We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$: $$Mh = \lambda h,$$ and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ Show that there exist $\pi \in \mathscr{P}$ and $h' \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ and $C > 0$ and $\gamma \in [0,1[$, such that $\pi M = \lambda \pi$ and for all $n \geqslant 0$, $$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i' \pi_j \right| \leqslant C\gamma^n.$$
We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$:
$$Mh = \lambda h,$$
and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$,
$$M_{i,j} \geqslant c\nu_j.$$
Show that there exist $\pi \in \mathscr{P}$ and $h' \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ and $C > 0$ and $\gamma \in [0,1[$, such that $\pi M = \lambda \pi$ and for all $n \geqslant 0$,
$$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i' \pi_j \right| \leqslant C\gamma^n.$$