  We consider a transitive action of a finitely generated group $G$ and the
Schreier graph $\Gamma$ defined by this action for some fixed generating set.
For a probability measure $\mu$ on $G$ with a finite first moment we show that
if the induced random walk is transient, it converges towards the space of ends
of $\Gamma$. As a corollary we obtain that for a probability measure with a
finite first moment on Thompson's group $F$, the support of which generates $F$
as a semigroup, the induced random walk on the dyadic numbers has a non-trivial
Poisson boundary. Some assumption on the moment of the measure is necessary as
follows from an example by Juschenko and Zheng.
