Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and by $\pi$ the orthogonal projection onto $E'$. We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $C_{t} = \pi(C \cap H_{t})$ where $H_t = E' + te_n$. For $t$ in $\{-1, 1\}$, we denote by $Y_{t}$ the projection of $X'$ onto the non-empty closed convex set $C_{t}$. Let $\lambda$ be a real such that $0 \leqslant \lambda \leqslant 1$.
Show that
$$d(X, C) \leqslant \left\|(1 - \lambda)(Y_{\varepsilon_{n}} + \varepsilon_{n} e_{n}) + \lambda(Y_{-\varepsilon_{n}} - \varepsilon_{n} e_{n}) - X\right\|$$

We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$.
Show that $p_{-} > 0$.

Justify that, for every natural integer $k$, $p _ { 1 } ^ { ( k ) } + \cdots + p _ { n } ^ { ( k ) } = 1$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. If $n \in \mathbb{N}^*$, we denote $\{n \mid X\}$ the event ``$n$ divides $X$'' and $\{n \nmid X\}$ the complementary event.
Let $r \geqslant 1$ be an integer. Show that $$P\left(\bigcap_{i=1}^{r}\left\{p_i \nmid X\right\}\right) = \prod_{i=1}^{r}\left(1 - p_i^{-s}\right).$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Deduce that $$\zeta(s)^{-1} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n}\left(1 - p_k^{-s}\right).$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
We denote $r(n)$ the number of divisors $d \geqslant 1$ of $n$. Show that the series $\sum_{n=1}^{+\infty} r(n) n^{-s}$ converges and that its sum equals $\zeta(s)^2$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. If $n \in \mathbb{N}^*$, we set $g(n) = r_1(n) - r_3(n)$ where $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Deduce that the series $\sum_{n=1}^{+\infty} g(n) n^{-s}$ 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 $\mu_\infty$ 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 $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$.

The functions $p _ { \alpha }$ are defined by $p_\alpha : t \mapsto t^\alpha$ for $\alpha \in \mathbb{R}_+^*$, and the inner product on $E$ is $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Is the family $\left( p _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ an orthogonal family of $E$?

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. We recall that $(p_i)_{i \in \mathbb{N}^*}$ denotes the sequence of prime numbers, ordered in increasing order. Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\bigcup_{i=1}^{n+1} \mathbb{N}^* r p_i = \left(\bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) \cup \left(\mathbb{N}^* r p_{n+1} \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_{n+1} p_i\right)$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We suppose that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$, where $\mathbb{N}^* r$ denotes the set of strictly positive multiples of $r$. Using the result of 17a, show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\mu_1\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) = \mu_2\left(\mathbb{N}^* r \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right)$$

With the notation of question 28, justify that, for all $j \in \llbracket 1 , n - 1 \rrbracket$, $$\mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) = \sum _ { \left( v _ { 1 } , \ldots , v _ { j } \right) \in \mathcal { V } _ { n , 1 } ^ { j } } \mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( v _ { 1 } , \ldots , v _ { j } \right) \right) \mathbb { P } \left( \left( C _ { 1 } = v _ { 1 } \right) \cap \cdots \cap \left( C _ { j } = v _ { j } \right) \right) .$$

With the notation of question 28, deduce that, for all $j \in \llbracket 1 , n - 1 \rrbracket$, $$\mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) \leqslant 2 ^ { j - n } .$$

With the notation of question 28, deduce that $$\mathbb { P } \left( M \in \mathcal { G } \ell _ { n } ( \mathbb { R } ) \right) \geqslant \frac { 1 } { 2 ^ { n - 1 } } .$$

Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$
By applying Markov's inequality to a suitably chosen random variable, prove that $$\forall t \in \mathbb { R } ^ { + } , \quad \mathbb { P } ( Z \geqslant \lambda ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } - \lambda t \right)$$

Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$
Deduce that $$\mathbb { P } ( | Z | \geqslant \lambda ) \leqslant 2 \exp \left( - \frac { \lambda ^ { 2 } } { 2 \sigma ^ { 2 } } \right)$$

For all integer $p \in \mathbb { N } ^ { * }$ and all $x > 0$, we set $P _ { p } ( x ) = x \mathrm { e } ^ { x } g _ { p } ^ { ( p ) } ( x )$, where $g _ { p } ( x ) = x ^ { p - 1 } \mathrm { e } ^ { - x }$. We recall that $P _ { p }$ is a polynomial function of degree $p$, that $P _ { p } \in E$, and that $P_p$ is an eigenvector of $U$ for the eigenvalue $1/p$. The inner product on $E$ is $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that the polynomials $P _ { p }$ are pairwise orthogonal in $E$.

Let $x, y, z$ be three distributions on $\mathbf{N}$. Prove the properties: $$\begin{gathered} 0 \leq d_{VT}(x, y) \leq 1 \\ d_{VT}(x, y) = 0 \Longleftrightarrow x = y \\ d_{VT}(y, x) = d_{VT}(x, y) \\ d_{VT}(x, z) \leq d_{VT}(x, y) + d_{VT}(y, z) \end{gathered}$$

If $x$ and $y$ are two probability distributions on $\mathbf{N}$, we define the application $x * y : \mathbf{N} \rightarrow \mathbf{R}_+$ by $$\forall k \in \mathbf{N} \quad (x * y)(k) = \sum_{i=0}^{k} x(i) y(k-i) = \sum_{i+j=k} x(i) y(j)$$ Show that $x * y$ is a distribution on $\mathbf{N}$.

Let $(x, y, u, v) \in \left(\mathcal{D}_{\mathbf{N}}\right)^4$. Show that, for all natural number $k$, $$|(x * y)(k) - (u * v)(k)| \leq \sum_{i+j=k} y(j)|x(i) - u(i)| + \sum_{i+j=k} u(i)|y(j) - v(j)|.$$

With the notation of the previous question, establish the inequality $$d_{VT}(x * y, u * v) \leq d_{VT}(x, u) + d_{VT}(y, v)$$

Let $G = ( S , A ) \in \Omega _ { n }$. Determine the probability $\mathbf { P } ( \{ G \} )$ of the elementary event $\{ G \}$ in terms of $p _ { n } , q _ { n } , N$ and $a = \operatorname { card } ( A )$. Then recover the fact that $\mathbf { P } \left( \Omega _ { n } \right) = 1$.

Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that $\mathbf{P}(X > 0) \leq \mathbf{E}(X)$.

Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that if $\mathbf{E}(X) \neq 0$, then $\mathbf{P}(X = 0) \leq \frac{\mathbf{V}(X)}{(\mathbf{E}(X))^{2}}$. Hint: note that $(X = 0) \subset (|X - \mathbf{E}(X)| \geq \mathbf{E}(X))$.