  We consider triangular arrays of Markov chains that converge weakly to a
diffusion process. Edgeworth type expansions of third order for transition
densities are proved. This is done for time horizons that converge to 0. For
this purpose we represent the transition density as a functional of densities
of sums of i.i.d. variables. This will be done by application of the parametrix
method. Then we apply Edgeworth expansions to the densities. The resulting
series gives our Edgeworth-type expansion for the transition density of Markov
chains. The research is motivated by applications to high frequency data that
are available on a very fine grid but are approximated by a diffusion model on
a more rough grid.
