Coalescent theory for a completely random mating monoecious population

Consider a large random mating monoecious diploid population that has N individuals in each generation. Let us assume that at time 0 a random sample of n ≪ N copies of a gene are taken from this population. It is also assumed that G1, … , GN, the numbers of successful gametes produced by parents 1, … , N, are exchangeable random variables. It is shown that if time is measured backward in units of 8N/E[G1(G1 − 1)] = 2Ne generations, where Ne is the effective population size, the separate copies of a gene ancestral to those observed at time 0 are almost certain to come from separate individuals as Ne → ∞. It is then possible to obtain a generalization of coalescent theory for haploid populations if the distribution of G1 has a finite second moment and E[G13]/N→0 as N → ∞.