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

We consider a family Xₙ^θ of discrete-time Markov processes indexed by a positive "step-size" parameter θ. The conditional expectations of Δ Xₙ^θ, (Δ Xₙ^θ)², and |Δ Xₙ^θ|³, given Xₙ^θ, are of the order of magnitude of θ, θ², and θ³, respectively. Previous work has shown that there are functions f and g such that (Xₙ^θ - f(nθ))/θ1/2 is asymptotically normally distributed, with mean 0 and variance $g(t)$, as θ → 0 and nθ → t < ∞. The present paper extends this result to t = ∞. The theory is illustrated by an application to the Wright-Fisher model for changes in gene frequency.