5 of [19], it suffices to show that lim n→∞ E[T n ] = ∞.
Fix > 0 such that (76) holds. Let Ξ 1 be the restriction of Ξ to ∆ , and let Ξ 2 be the restriction of Ξ to ∆ \ ∆ . By (76) and Proposition 29, the Ξ 2 -coalescent is a jump-hold Markov process with bounded transition rates. Therefore, by the argument used to prove Lemma 8 of [19], it suffices to show that the Ξ 1 -coalescent stays infinite. We will therefore assume for the remainder of the proof that Ξ 2 = 0.
We now follow the idea of the proof of Lemma 7 of [19].  − (b − d)). This is greater than or equal to the probability that the sum of M l independent Bernoulli random variables with success probability is at least M l−1 + 1. Since > 1/M , this probability approaches 1 as l → ∞ and is therefore bounded below by some constant C > 0 which does not depend on b, d, or l. Therefore, for all x ∈ ∆ . Also, if z ij ∈ R b,l , then since at least b − 1 of the first b coordinates of z ij are distinct, we must have z ij ∈ S b,l . Thus, since C ≤ 1, we have for all b and l such that b > M l . Now fix n ∈ N . For l such that M l ≤ n, let D l be the event that M l−1 + 1 ≤ #Π n (t) ≤ M l for some t. Assume for now that M l < n. For all b ≥ 1, let U b = inf{t : #Π n (t) ≤ b}. When Π n (t) has b blocks and b > M l , the total rate of all collisions that take Π n down to M l or fewer blocks is L(R b,l ). The total rate of all collisions that take Π n down to between M l−1 + 1 and M l blocks is L(S b,l ). Therefore, for all b > M l , the strong Markov property and part (d) of Lemma 41 of appendix B imply that if P (#Π n (U b ) = b) > 0, then and Note that the event {#Π n (U M l −) = b} is the same as Also, the events D l ∩{#Π n (U M l −) = b} and {#Π n (U b ) = b}∩{M l−1 +1 ≤ #Π n (U b−1 ) ≤ M l } are the same. Therefore, by (77) and (78), As in the proof of Lemma 6 of [19], we recursively define times R 0 , R 1 , . . . , R n−1 by: Note that R n−1 = T n . For i = 0, 1, . . . , n − 1, let N i = #Π n (R i ). For i = 1, 2, . . . , n − 1, define For j = 2, 3, . . . , n, let L n (j) = min{s ≥ j : #Π n (t) = s for some t}.
Therefore, using equation (12) of [19], which is valid for the Ξ-coalescent as well as for the Λ-coalescent, to get the first equality, we have Since (γ b ) ∞ b=2 is increasing by Lemma 28 and L n (j) ≤ M l+1 on D l+1 for all j ≤ M l , we have Therefore, using the monotonicity of (T n ) ∞ n=1 for the first equality, we have which completes the proof.
Suppose there exists K < ∞ such that Ξ is concentrated on {x : x i = 0 for all i > K}. Then almost surely the Ξ-coalescent never undergoes more than K multiple collisions at one time.
To show that the conclusion of Proposition 33 does not necessarily hold when (76) fails, we give the following example of a Ξ-coalescent for which Ξ(∆ f ) = 0 and ∞ b=2 γ −1 b = ∞ but the Ξ-coalescent comes down from infinity.
Example 34 For all n ∈ N , let y n ∈ ∆ be the point whose first 2 n − 1 coordinates equal 2 −n and whose remaining coordinates equal zero. Let Ξ be the probability measure on ∆ with an atom of size 2 −n at y n for all n ∈ N . Define A b and A k,l as in (15) and (16) respectively. For all Since Ξ has no atom at zero, (14) implies that for all b ≥ 2 we have We have γ b ≤ bλ b for all b ≥ 2 by (69) and (71). Therefore, We now show that the Ξ-coalescent comes down from infinity. For all b ≥ 2, define the set has distribution P yn , then P (ξ i < 0) = 2 −n for all i. Therefore, under P yn , the expected number of ξ 1 , . . . , ξ b that are negative is Therefore, by Markov's inequality, we have Under P yn the random variables ξ i can take on only 2 n − 1 different positive values. Therefore, . Thus, if a Ξ-coalescent has b ≥ M blocks, then the total rate of all collisions that take the coalescent down to b 3/4 or fewer blocks is where for the last inequality we used the fact that b ≥ 64. For k ∈ N , let Then, if a Ξ-coalescent has b ∈ S k blocks, the expected time before the number of blocks is no longer in S k is at most For a standard Ξ-coalescent Π ∞ , let T n = inf{t : #R n Π ∞ (t) = 1}. Then, for all n, Thus, (E[T n ]) ∞ n=1 is bounded, which means E[T ∞ ] < ∞. Hence, by Lemma 31, the Ξ-coalescent comes down from infinity.

The discrete-time Ξ-coalescent
We have shown that the continuous-time processes (Ψ n,∞ (t)) t≥0 obtained in [13] as limits of ancestral process all have the same distribution as some Ξ-coalescent restricted to {1, . . . , n}. However, Möhle and Sagitov also obtain some discrete-time Markov chains as limits. They show, as part of Theorem 2.1 of [13], that if the conditions of Proposition 1 are satisfied except that lim N →∞ c N = c > 0, then the processes (Ψ n,N ( t/c N )) t≥0 converge as N → ∞ to a Markov chain (Ψ n,∞ (t)) t≥0 that jumps only at times t = cm for m ∈ N . Suppose η and θ are partitions (81) for all r ≥ 1, k 1 , . . . , k r ≥ 2, s ≥ 0, and b = r j=1 k j + s.
Proof. Suppose the nonnegative real numbers p b;k 1 ,...,kr;s are defined such that there exists a Markov chain (Y m ) ∞ m=0 satisfying C1 and C2. Let J 0 = 0, and let (J i ) ∞ i=1 be a sequence of independent random variables, each having an exponential distribution with rate 1. For all t ≥ 0, define K t = max{i : J 0 + J 1 + . . . + J i ≤ t}. Define a P ∞ -valued process (Π ∞ (t)) t≥0 by Π ∞ (t) = Y Kt . Then Π ∞ (0) is the partition of N into singletons. Also, (R n Π ∞ (t)) t≥0 is a jump-hold Markov process such that when R n Π ∞ (t) has b blocks, each (b; k 1 , . . . , k r )-collision is occurring at rate p b;k 1 ,...,kr;s . By Theorem 2, there exists a finite measure Ξ on ∆ such that the rates p b;k 1 ,...,kr;s are given by (11). We also have η =θ for all n ∈ N and θ ∈ P ∞ . If θ has b blocks, then the left-hand side of (82) equals blocks. Thus, we have λ b ≤ 1 for all b ≥ 2. Since (λ b ) ∞ b=2 is increasing, we have lim b→∞ λ b ≤ 1. Let L be defined from Ξ by (14). Then, λ b = L(A b ), where A b is defined by (15). Since the sets A b increase to the set A ∞ defined in (19), we have lim b→∞ λ b = lim b→∞ L(A b ) = L(A ∞ ). The expression for L(A ∞ ) in (74) implies that lim b→∞ λ b ≤ 1 if and only if Ξ has no atom at zero and (80) holds. Since Ξ has no atom at zero when (80) holds, the expression for p b;k 1 ,...,kr;s can be simplified to the right-hand side of (81).
Conversely, suppose there is a finite measure Ξ on the infinite simplex ∆ with no atom at zero such that (80) and (81) hold. Let Π ∞ be a standard Ξ-coalescent, derived from a Poisson point process (e(t)) t≥0 with characteristic measure L. By (74)  Let (I i ) ∞ i=1 be a sequence of independent Bernoulli random variables that take on the value 1 with probability L(A ∞ ). Assume the sequence is independent of (Z m ) ∞ m=0 . Let V m = I 1 + . . . + I m . Define a Markov chain (Y m ) ∞ m=0 by Y m = Z Vm for all m. Then P (R n Y m+1 = η|R n Y m = θ) = L(A ∞ )P (R n Z Vm+1 = η|R n Z Vm = θ) = p b;k 1 ,...,kr;s , which completes the proof.

Definition 36
We call a discrete-time Markov chain satisfying C2 with transition probabilities given by (81) for a particular finite measure Ξ a discrete-time Ξ-coalescent. A discrete-time Ξ-coalescent satisfying C1 is called a standard discrete-time Ξ-coalescent.
Propositions 29 and 35 imply that if Ξ is a finite measure on ∆ for which a standard discrete-time Ξ-coalescent exists, then a standard (continuous-time) Ξ-coalescent Π ∞ is a jump-hold Markov process. Therefore, we can define the jump chain (X m ) ∞ m=0 associated with Π ∞ by defining X m to be the value of Π ∞ at the time of its mth jump, unless Π ∞ consists of just a single block after fewer than m jumps, in which case X m is defined to be {N }. Then X m+1 = X m a.s. on {#X m > 2}. This chain is different from the standard discrete-time Ξ-coalescent Finally, we prove the analog of Proposition 7 for discrete-time Ξ-coalescents. Note that although the time-scaling conventions in [13] are such that we can only obtain a continuous-time Ξcoalescent as a limit of ancestral processes when Ξ is a probability measure, the proposition below shows that any nontrivial discrete-time Ξ-coalescent arises as a limit of ancestral processes in a population model of the type studied in [13].
Proposition 37 Let Ξ be a finite nonzero measure on ∆ with no atom at zero such that (80) holds. Then there exists a sequence (µ N ) ∞ N =1 such that each µ N is a probability distribution on {0, 1, 2, . . . , } N that is exchangeable with respect to the N coordinates with the property that if for all N , µ N is the distribution of family sizes in the population model described in the introduction, then the processes (Ψ n,N ( t/c N )) t≥0 converge as N → ∞ in the Skorohod topology to a process (Ψ n,∞ (t)) t≥0 satisfying: (a) For all n, (Ψ n,∞ (t)) t≥0 jumps only at times cm for m ∈ N , where c = lim N →∞ c N .
Denote the transition probabilities for the discrete-time Ξ-coalescent by p b;k 1 ,...,kr;s . These are the same as the transition rates for the the continuous-time Ξ-coalescent. Therefore, we can follow the argument in the proof of Proposition 7 to see that (51) still holds in the setting of Proposition 37 when λ b;k 1 ,...,kr;0 is replaced by p b;k 1 ,...,kr;0 on the right-hand side. Equation (52) remains true if we make this change and also replace the factor of 1/N on the right-hand side, which comes from the definition of V N , by L(A ∞ ). That is, we have Therefore, Lemma 39 Let (Z i ) ∞ i=1 be an exchangeable sequence with limiting empirical distribution µ. Suppose V is a random variable such that (V, Z 1 , Z 2 , . . .) and (V, Z σ(1) , Z σ (2) , . . .) have the same distribution for all finite permutations σ of N . Then (Z i ) ∞ i=1 and V are conditionally independent given the σ-field generated by µ.
Given a partition π of a finite or countable set S and a finite permutation σ of S, letσπ be the partition of S such that σ(i) and σ(j) are in the same block ofσπ if and only if i and j are in the same block of π. Following [10], we say that a random partition Π of S is exchangeable if σΠ has the same distribution as Π for all finite permutations σ of S.
Given a point x = (x 1 , x 2 , . . .) in the infinite simplex ∆, let P x be the distribution of a random partition Π obtained by first defining an i.i.d. sequence of random variables (Z i ) ∞ i=1 such that P (Z i = j) = x j for j ≥ 1 and P (Z i = 0) = 1 − ∞ j=1 x j , and then declaring i and j to be in the same block of Π if and only if Z i = Z j ≥ 1. In [9] and [10], Kingman establishes that all exchangeable random partitions are mixtures of random partitions that can be constructed in this way. A simpler proof of Kingman's result, using de Finetti's Theorem, is given in section of 11 of [1]. We state below a version of this result, which is essentially Theorem 2 of [10]. exists almost surely and is called the limiting relative frequency of the block B i . Let X 1 ≥ X 2 ≥ . . . be the ranked sequence of limiting relative frequencies of distinct blocks of Π, where X n is defined to be zero if Π has fewer than n blocks with nonzero limiting relative frequencies. Then X = (X 1 , X 2 , . . .) is almost surely in the infinite simplex ∆. Moreover, the conditional distribution of Π given X is P X , and therefore for all Borel subsets B of P ∞ , where G is the distribution of X.
In [10], Kingman defines X r = lim n→∞ n −1 λ r (n), where λ r (n) is the size of the rth-largest block of R n Π. The observation that this definition is equivalent to the one given in Lemma 40 above is made in the introduction of [15].

Poisson Point Processes.
We review here some basic facts about Poisson point processes, most of which are stated in section 0.5 of [2]. Let L be a σ-finite measure on a Polish space E. We can construct a Poisson random measure X on [0, ∞) × E with intensity measure λ × L, where λ denotes Lebesgue measure on [0, ∞). Almost surely X({t} × E) equals 0 or 1 for all t ≥ 0. Therefore, we can define a process (e(t)) t≥0 taking values in E ∪ {δ}, where δ is an isolated point that we add to the state space, by defining e(t) = δ if X({t} × E) = 0 and e(t) = x if the restriction of X to {t} × E is a unit mass at x. The process (e(t)) t≥0 is called a Poisson point process with characteristic measure L. We record below some useful facts about Poisson point processes.   (e) The process (e (t)) t≥0 defined such that e (t) ∈ δ if e(t) ∈ A and e (t) = e(t) otherwise is a Poisson point process whose characteristic measure is the restriction of L to A c . Also, (e (t)) t≥0 is independent of (T A , e(T A )). Thus, e(t) ∈ A for some t < a.s., and so T A = 0 a.s. Next, suppose 0 < L(A) < ∞. Conditions (a), (d), and (e) are part of Proposition 2 in section 0.5 of [2]. To prove (b), fix t < ∞. Since (λ × L)([0, t) × A) < ∞, we have X([0, t) × A) < ∞ a.s., so e(s) ∈ A for only finitely many s < t. Finally, to prove (c), note that for all > 0, we almost surely do not have e(s) ∈ A for infinitely many values of s in (T A , T A + ). The definition of T A thus implies that e(T A ) ∈ A a.s.