Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and $$x_{n+1} = x_n P.$$ Show that the series $\sum_{n \geqslant 0} \|x_{n+1} - x_n\|_1$ is convergent.
Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1:
$$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$
We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$,
$$\sum_{j=1}^{d} P_{i,j} = 1$$
We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$,
$$P_{i,j} \geqslant c\nu_j.$$
Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and
$$x_{n+1} = x_n P.$$
Show that the series $\sum_{n \geqslant 0} \|x_{n+1} - x_n\|_1$ is convergent.