We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ with $Mh = \lambda h$, and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that $M_{i,j} \geqslant c\nu_j$ for all $i,j$. Let $\pi \in \mathscr{P}$ be such that $\pi M = \lambda \pi$, and let $C > 0$, $\gamma \in [0,1[$ be as in question 17.
(a) Show that for all $n \geqslant 0$ and $u \in \mathscr{M}_{d,1}(\mathbb{R})$ such that $\langle u, \pi \rangle = 0$, $$\left\| M^n u \right\|_1 \leqslant C(\lambda\gamma)^n \|u\|_1.$$
(b) Deduce that there exists $C_1 \geqslant 0$ such that for all $n \geqslant 0$ and $u \in \mathscr{M}_{d,1}(\mathbb{R})$ column vector such that $\langle u, \pi \rangle = 0$, $$\mathbb{E}\left(\langle X_n, u \rangle^2\right) \leqslant C_1 \|u\|_1^2 \left(\lambda^{2n} \left(\sum_{k=0}^{n-1} \lambda^{-k} \gamma^{2n-2k}\right) + (\lambda\gamma)^{2n}\right).$$