  Assume that one observes the $k$th, $2k$th$,\ldots,nk$th value of a Markov
chain $X_{1,h},\ldots,X_{nk,h}$. That means we assume that a high frequency
Markov chain runs in the background on a very fine time grid but that it is
only observed on a coarser grid. This asymptotics reflects a set up occurring
in the high frequency statistical analysis for financial data where diffusion
approximations are used only for coarser time scales. In this paper, we show
that under appropriate conditions the L$_1$-distance between the joint
distribution of the Markov chain and the distribution of the discretized
diffusion limit converges to zero. The result implies that the LeCam deficiency
distance between the statistical Markov experiment and its diffusion limit
converges to zero. This result can be applied to Euler approximations for the
joint distribution of diffusions observed at points
$\Delta,2\Delta,\ldots,n\Delta$. The joint distribution can be approximated by
generating Euler approximations at the points $\Delta k^{-1},2\Delta
k^{-1},\ldots,n\Delta$. Our result implies that under our regularity conditions
the Euler approximation is consistent for $n\to\infty$ if $nk^{-2}\to0$.
