Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. Show that, for every real number $\delta$, $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$ is equivalent to $\frac{\delta^{2}}{\tau}$, as $\tau$ tends to 0 from above.

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $n \in \mathbb{N}^{*}$, denote by $B_{n}$ the event $B_{n} = \bigcap_{k=1}^{n} (X \notin p_{k}\mathbb{N}^{*})$, where $p_1 < p_2 < \cdots$ are the prime numbers in increasing order.
Show that $\lim_{n \rightarrow \infty} \mathbb{P}(B_{n}) = \mathbb{P}(X = 1)$. Deduce that $$\forall x \in {]1,+\infty[}, \quad \frac{1}{\zeta(x)} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \left(1 - \frac{1}{p_{k}^{x}}\right)$$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$, and $W_{n} = U_{n} \wedge V_{n}$. We admit that, for all $k \in \mathbb{N}^{*}$, the sequence $(\mathbb{P}(W_{n} = k))_{n \in \mathbb{N}^{*}}$ converges to a real number $\ell_{k}$.
Show that $$\forall \varepsilon > 0, \quad \exists M \in \mathbb{N}^{*} \text{ such that } \forall m \in \mathbb{N}^{*},\ m \geqslant M \Longrightarrow 1 - \varepsilon \leqslant \sum_{k=1}^{m} \ell_{k} \leqslant 1$$

For every integer $n \geqslant 2$, equip the set $\Omega_n$ of permutations of $\llbracket 1, n \rrbracket$ with the uniform probability. Let $p_i$ denote the probability that a permutation is alternating up (with $p_0 = p_1 = 1$). Define the random variable $M_n$ on $\Omega_n$ by: $M_n(\sigma) = k+1$ where $k$ is the largest integer such that $(\sigma(1), \ldots, \sigma(k))$ is alternating up. Express $\mathbb{E}(M_n)$ as a function of $p_0, p_1, \ldots, p_n$. Deduce $\lim_{n \rightarrow \infty} \mathbb{E}(M_n) = \frac{\sin(1) + 1}{\cos(1)}$.

In this part, $d$ equals 1 and we simply denote $0_d = 0$. Moreover, $p$ is an element of $]0,1[$, $q = 1 - p$ and the distribution of $X$ is given by $$P ( X = 1 ) = p \quad \text{and} \quad P ( X = - 1 ) = q .$$ We assume that $$p = q = \frac { 1 } { 2 }$$ Give a simple equivalent of $P ( R = 2 n )$ as $n$ tends to $+ \infty$. Deduce a simple equivalent of $E \left( N _ { n } \right)$ as $n$ tends to $+ \infty$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
For $s > 1$ fixed, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}.$$
Let $s \geqslant 2$ be an integer. Let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$, and let $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ be their gcd.
Deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. For fixed $s > 1$, we define a probability distribution $\mu_s$ on $\mathbb{N}^*$ by setting, for $n \in \mathbb{N}^*$, $$\mu_s(\{n\}) = \frac{1}{\zeta(s) n^s}$$ Let $s \geqslant 2$ be an integer. Let $Z_n^{(s)}$ be the gcd of $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. Using the results of questions 18, 19, and 20a, deduce that the sequence $(\mu_{Z_n^{(s)}})_{n \in \mathbb{N}}$ converges in $\mathscr{B}(\mathscr{P}(\mathbb{N}^*), \mathbb{R})$ to $\mu_s$.

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Consider a pair $( u , v )$ of real numbers such that $u < v$, and denote $$J _ { n } = \left\{ j \in \llbracket 0 , n \rrbracket \left\lvert \, \frac { n + u \sqrt { n } } { 2 } \leqslant j \leqslant \frac { n + v \sqrt { n } } { 2 } \right. \right\}$$
Justify that $$\mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \sum _ { j \in J _ { n } } \mathbb { P } \left( \left\{ T _ { n } = j \right\} \right)$$

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_{n}, \mathscr{P}(\mathfrak{D}_{n}))$ equipped with the uniform probability. We define a random variable $Y_{n}$ by $Y_{n}(\sigma) = \varepsilon(\sigma)$.
Calculate, for all $\varepsilon \in \{-1, 1\}$, $\lim_{n \rightarrow +\infty} \mathbb{P}\left(Y_{n} = \varepsilon\right)$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{D}_n, \mathscr{P}(\mathfrak{D}_n))$ equipped with the uniform probability. We define a random variable $Y_n$ by $Y_n(\sigma) = \varepsilon(\sigma)$.
Calculate, for all $\varepsilon \in \{-1, 1\}$, $\lim_{n \rightarrow +\infty} \mathbb{P}(Y_n = \varepsilon)$.

In a room with $n \geqslant 2$ people, each pair shakes hands between themselves with probability $\frac{2}{n^2}$ and independently of other pairs. If $p_n$ is the probability that the total number of handshakes is at most 1, then $\lim_{n \rightarrow \infty} p_n$ is equal to
(A) 0
(B) 1
(C) $e^{-1}$
(D) $2e^{-1}$