  Let $G$ be a real Lie group, $\Lambda\subseteq G$ a lattice, and
$X=G/\Lambda$. We fix a probability measure $\mu$ on $G$ and consider the left
random walk induced on $X$. It is assumed that $\mu$ is aperiodic, has a finite
first moment, spans a semisimple algebraic group without compact factors, and
has two non mutually singular convolution powers. We show that for every
starting point $x\in X$, the $n$-th step distribution $\mu^n*\delta_{x}$ of the
walk weak-$\ast$ converges toward some homogeneous probability measure on $X$.
