Forward and backward diffusion approximations for haploid exchangeable population models.
A class of haploid population models is considered with non-overlapping generations and family sizes \(\nu _1,\dots ,\nu _N\) which are exchangeable random variables, \(\nu _i\) being the number of offsprings of the \(i\)-th individual, where the population size \(N=\nu _1+\dots +\nu _N\) is fixed. A criterion for weak convergence in the Skorohod sense is given for proper time and space scales, counting the number of descendants forward in time. The generator of the limit process is constructed by means of the joint moments of the offspring variables \(\nu _1,\dots ,\nu _N\). In particular, the Wright-Fischer diffusion appears in the limit if and only if the condition \(\lim _{N\to \infty }E((\nu _1-1)^3) /(N\,\text{Var}(\nu _1)) =0\) is satisfied. By duality the convergence results are compared with the limit theorems known for models considered backward in time.
scientific article; zbMATH DE number 2138892
scientific article