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 $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$ We assume, in this question only, that $\lambda \in ]0,1[$. Show then that $\mathbb{E}\left(\|X_n\|_1\right)$ tends to $0$ as $n$ tends to infinity and $\mathbb{P}\left(\exists n \geqslant 0 : X_n = 0\right) = 1$. We say that the population becomes extinct almost surely in finite time.
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 $i,j \in \{1,\ldots,d\}$,
$$M_{i,j} \geqslant c\nu_j.$$
We assume, in this question only, that $\lambda \in ]0,1[$. Show then that $\mathbb{E}\left(\|X_n\|_1\right)$ tends to $0$ as $n$ tends to infinity and $\mathbb{P}\left(\exists n \geqslant 0 : X_n = 0\right) = 1$. We say that the population becomes extinct almost surely in finite time.