Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent

The class of coalescent processes with simultaneous multiple collisions ([Xi]-coalescents) without proper frequencies is considered. We study the asymptotic behavior of the external branch length, the total branch length and the number of mutations on the genealogical tree as the sample size n tends to infinity. The limiting random variables arising are characterized via exponential integrals of the subordinator associated with the frequency of singletons of the coalescent. The proofs are based on decompositions into external and internal branches. The asymptotics of the external branches is treated via the method of moments. The internal branches do not contribute to the limiting variables since the number Cn of collisions for coalescents without proper frequencies is asymptotically negligible compared to n. The results are applied to the two-parameter Poisson-Dirichlet coalescent indicating that this particular class of coalescent processes in many respects behaves approximately as the star-shaped coalescent.