  In a first part, using the recent measure classification results of
Eskin--Lindenstrauss, we give a criterion to ensure a.s. equidistribution of
empirical measures of an i.i.d. random walk on a homogeneous space $G/\Gamma$.
Employing renewal and joint equidistribution arguments, this result is
generalized in the second part to random walks with Markovian dependence.
Finally, following a strategy of Simmons--Weiss, we apply these results to
Diophantine approximation problems on fractals and show that almost every point
with respect to Hausdorff measure on a graph directed self-similar set is of
generic type, so in particular, well approximable.
