Infinite products of nonnegative $2\times 2$ matrices by nonnegative vectors

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 $\forall n,
M_{\omega_1}\dots M_{\omega_n}V\ne\begin{pmatrix}0\0\end{pmatrix}$.