= Diffusion approximation theorem
{title2=$n\mathbb E(\Delta X^n\mid x)\to b(x),\quad n\mathbb E((\Delta X^n)^2\mid x)\to a(x)$}
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.
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