Ornstein-Uhlenbeck fluctuation limit at a stable population equilibrium ($d{X}_t=-\alpha X_{t}dt+\alpha \sqrt{2/\beta }d{B}_t$)

For a unit-jump population chain with one-step probabilities $\alpha k/n$ and $\beta k^2/n^2$, its equilibrium density is $u_0=\alpha /\beta$. Speed time by $n$ and center and scale the population as $(k-n{u}_0)/\sqrt{n}$. The accelerated conditional drift is $-\alpha x-\beta x^2/\sqrt{n}$, and the second moment is $2{\alpha }^2/\beta +3\alpha x/\sqrt{n}+\beta x^2/n$. Their compact-set limits give the displayed Ornstein-Uhlenbeck process. If $2{\alpha }^2<\beta$, the invalid-probability threshold stays a fixed positive density distance above equilibrium. Before that threshold, the generator of $x^2$ is at most one because its drift contribution is nonpositive. The stopped Lyapunov function bound proves compact containment and makes the probability of hitting that threshold on a fixed accelerated horizon tend to zero. This supplies the global hypotheses of the diffusion approximation theorem.