Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Let $(\varphi_k)_{k \in \mathbb{N}^*}$ be the sequence of strictly increasing maps from 12a. Show that for all $i \in \mathbb{N}^*$ and all integer $k \geqslant i$, the limit of the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ depends only on $i$ and not on $k$. We denote this limit by $\mu_\infty(x_i)$.
Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Let $(\varphi_k)_{k \in \mathbb{N}^*}$ be the sequence of strictly increasing maps from 12a.
Show that for all $i \in \mathbb{N}^*$ and all integer $k \geqslant i$, the limit of the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ depends only on $i$ and not on $k$. We denote this limit by $\mu_\infty(x_i)$.