Forward and backward diffusion approximations for haploid exchangeable population models

Abstract

The class of haploid population models with non-overlapping generations and fixed population size N is considered such that the family sizes ν 1,…, ν N within a generation are exchangeable random variables. A criterion for weak convergence in the Skorohod sense is established for a properly time- and space-scaled process counting the number of descendants forward in time. The generator A of the limit process X is constructed using the joint moments of the offspring variables ν 1,…, ν N . In particular, the Wright–Fisher diffusion with generator Af(x)= 1 2 x(1−x)f″(x) appears in the limit as the population size N tends to infinity if and only if the condition lim N→∞ E((ν 1−1) 3)/(N Var(ν 1))=0 is satisfied. Using the concept of duality, these convergence results are compared with the limit theorems known for the coalescent processes with simultaneous and multiple collisions arising when the models are considered backward in time. In particular the Wright–Fisher diffusion appears forward in time if and only if the Kingman coalescent appears backward in time as N tends to infinity. A commutative diagram leads to a full understanding of the model considered forward and backward in time for finite population size and in the limit as N tends to infinity.