Coalescent patterns in diploid exchangeable population models.

A class of two-sex population models is considered with N females and equal number N of males constituting each generation. Reproduction is assumed to undergo three stages: 1) random mating, 2) exchangeable reproduction, 3) random sex assignment. Treating individuals as pairs of genes at a certain locus we introduce the diploid ancestral process (the past genealogical tree) for n such genes sampled in the current generation. Neither mutation nor selection are assumed. A convergence criterium for the diploid ancestral process is proved as N goes to infinity while n remains unchanged. Conditions are specified when the limiting process (coalescent) is the Kingman coalescent and situations are discussed when the coalescent allows for multiple mergers of ancestral lines.