  Let $G$ be a real linear semisimple algebraic group without compact factors
and $\Gamma$ a Zariski dense subgroup of $G$. In this paper, we use a
probabilistic counting in order to study the asymptotic properties of $\Gamma$
acting on the Furstenberg boundary of $G$. First, we show that the $K$
components of the elements of $\Gamma$ in the KAK decomposition of $G$ become
asymptotically independent. This result is an analog of a result of Gorodnik-Oh
in the context of the Archimedean counting. Then, we give a new proof of a
result of Guivarc'h concerning the positivity of the Hausdorff dimension of the
unique stationary probability measure on the Furstenberg Boundary of $G$.
Finally, we show how these results can be combined to give a probabilistic
proof of the Tit's alternative; namely that two independent random walks on
$\Gamma$ will eventually generate a free subgroup. This result answered a
question of Guivarc'h and was published earlier by the author. Since we're
working with the field of real numbers, we give here a more direct proof and a
more general statement.
