Diffusion approximation theorem ($n\mathbb{E}(\Delta X^n\mid x)\to b(x),n\mathbb{E}((\Delta X^n{)}^2\mid x)\to a(x)$)

For discrete-time Markov chains run at time step $1/n$, suppose the displayed conditional moments converge uniformly on compact sets to continuous coefficients, the local conditional large-jump second moments vanish, initial laws converge, and compact containment holds on each finite horizon. If the limiting diffusion martingale problem is well-posed and nonexplosive, the chain laws converge to its solution. Vanishing jumps give a continuous limit; linearly interpolated paths converge in the uniform path topology and step paths in Skorokhod J1 topology. Local moment bounds give tightness after exit-time localization. Taylor expansion identifies the limiting test-function martingales, and uniqueness of the martingale problem makes every subsequential limit the same. The diffusion approximation of a birth-death process is an important example.