The external lengths in Kingman’s coalescent 1 Svante Janson∗ Go¨tz Kersting† 1 0 2 17 January, 2011 n a J 7 Abstract 1 In this paper we prove asymptotic normality of the total length of ] external branches in Kingman’s coalescent. The proof uses an embedded R Markovchain,whichcanbedescribedasfollows: Takeanurnwithnblack P balls. Empty it in n steps according to the rule: In each step remove a . randomly chosen pair of balls and replace it by one red ball. Finally h t remove the last remaining ball. Then the numbers Uk, 0 ≤ k ≤ n, of a red balls after k steps exhibit an unexpected property: (U0,...,Un) and m (Un,...,U0) are equal in distribution. [ MSC 2000 subject classifications. 60K35, 60F05, 60J10 2 Key words and phrases. coalescent, external branch,time reversibility, urn model v 1 1 1 Introduction and results 0 5 OurmainresultinthispaperisthatthetotallengthL ofallexternalbranches . n 4 in Kingman’s coalescent with n external branches is asymptotically normal for 0 n . 0 →∞ Kingman’scoalescent(1982)consistsoftwocomponents. Firsttherearethe 1 : coalescent times T1 >T2 > >Tn =0. They are such that v ··· i X k (T T ) , k =2,...,n k 1 k r (cid:18)2(cid:19) − − a are independent, exponential random variables with expectation 1. Second there are partitions π = 1,...,n ,π ,...,π = 1 ,..., n of the set 1 2 n { } { } { } 1,...,n , where the set π containes k disjoint subsets of 1,...,n and π k k 1 { } (cid:8) (cid:9) (cid:8) { (cid:9)} − evolves from π by merging two randomly chosen elements of π . Moreover, k k (T ,...,T ) and (π ,...,π ) are independent. For convenience we put π := . n 1 n 1 0 ∅ ∗DepartmentofMathematics, UppsalaUniversity,POBox480,SE-75106Uppsala,Swe- den. svante.janson@math.uu.se †FachbereichInformatikundMathematik,UniversitätFrankfurt,Fach187,D-60054Frank- furtamMain,Germany. kersting@math.uni-frankfurt.de 1 As is customary the coalescent can be represented by a tree with n leaves labelled from 1 to n. Each of these leaves corresponds to an external branch of the tree. The other node of the branch with label i is located at level ρ(i):=max k 1: i π k { ≥ { }6∈ } within the coalescent. The length of this branch is T , The total external ρ(i) length of the coalescent is given by n L := T . n ρ(i) i=1 X This quantity is of a certain statistical interest. Coalescent trees have been introduced by Kingman as a model for the genealogic relationshipof n individuals, down to their most recent common ancestor. Mutations can be located everywhere on the branches. Then mutations on external branches affect only single individuals. This fact was used by Fu and Li (1993) in designing their D-statisticandprovidingatestwhetherornotdatafittoKingman’scoalescent. Otherwise single external branches have mainly been studied in the litera- ture. The asymptotic distribution of T has been obtained by Caliebe et al ρ(i) (2007),usingarepresentationofitsLaplacetransformduetoBlumandFrançois (2005). We address this issue in Section 6 below. Freund and Mo¨hle (2009) in- vestigated the external branch length of the Bolthausen-Snitman coalescent, and Gnedin et al (2008) the Λ-coalescent. Here is our main result. Theorem 1. As n , →∞ 1 n d L 2 N(0,1) . n 2 logn − → r (cid:0) (cid:1) The proof will show that the limiting normal distribution originates from the random partitions and not from the exponential waiting times. A second glance on this result reveals a peculiarity: The normalization of L is carriedout using its expectation, but only half of its variance. These two n terms have been determined by Fu and Li (1993) (with a correction given by Durrett (2002)). They obtained 8nh 16n+8 8logn E(L )=2 , Var(L )= n− n n (n 1)(n 2) ∼ n − − with h :=1+1+ + 1, the n-th harmonicnumber. Belowwe derive a more n 2 ··· n general result. Touncoverthispeculiarityweshallstudytheexternallengthsinmoredetail. First we look at the point processes η on (0, ), given by η = n δ , n ∞ n i=1 √nTρ(i) i.e. P η (B):=# i:√nT B (1) n ρ(i) { ∈ } for Borel sets B (0, ). ⊆ ∞ 2 Theorem 2. As n the point process η converges in distribution, as n → ∞ point processes on (0, ], to a Poisson point process η on (0, ) with intensity ∞ ∞ measure λ(dx)=8x 3dx. − We use (0, ] in the statement of Theorem 2 instead of (0, ) since it is ∞ ∞ d stronger, including for example η (a, ) η(a, ) for every a > 0. The n ∞ → ∞ significance is that, as n , there will be points clustering at 0 but not at → ∞ . (Below in the proof we recall the definition of convergence in distribution ∞ of point processes.) Theorem 2 permits a first orientation. Since √nL = xη (dx), one is n n tempted to resort to infinitely divisible distributions. However, the intensity R measure λ(dx) is slightly outside the range of the Lévy-Chintchin formula. Shortly speaking this means that small points of η have a dominant influ- n ence on the distribution of L and we are within the domain of the normal n distribution. Thus let us look in more detail on the external lengths and focus on Lα,β := T , 0 α<β 1 , n ρ(i) ≤ ≤ nα≤Xρ(i) 0. Then ≤ − Var(M ) (n k 1) 4Var(U ) = O(n 3), and it follows from Doob’s in- k − k − ≤ − − equality that max U E(U ) =O (n1/2). k k P k | − | 9 (Using Theorem 7 we see that this extends to 0 k n.) A straight- ≤ ≤ forward computation of the conditional quadratic variation M,M := m h i E (M M )2 U ) shows that, uniformly in 0 t 1 δ, k0 is arbitrary,this yields convergence in D[0,1). By time-reversal and Theorem 7, we also have convergence in D(0,1], and together these imply convergence in D[0,1], see e.g. the proof in Janson (2009). 3 Proof of Proposition 3 We use the representation Lα,β = T X , n k k nα≤Xk0, independent of n. Thus from independence 1 1 2k E(Lα,β)= 2 . n k − n n 1 nα≤Xk

The external lengths in Kingman's coalescent

Similar The external lengths in Kingman's coalescent