On the Stationary Distribution of the Neutral Diffusion Model in Population Genetics

Let S be a compact metric space, let .>0, and let P(x,dy) be a one-step Feller transition function on S.B(S) corresponding to a weakly ergodic Markov chain in S with unique stationary distribution .0. The neutral diffusion model, or Fleming-Viot process, with type space S, mutation intensity 12. and mutation transition function P(x,dy), assumes values in P(S), the set of Borel probability measures on S with the topology of weak convergence, and is known to be weakly ergodic and have a unique stationary distribution ..P(P(S)). Define the Markov chain {X(.),..Z+} in S2.S3.. as follows. Let X(0)=(.,.).S2, where . is an S-valued random variable with distribution .0. From state (x1,.,xn).Sn, where n.2, one of two types of transitions occurs. With probability ./(n(n.1+.)) a transition to state (x1,.,xi.1,.i,xi+1,.,xn).Sn occurs (1.i.n), where .i is distributed according to P(xi,dy). With probability (n.1)/((n+1)n(n.1+.)) a transition to state (x1,.,xj.1,xi,xj,.,xn).Sn+1 occurs (1.i.n,1.j.n+1). Letting .n denote the hitting time of Sn, we show that the empirical measure determined by the n coordinates of X(.n+1.1) converges almost surely as n.. to a P(S)-valued random variable with distribution ..