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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)
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)
A class of haploid population models with non-overlapping generations and exchangeable offspring distribution is considered. Based on an analysis of the discrete ancestral process, we present solutions, algorithms and strong upper bounds for the expected time back to the most recent common ancestor. New insights into the asymptotical behavior of the(More)
A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Λ-coalescents) is provided. This recursion is used to derive asymptotic expansions as the sample size n tends to infinity for the moments of the total external branch length of the Bolthausen–Sznitman coalescent. The proof is based on an elementary(More)
Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radius r of convergence. It is shown(More)
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)
A special stochastic process, called the coalescent, is of fundamental interest in population genetics. For a large class of population models this process is the appropriate tool to analyse the ancestral structure of a sample of n individuals or genes, if the total number of individuals in the population is sufficiently large. A corresponding convergence(More)
We study the asymptotics of the extended Moran model as the total population size N tends to infinity. Two convergence results are provided, the first result leading to discrete-time limiting coalescent processes and the second result leading to continuous-time limiting coalescent processes. The limiting coalescent processes allow for multiple mergers of(More)