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

Abstract

Let $S$ be a compact metric space, let $\theta > 0$, and let $P(x,dy)$ be a one-step Feller transition function on $S \times \mathscr{B}(S)$ corresponding to a weakly ergodic Markov chain in $S$ with unique stationary distribution $\nu_0$. The neutral diffusion model, or Fleming-Viot process, with type space $S$, mutation intensity $\frac{1}{2}\theta$ and mutation transition function $P(x,dy)$, assumes values in $\mathscr{P}(S)$, 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 $\Pi \in \mathscr{P}(\mathscr{P}(S))$. Define the Markov chain $\{X(\tau), \tau \in \mathbf{Z}_+\}$ in $S^2 \cup S^3 \cup \cdots$ as follows. Let $X(0) = (\xi,\xi) \in S^2$, where $\xi$ is an $S$-valued random variable with distribution $\nu_0$. From state $(x_1, \ldots, x_n) \in S^n$, where $n \geq 2$, one of two types of transitions occurs. With probability $\theta/(n(n - 1 + \theta))$ a transition to state $(x_1,\ldots,x_{i - 1},\xi_i,x_{i + 1}, \ldots, x_n) \in S^n$ occurs $(1 \leq i \leq n)$, where $\xi_i$ is distributed according to $P(x_i,dy)$. With probability $(n - 1)/((n + 1)n(n - 1 + \theta))$ a transition to state $(x_1, \ldots ,x_{j - 1},x_i,x_j, \ldots, x_n) \in S^{n + 1}$ occurs $(1 \leq i \leq n, 1 \leq j \leq n + 1)$. Letting $\tau_n$ denote the hitting time of $S^n$, we show that the empirical measure determined by the $n$ coordinates of $X(\tau_{n + 1} - 1)$ converges almost surely as $n \rightarrow \infty$ to a $\mathscr{P}(S)$-valued random variable with distribution $\Pi$.