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Let Λ be a finite measure on the unit interval. A Λ-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions (Λ-coalescent) in analogy to the duality known for the classical Fleming-Viot process and Kingman’s coalescent, where Λ is the Dirac measure in 0. We explicitly construct a dual… (More)

- M. Birkner, J. Blath, +4 authors A. Wakolbinger
- 2004

We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from α-stable branching mechanisms. The random ancestral partition is then a time-changed Λ-coalescent, where Λ is the Beta-distribution with parameters 2− α and α, and the… (More)

- Michael Drmota, Alex Iksanov, Martin Moehle, Uwe Roesler
- Random Struct. Algorithms
- 2009

Abstract. Let Xn be the number of cuts needed to isolate the root in a random recursive tree with n vertices. We provide a weak convergence result for Xn. The basic observation for its proof is that the probability distributions of {Xn : n = 2, 3, . . .} are recursively defined by Xn d = Xn−Dn + 1, n = 2, 3, . . ., X1 = 0, where Dn is a discrete random… (More)

- Alex Iksanov, Martin Möhle
- 2007

Let X1, X2, . . . be a sequence of random variables satisfying the distributional recursion X1 = 0 and Xn d = Xn−In + 1 for n = 2, 3, . . ., where In is a random variable with values in {1, . . . , n − 1} which is independent of X2, . . . , Xn−1. The random variable Xn can be interpreted as the absorption time of a suitable death Markov chain with state… (More)

- Martin Möhle, Helmut Pitters
- 2014

A spectral decomposition for the generator and the transition probabilities of the block counting process of the Bolthausen–Sznitman coalescent is derived. This decomposition is closely related to the Stirling numbers of the first and second kind. The proof is based on generating functions and exploits a certain factorization property of the… (More)

Abstract We study the total branch length Ln of the Bolthausen-Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are… (More)

- M. Möhle
- 2003

A new scaling for the standard discrete Luria-Delbrück distribution is provided which leads to a weak convergence result as the parameter of the distribution tends to infinity. We show that the stable limiting probability measure has the Fourier transform t → exp(−π|t|/2−it log |t|). For the corresponding density an integral representation is derived, which… (More)

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b =… (More)

- MARTIN MÖHLE
- 2009

Expansions are provided for the moments of the number of collisions Xn in the β(2, b)-coalescent restricted to the set {1, . . . , n}. We verify that Xn/EXn converges almost surely to one and that Xn, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in β(a,… (More)

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