Infinite products of nonnegative $2\times2$ 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 $(ω_n)\in\{0,\dots,d-1\}^\mathbb N$ the sequence of the column-vectors $\displaystyle{M_{ω_1}\dots M_{ω_n}V\over\Vert M_{ω_1}\dots M_{ω_n}V\Vert}$. We give the necessary and sufficient condition on the matrices $M_k$ and the vector $V$ for this sequence to converge for all \hbox{$(ω_n)\in\{0,\dots,d-1\}^\mathbb N$} such that $\forall n,\ M_{ω_1}\dots M_{ω_n}V\ne\begin{pmatrix}0\\0\end{pmatrix}$.

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