A CENTRAL LIMIT THEOREM ASSOCIATED WITH THE TRANSFORMED TWO-PARAMETER POISSON-DIRICHLET DISTRIBUTION

In this paper we introduce the transformed two-parameter Poisson-Dirichlet distribution Pi(sigma)(theta,alpha) on the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rate theta and the selection rate sigma become large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength of sigma and theta.