[32, Proposition 30] No matter what Λ, the process f1 is an increasing pure jump process with càdlàg paths, f1(0)=0 and limt→∞f1(t)=1 . If μ−1=∞ then almost surely f1(t)>0 0$]]> for all t>0 0$]]> and limt↘0f1(t)=0 . If μ−1<∞ then f1 starts by holding at zero until an exponential time with rate μ−1 , when it enters (0,1] by a jump, and proceeds thereafter by a succession of holds and jumps, with holding rates bounded above by μ−1 .

In any Ξ-coalescent Π with dust and Ξ(Δf)=0 , the asymptotic frequency process f1:=(f1(t))t≥0 of the block of 1, defined by Eq. (1), is an increasing pure jump process with càdlàg paths, f1(0)=0 and limt→∞f1(t)=1 , but f1(t)<1 for t>0 0$]]> almost surely. The waiting times between almost surely infinitely many jumps are distributed as independent Exp(μ−1) random variables. Its jump chain (f1[k])k∈N can be expressed via stick-breaking (5) f1[k]= ∑i=1kXi ∏j=1i−1(1−Xj), where (Xj)j∈N are pairwise uncorrelated, Xj>0 0$]]> almost surely and E(Xj)=γ for all j∈N . In particular, E(f1[k])=1−(1−γ)k . In general, (Xj)j∈N are neither independent nor identically distributed.

From Theorem 1, the dependence between f1 and its jump times is readily seen as follows. Recall [32, Eq. (51)] that E(f1(t))=1−e−t for any Λ-coalescent with Λ([0,1])=1 . If we would have independence, integrating E(f1(t)) over the waiting time distribution Exp(μ−1) for the first jump of f1 would yield E(f1[1])=(1+μ−1)−1 , in contradiction to E(f1[1])=1/μ−1 by Theorem 1.

Let Λ=δp , p∈[12,1) and q:=1−p . f1 takes values in the set (6) Mp:={∑i∈Nbipqi−1:bi∈{0,1},1≤∑i∈Nbi<∞}. For x=∑i∈Nbipqi−1∈Mp , we have (7) P(f1[1]=x)=pqj−1∏i∈J∖{j}P(Y+i∈J)∏i∈[j−1]∖JP(Y+i∉J)>0, 0,\]]]> where Y=dGeo(p) , J:={i∈N|bi=1} and j:=maxJ . The process f1 is not Markovian whereas its jump chain (f1[k])k∈N is Markovian.

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The law of f1[1] is a discrete measure on [0,1] for Dirac coalescents. Surprisingly different properties arise for different values of p. For instance, M2/3={∑i∈Nbi3−i:bi∈{0,2},1≤∑i∈Nbi<∞} is a subset of the ternary Cantor set which is nowhere dense in [0,1] , whereas M1/2 , the set of all x∈[0,1] with finite 2-adic expansion, is dense in [0,1] .

Let Π be a Ξ-coalescent with dust and Π(n) its restriction on [n] . Let Mn be the minimal clade size, i.e. the size of the block of 1 at its first merger in Π(n) . Then, Mn/n→f1[1] almost surely, f1[1]>0 0$]]> almost surely and E(f1[1])=γ .

Let Λ fulfil (3). Then, (9) f1[1]= ∑i=1CBi(C)Pi∏j∈[i−1](1−Pj)=∑i∈NPi∏j∈[i−1](1−Pj)∑k≥iBi(k)1{C=k}, where (Pi)i∈N are i.i.d. with Pi=dμ−2−1x−2Λ(dx) . We have P(C=k|(Pi)i∈N)=Pk∏j∈[k−1](1−Pj),CisGeo(α)-distributed. Given (Pi)i∈N , C and (Bi(k))k∈N,i∈[k] are independent and (Bi(k))k∈N,i∈[k] is defined as (10) P((B1(j),…,Bj(j))=b|(Pi)i∈N)=∏i∈J∖{j}P(I(i)∈J|(Pi)i∈N)∏i∈[j−1]∖JP(I(i)∉J|(Pi)i∈N)almost surely, where b=(b1,…,bj)∈{0,1}j−1×{1} , J:={i∈[j]|bi=1} and, for each i∈N , I(i):=min{j≥i+1:Bi(j)=1} . We have (i)

P(I(i)=i+k|(Pi)i∈N)=Pi+k∏l=i+1i+k−1(1−Pl) almost surely for k∈N .

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The distribution of C is known from [16, Proposition 3.1].

For any finite measure Ξ on Δ fulfilling (2), P splits into two independent Poisson processes P1:={(T,(Ki)i∈N)∈P:K1>0}andP2:={(T,(Ki)i∈N)∈P:K1=0}. 0\big\}\hspace{1em}\textit{and}\hspace{1em}\mathcal{P}_{2}:=\big\{\big(T,(K_{i})_{i\in \mathbb{N}}\big)\in \mathcal{P}:K_{1}=0\big\}.\]]]> P1 has almost surely finitely many points on any set [0,t]×N0∞ , thus we can order P1=((Tj,(Ki(j))i∈N))j∈N, where Tj<Tj+1 almost surely for j∈N .

(Tj)j∈N is a homogeneous Poisson process on [0,∞) with intensity μ−1 .

((Ki(j))i∈N)j∈N is an i.i.d. sequence in j and (1{K1(1)=Ki(1)})i≥2 are exchangeable(Q) indicators with Q:=1μ−1∫Δ∑i∈Nxiδxiν0(dx), which is a probability measure on [0,1] . For X=dQ , we have X>0 0$]]> almost surely and E(X)=γ .

P1 and P2 are obtained by restricting P on the disjoint subsets A1:=[0,∞)×N×N0∞ and A2:=[0,∞)×{0}×N0∞ of A. Thus, P1 and P2 are independent Poisson processes (restriction theorem [25, p.17]) with intensity measures ν1=ν(·∩A1) and ν2=ν(·∩A2) . For any Borel set B⊆[0,∞) and λ being the Lebesgue measure, (12) ν1(B×N0∞)=λ(B)∫ΔP(x)(N)︸=|x|∏i≥2P(x)(N0)︸=1ν0(dx)=λ(B)μ−1. Thus, on any bounded set B, P1 has almost surely finitely many points, which can be ordered as described. Projecting P1 on the first coordinate t of A yields a Poisson process with intensity measure μ−1dt (mapping theorem [25, p.18]).

Now, we project the points of P1 on the coordinate of (Ki(j))i∈N . Recall the construction of a Poisson process as a collection of i.i.d. variables with distribution (μ(C))−1μ on sets of finite mass C of the intensity measure μ, e.g. [25, p.23]. It shows that we can treat the collection of (Tj,(Ki(j))i∈N) with, for instance, Ti∈[k,k+1) for k∈N as a random number of i.i.d. random variables with distribution (1·μ−1)−1ν1 . Since ν1 has a product structure on A1 , we have that ((Ki(j))i∈N)j∈N are i.i.d. in j and have distribution, for m∈N , (13) P((Ki(1)=li)i∈[m])=1μ−1∫Δ∏i∈[m]P(x)(li)ν0(dx)=1μ−1∫Δ∏i∈[m]xliν0(dx) for l1∈N and l2,…,lm∈N0 . Consider (1{K1(1)=Ki(1)})i≥2 . To show that they are exchangeable(Q) indicators, [32, Eq. (27)] has to be fulfilled, i.e. we need to show P({i∈[m]:Ki(1)=K1(1)}=M)=E(X|M|−1(1−X)m−|M|) for X=dQ and any M⊆[m] with 1∈M . Using Eq. (13) we compute P({i∈[m]:Ki(1)=K1(1)}=M)=∑j∈NP({i∈[m]:Ki(1)=j}=M,K1(1)=j)=1μ−1∫Δ∑j∈Nxj|M|(1−xj)m−|M|ν0(dx)=E(X|M|−1(1−X)m−|M|). Clearly, P(X>0)=1 0)=1$]]> since Ξ({0})=0 and E(X)=μ−1−1∫Δ(x,x)ν0(dx)=γ .  □

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The properties of the exchangeable(Q) indicators remind of [32, Lemma 21, Theorem 4] and [33, Proposition 6]. Restricting P to points with K1=K2>0 0$]]> we can reproduce their results analogously to the proof of Lemma 1.

Let Λ be a finite measure on [0,1] fulfilling (3). For the Poisson process P=(Tj,(Ki(j))i∈N)j∈N , ((Ki(j))i∈N)j∈N is an i.i.d. sequence (in j) of sequences of exchangeable(Pj) indicators, where (Pj)j∈N are i.i.d. with P1=dμ−2−1x−2Λ(dx) . In particular, E(Pi)=α .

These properties essentially follow from the de Finetti representation of an infinite series of exchangeable variables as conditionally i.i.d. variables. The lemma is a collection of well-known properties as e.g. described in [3, Sections 2 and 3], arguments of which we use in the following.

An infinite exchangeable sequence is conditionally i.i.d. given an almost surely unique random measure α. This measure is the weak limit of the empirical measures, in our case, n−1∑i=1nδKi , which has limit X′δ1+(1−X′)δ0 for some random variable X′ with values in [0,1] . Given α, the indicators are α-distributed. However, since X gives the success probability of each Bernoulli coin, we have X=X′ almost surely, so X is almost surely unique. The rest of a) is just the strong law of large numbers e.g. from [3, 2.24] ( E(K1)≤1 ), the limit is X′ . Part b) follows from measure theory since the limit is measurable in the σ-field spanned by the summed variables. For c), we again check Pitman’s condition [32, Eq. 27]. Let M⊆[m] . We have that X, Y are independent from b). With P(Ki=Li=1|X,Y)=XY almost surely, P({i∈[m]:KiLi=1}=M)=E(P({i∈[m]:KiLi=1}=M|X,Y))=E((XY)|M|(1−XY)m−|M|), since given X, Y, both (Ki)i∈N and (Li)i∈N are independent. This shows c).  □

Let Δ∗:={x∈Δ:|x|=1} . All Ξ-coalescents considered are constructed via the Poisson construction with Poisson point process P .

First, assume Ξ(Δ∗)=0 . We recall the (well-known) property that for a Ξ-coalescent with dust Ξ(Δ∗)=0 is equivalent to P(St>0∀t)=1 0\hspace{2.5pt}\forall t)=1$]]>, where St is the asymptotic frequency of the dust component. We use the remark on [12, p.1091]: For Ξ-coalescents with dust, (−logSt)t≥0 is a subordinator. The subordinator jumps to ∞ (corresponds to St=0 ) if and only if for its Laplace exponent Φ, we have limη↘0Φ(η)>0 0$]]>. For a Ξ-coalescent with dust we have limη↘0Φ(η)=∫Δ∗ν0(dx) . Hence, Ξ(Δ∗)=0 almost surely guarantees infinitely many singleton blocks for all t≥0 , so the corresponding Ξ coalescent stays infinite.

Now assume Ξ(Δ∗)>0 0$]]>. The subordinator (−logSt)t≥0 jumps from finite values ( St>0 0$]]>) to ∞ ( St=0 ) after an exponential time with rate ν0(Δ∗) . This shows that the Ξ-coalescent does not come down from infinity. Assume further that Ξ(Δf)=0 . Then, [33, Lemma 31] shows that the Ξ-coalescent either comes down from infinity or stays infinite, so it stays infinite.

Finally, assume Ξ(Δf)>0 0$]]>. Split P into independent Poisson processes P1′:={(T,(Ki)i∈N)∈P:κ∈Δf} and P2′:={(T,(Ki)i∈N)∈P:κ∉Δf} , where κ:=(limn→∞n−1∑i∈[n]1{Ki=j})j∈N (again restriction theorem [25, p.17], Lemma 3 shows κ exists almost surely). Their intensity measures are defined as in Eq. (11), but using ν1′(·):=ν0(·∩Δf) and ν2′:=ν0−ν1′ instead of ν0 . Since ν1′(Δf)≤μ−1<∞ , for any t>0 0$]]> there are almost surely finitely many P∈P1′ with T<t . Consider such P=(T,(Ki)i∈N) with T smallest. Observe that until T, we can construct the Ξ-coalescent using only the points of P2′ , which is the construction of a Ξ′ -coalescent with Ξ′(dx):=(x,x)ν2′(dx) . Since ∫Δ|x|ν2′(dx)<μ−1<∞ and Ξ′(Δf)=0 , the proof steps above show that the Ξ-coalescent has infinitely many blocks until T. Now consider the merger at time T. The form of ν1′ ensures that (Ki)i∈N can only take finitely many values, and Lemma 3a) ensures that infinitely many Ki ’s show each value. Thus, all blocks present before time T are merged at T into a finite number of blocks (given by which Ki ’s show the same number). This shows that if Ξ(Δf)>0 0$]]>, the Ξ-coalescent stays neither infinite nor comes down from infinity. Additionally, this shows that either the block of 1 already merged at least once before T or it merges at T, thus there are infinitely many blocks before the first merger of 1.  □

As in Lemma 1, split the Poisson point process P used to construct the Ξ-coalescent in P1 and P2 . We also use the notation from Lemma 1 and its proof. The block of 1 in the Ξ-n-coalescent for any n∈N can only merge at times t for which there exists a Poisson point (T,(Ki)i∈N)∈P1 . Lemma 1 states that the set of times T forms a homogeneous Poisson process with rate μ−1 . This shows that potential jump times are separated by countably many independent Exp(μ−1) random variables. Kingman’s correspondence yields that f1 exists almost surely at each potential jump time. To see this, observe that even though the partition of N induced by the Poisson construction is not exchangeable, the partition on N∖{1} is, and the asymptotic frequencies of the former and the latter coincide. Since f1 is by definition constant between these jump points, f1 has càdlàg paths almost surely. Since any blocks change by mergers, f1 is increasing.

The value of f1 at 0 follows by definition. Since Π stays infinite (see Lemma 4), at each P∈P1 infinitely many blocks can potentially merge. Lemma 1 shows that the indicators of whether blocks present immediately before P merge with the block of 1 are exchangeable(X) indicators with X>0 0$]]> almost surely. Then, Lemma 3 ensures that a positive fraction of them almost surely does, causing f1 to jump (since a positive fraction of merging blocks has positive frequency). Thus, every Poisson point leads to a merger almost surely, which shows that f1 jumps at all potential jump times described above. Since, for all t, either St>0 0$]]> or non-dust blocks not including 1 exist (having asymptotical frequency >0 0$]]>), f1(t)<1 for all t≥0 .