  We consider triangular arrays of Markov chains that converge weakly to a
diffusion process. Second order Edgeworth type expansions for transition
densities are proved. The paper differs from recent results in two respects. We
allow nonhomogeneous diffusion limits and we treat transition densities with
time lag converging to zero. Small time asymptotics are motivated by
statistical applications and by resulting approximations for the joint density
of diffusion values at an increasing grid of points.
