On the external branches of coalescents with multiple collisions

A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions ($\Lambda$-coalescents) is provided. This recursion is used to derive asymptotic results as the sample size $n$ tends to infinity for the joint moments of the external branch lengths and for the moments of the total external branch length of the Bolthausen-Sznitman coalescent. These asymptotic results are based on a differential equation approach, which is as well useful to obtain exact solutions for the joint moments of the external branch lengths for the Bolthausen-Sznitman coalescent. The results for example show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen--Sznitman coalescent, whereas they are negatively correlated for the Kingman coalescent provided that $n\ge 4$.

Remarks.The recursion (4) works as follows.Let us call d := k 1 + • • • + k j the order (or degree) of the moment µ n (k 1 , . . ., k j ).Provided that all the moments of order d − 1 are already known, ( 4) is a recursion on n for the joint moments of order d, which can be solved iteratively.So one starts with d = 1 (and hence j = 1), in which case (4) reduces to ), this recursion determines the moments of order 1 completely.Now choose d = 2 in (4) leading to a recursion for the second order moments.Iteratively, one can move to higher orders.Note that for j = 2 and k 1 = k 2 = 1 the recursion (4) reduces to Note that Theorem 1.1 holds for arbitrary Λ-coalescents.For particular Λ-coalescents the recursion (4) can be used to derive exact solutions and asymptotic expansions for the joint moments of the lengths of the external branches.In the following we briefly discuss the star-shaped coalescent and the Kingman coalescent.Afterwards we intensively study the Bolthausen-Sznitman coalescent.For related results on external branches for beta-coalescents we refer the reader to [8], [9] and [19].
Since h m+1 − h m = 1/(m + 1), the last sum simplifies considerably to Thus, the third moment of τ n,1 is For the fourth moment we obtain , a formula which does not seem to simplify much further.One may also introduce the generating functions so these generating functions satisfy the recursion Using this recursion, g k (t) can be computed iteratively, however, the expressions become quite involved with increasing k.For example, g the dilogarithm function.In principle higher order moments and as well joint moments can be calculated analogously, however the expressions become more and more nasty with increasing order.In the following we exemplary derive an exact formula for µ n (1, 1) = E(τ n,1 τ n,2 ).The recursion (4) for j = 2 and k 1 = k 2 = 1 reduces to (see ( 5)) It is readily checked by induction on n that this recursion is solved by µ 2 (1, 1) = 2 and In particular, Thus, for the Kingman coalescent, the lengths of two randomly chosen external branches are (slightly) negatively correlated for all n ≥ 4. We have used the derived formulas to compute the following table.1: Covariance of τ n,1 and τ n,2 for the Kingman coalescent In the following we focus on the Bolthausen-Sznitman coalescent [5], where Λ is the uniform distribution on [0, 1].Our second main result (Theorem 1.2) provides the asymptotics of all the joint moments of the external branch lengths for the Bolthausen-Sznitman coalescent.
Theorem 1.2 (Asymptotics of the joint moments of the external branch lengths) For the Bolthausen-Sznitman coalescent, the joint moments 0 , of the lengths τ n,1 , . . ., τ n,n of the external branches satisfy Remark.For j = 2 and With some more effort (see Corollary 3.2 and the remark thereafter) exact solutions for E(τ n,1 ) and E(τ n,1 τ n,2 ) are obtained and it follows that τ n,1 and τ n,2 are positively correlated for all n ≥ 2, in contrast to the situation for the Kingman coalescent, where τ n,1 and τ n,2 are slightly negatively correlated for all n ≥ 4.
The following two corollaries are a direct consequence of Theorem 1.2.
In particular, Remark.The same scaling and, except for the additional shift −1 on the right hand side in (8), the same limiting law as in ( 8) is known for the number of cuts needed to isolate the root of a random recursive tree ( [11], [16]).Essentially the same scaling and convergence result has been obtained for random records and cuttings in binary search trees by Holmgren [14, Theorem 1.1] and more generally in split trees (Holmgren [13, Theorem 1.1] and [15, Theorem 1.1]) introduced by Devroye [7].The logarithmic height of the involved trees seems to be one of the main sources for the occurrence of such scalings and of 1-stable limiting laws.To the best of the authors knowledge the distributional limiting behavior of L internal n , properly centered and scaled, is so far unknown for the Bolthausen-Sznitman coalescent.

Proof of Theorem 1.1
Let T = T n denote the time of the first jump of the block counting process N (n) and let I = I n denote the state of N (n) at its first jump.Note that T and I are independent, T is exponentially distributed with parameter g n and p nm := P(I = m) = g nm /g n , m ∈ {1, . . ., n − 1}.For i ∈ {1, . . ., n} and h > 0 define τ ′ i := τ n,i − h ∧ T .By the Markov property, for h → 0, Also for h → 0, ).Now at time T either the event A := {one of the individuals 1 to j is involved in the first collision} occurs, in which case τ ′ i = 0 for some i ∈ {1, . . ., j}, and the above expectation vanishes since k 1 , . . ., k j > 0, or none of these j individuals is involved in the first collision.Then, by the strong Markov property, ), where A c denotes the complement of A. Adding both expectations yields Collecting both terms involving E(τ k1 n,1 • • • τ kj n,j ) on the left hand side and letting h → 0 gives the claim, since P(T ≤ h) = 1 − e −gnh ∼ g n h as h → 0. ✷

Differential equations approach
A differential equations approach is provided, which is used in the proof of Theorem 1.2 given in the following Section 4. This approach furthermore yields for example an exact expression for E(τ n,1 τ n,2 ) in terms of Stirling numbers (see Corollary 3.2).Let D := {z ∈ C : |z| < 1} denote the open unit disc in the complex plane.For j ∈ N and k = (k 1 , . . ., k j ) ∈ N j 0 define the generating function where, for n ≥ j, we use the abbreviation ) for convenience.Note that, due to the natural coupling property of n-coalescents, the sequence (a n ) n≥j is non-increasing.Thus, f k and all its derivatives f ′ k , f ′′ k , . . .are analytic functions on D. In order to state the following result it is convenient to introduce L(z) := − log(1 − z), z ∈ D, and to define the functions for all z ∈ D and all k = (k 1 , . . ., k j ) ∈ N j satisfying k 1 + • • • + k j > 1, where e i , i ∈ {1, . . ., j}, denotes the ith unit vector in R j .

log n n L external n → 1 −
in probability as n → ∞.The moments of L external n do not provide much information on the distributional limiting behavior of L external n as n → ∞.Let L n denote the total branch length (the sum of the lengths of all branches) of the Bolthausen-Sznitman n-coalescent.Kersting et al. [18, Theorem 1.1] recently showed that the internal branch length L internal n := L n − L external n Combining this result with [10, Theorem 5.2] it follows that (see [18, Corollary 1.log n − log log n → L − 1 (8)in distribution as n → ∞, where L is a 1-stable random variable with characteristic function t → exp(it log |t| − π|t|/2), t ∈ R.