A Central Limit theorem for Markov Processes that Move by Small Steps

We consider a family $X_n^\theta$ of discrete-time Markov processes indexed by a positive "step-size" parameter $\theta$. The conditional expectations of $\Delta X_n^\theta, (\Delta X_n^\theta)^2$, and $|\Delta X_n^\theta|^3$, given $X_n^\theta$, are of the order of magnitude of $\theta, \theta^2$, and $\theta^3$, respectively. Previous work has shown that there are functions $f$ and $g$ such that $(X_n^\theta - f(n\theta))/\theta^{\frac{1}{2}}$ is asymptotically normally distributed, with mean 0 and variance $g(t)$, as $\theta \rightarrow 0$ and $n\theta \rightarrow t < \infty$. The present paper extends this result to $t = \infty$. The theory is illustrated by an application to the Wright-Fisher model for changes in gene frequency.