Convergence to the coalescent with simultaneous multiple mergers

The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized by Mohle and Sagitov (2001) in terms of a sequence of measures defined on the finite-dimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure (xi) defined on the infinite simplex (DELTA) was suggested by Schweinsberg (2000). This paper presents a simple criterion of weak convergence to the (xi)-coalescent. In contrast to the earlier criterion of Mohle and Sagitov (2001) based on the moment conditions, the key condition here is expressed in terms of the joint distribution of the ranked offspring sizes. This criterion interprets a vector in (DELTA) as the ranked fractions of the total population size assigned to sibling groups constituting a (rare) generation, where a merger might occur. An example of the general coalescent is developed on the basis of the Poisson-Dirichlet distribution. It suggests a simple algorithm of simulating the Kingman coalescent with occasional (simultaneous) multiple mergers.