CONVERGENCE TO FLEMING-VIOT PROCESSES IN THE WEAK ATOMIC TOPOLOGY

Stochastic models for gene frequencies can be viewed as probability-measure-valued processes. Fleming and Viot introduced a class of processes that arise as limits of genetic models as the population size and the number of possible genetic types tend to infinity. In general, the topology on the process values in which these limits exist is the topology of weak convergence; however, convergence in the weak topology is not strong enough for many genetic applications. A new topology on the space of finite measures is introduced in which convergence implies convergence of the sizes and locations of atoms, and conditions are given under which genetic models converge in this topology. As an application, Kingman's Poisson-Dirichlet limit is extended to models with selection.