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 $\mathscr{M}(E)$.
Show that there exists a sequence $(\varphi_k)_{k \in \mathbb{N}^*}$ of strictly increasing maps from $\mathbb{N}^*$ to $\mathbb{N}^*$ such that, for all $k \in \mathbb{N}^*$ and for all integer $1 \leqslant i \leqslant k$, the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ converges.

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, and let $\mu_\infty(x_i)$ be the limits defined in 12b.
Show that the map $$\begin{array}{rcl} \psi : \mathbb{N}^* & \longrightarrow & \mathbb{N}^* \\ k & \longmapsto & \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{array}$$ is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.

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 $\mu_\infty(x_i)$ be the limits defined in 12b.
Show that $\mu_\infty(x_i) \geqslant 0$ for all $i$ in $\mathbb{N}^*$, and that $\sum_{i=1}^{\infty} \mu_\infty(x_i) \leqslant 1$.

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 $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Let $\mu_\infty$ be defined as in 12b and 12d.
We say that the sequence $(\mu_n)_{n \in \mathbb{N}}$ of elements of $\mathscr{M}(E)$ is tight if for every real number $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of $E$ such that $\mu_n(F_\varepsilon) \geqslant 1 - \varepsilon$ for all natural integer $n$.
Suppose further that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight. Show then that $\mu_\infty$ defines an element of $\mathscr{M}(E)$ which is a cluster point of the sequence $(\mu_n)_{n \in \mathbb{N}}$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:

1. [i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
2. [ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} P(r \mid X_n) = P(r \mid X)$.

Show that then the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$.
We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$.
For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $P(r \mid X_n^{(i)})$ and show that $P(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} P\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}.$$

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$. Show that there exists a sequence $(\varphi_k)_{k \in \mathbb{N}^*}$ of strictly increasing applications from $\mathbb{N}^*$ to $\mathbb{N}^*$ such that, for all $k \in \mathbb{N}^*$ and for all integer $1 \leqslant i \leqslant k$, the sequence $\left(\mu_{\varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(n)}(x_i)\right)_{n \in \mathbb{N}^*}$ converges.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications 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 $\mu_\infty(x_i)$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $(\varphi_k)_{k \in \mathbb{N}^*}$ the sequence of strictly increasing applications from 12a, with limits $\mu_\infty(x_i)$ as defined in 12b. Show that the application $$\begin{aligned} \psi : \mathbb{N}^* &\longrightarrow \mathbb{N}^* \\ k &\longmapsto \varphi_1 \circ \varphi_2 \circ \ldots \circ \varphi_k(k) \end{aligned}$$ is strictly increasing, and that, for all integer $i \in \mathbb{N}^*$, the sequence $\left(\mu_{\psi(k)}(x_i)\right)_{k \in \mathbb{N}^*}$ converges to $\mu_\infty(x_i)$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $\mu_\infty$ as defined in 12b. Show that $\mu_\infty(x_i) \geqslant 0$ for all $i$ in $\mathbb{N}^*$, and that $\sum_{i=1}^{\infty} \mu_\infty(x_i) \leqslant 1$.

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Let $(\mu_n)_{n \in \mathbb{N}}$ be a sequence of elements of $\mathscr{M}(E)$ and $\mu_\infty$ as defined in 12b. We say that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight if for all real $\varepsilon > 0$, there exists a finite subset $F_\varepsilon$ of $E$ such that $\mu_n(F_\varepsilon) \geqslant 1 - \varepsilon$ for all natural integer $n$. Suppose further that the sequence $(\mu_n)_{n \in \mathbb{N}}$ is tight. Show then that $\mu_\infty$ defines an element of $\mathscr{M}(E)$ which is a cluster point of the sequence $(\mu_n)_{n \in \mathbb{N}}$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$ and let $X$ be a random variable defined on $(\Omega, \mathscr{A}, P)$ and taking values in $\mathbb{N}^*$. Assume that:

1. [i.] The sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ is tight.
2. [ii.] For all $r \in \mathbb{N}^*$, $\lim_{n \rightarrow +\infty} \mathbf{P}(r \mid X_n) = \mathbf{P}(r \mid X)$.

Show that the sequence $(\mu_{X_n})_{n \in \mathbb{N}}$ converges to $\mu_X$ in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$.

Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$. For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $\mathbf{P}(r \mid X_n^{(i)})$ and show that $\mathbf{P}(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} \mathbf{P}\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}$$