  Given a finite set $\{M_0,\dots,M_{d-1}\}$ of nonnegative $2\times 2$
matrices and a nonnegative column-vector $V$, we associate to each
$(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$ the sequence of the column-vectors
$\displaystyle{M_{\omega_1}\dots M_{\omega_n}V\over\Vert M_{\omega_1}\dots
M_{\omega_n}V\Vert}$. We give the necessary and sufficient condition on the
matrices $M_k$ and the vector $V$ for this sequence to converge for all
\hbox{$(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$} such that $\forall n,\
M_{\omega_1}\dots M_{\omega_n}V\ne\begin{pmatrix}0\\0\end{pmatrix}$.
