Let $S$ and $T$ be two finite real random variables that are independent and defined on $(\Omega, \mathcal{A}, P)$. We assume that $T$ and $-T$ have the same distribution.
Show that $E(\cos(S + T)) = E(\cos(S)) E(\cos(T))$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$.
We consider the function $\varphi_{n}$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\varphi_{n}(t) = E\left(\cos\left(S_{n} t\right)\right)$ for all real $t$.
Show that $\varphi_{n}(t) = (\cos t)^{n}$ for all integers $n \in \mathbb{N}^{*}$ and all real $t$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$ and $U_{n} = \left(\frac{S_{n}}{n}\right)^{4}$.
Show that $E\left(S_{n}^{4}\right) = 3n^{2} - 2n$ for all $n \in \mathbb{N}^{*}$.

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
We assume $a_{1} \geqslant a_{2} + \cdots + a_{n}$. Show $E\left(\left|T_{n}\right|\right) = E\left(\left|T_{1}\right|\right) = a_{1}$.

Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that if $g = \mathbb{1}_{[0,K]}$, then $f(x) = \mathbb{E}(N(x-K, x))$.

Let $Y$ be a discrete random variable, independent of $X$, and $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a bounded function. Show that $$\mathbb{E}(\varphi(X, Y)) = \sum_{i=0}^{+\infty} p_i \mathbb{E}\left(\varphi\left(x_i, Y\right)\right)$$

Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$. Show that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, we have $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$.

We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$, $$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Using the result of Q26, deduce the variance of $X$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. Let $(A_{1}, \ldots, A_{m})$ be a complete system of events with non-zero probabilities. Show that
$$\mathbb{E}(X) = \sum_{i=1}^{m} \mathbb{P}(A_{i}) \cdot \mathbb{E}(X \mid A_{i})$$

We consider $g(X)$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable with independent Rademacher coefficients and $g(M) = \|M \cdot u\|$ for a fixed unit vector $u$. Show that $\mathbb{E}\left(g(X)^{2}\right) = k$, and deduce that $\mathbb{E}(g(X)) \leqslant \sqrt{k}$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. Let $(A_{1}, \ldots, A_{m})$ be a complete system of events with non-zero probabilities. Show that
$$\mathbb{E}(X) = \sum_{i=1}^{m} \mathbb{P}(A_{i}) \cdot \mathbb{E}(X \mid A_{i})$$

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables.
Show that $\mathbb{E}\left(g(X)^{2}\right) = k$, and deduce that $\mathbb{E}(g(X)) \leqslant \sqrt{k}$.

Using the result that $X_n$ and $-X_n$ have the same distribution, deduce the pointwise limit of the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$

We assume in this question that $X ( \Omega )$ is a countable set. We denote $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ where the $x _ { n }$ are pairwise distinct. For all $n \in \mathbb { N }$, we set $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Show that $\phi _ { X }$ is defined on $\mathbb { R }$ and that, for all real $t , \phi _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \mathrm { e } ^ { \mathrm { i } t x _ { n } }$.

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 $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Show that $E(g(X)) = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} E\left(g\left(p_k^{\nu_{p_k}(X)}\right)\right)$.

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 $s$.
Show that if $p$ is a prime number such that $p \equiv 1 [4]$, we have $$E\left(g\left(p^{\nu_p(X)}\right)\right) = \frac{1}{1 - p^{-s}}.$$

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 $s$.
Calculate $E\left(g\left(p^{\nu_p(X)}\right)\right)$ if $p$ is a prime number satisfying $p \equiv 3 [4]$.

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 $$E(g(X)) = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \frac{1}{1 - \chi_4\left(p_k\right) p_k^{-s}}.$$

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 recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$.
Show that, if $p$ is a prime number, $$E\left(\chi_4\left(p^{\nu_p(X)}\right)\right) = \frac{1 - p^{-s}}{1 - \chi_4(p) p^{-s}}.$$

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 recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$.
Show that $$E\left(\chi_4(X)\right) = \frac{1}{\zeta(s)} \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \frac{1}{1 - \chi_4\left(p_k\right) p_k^{-s}}.$$

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show, for all real $u$, the inequality
$$\left| e ^ { i u } - 1 - i u + \frac { u ^ { 2 } } { 2 } \right| \leq \frac { | u | ^ { 3 } } { 6 }$$
Deduce that for all real $\theta$,
$$\left| \Phi _ { Y } ( \theta ) - 1 + \frac { \mathbf { E } \left( Y ^ { 2 } \right) \theta ^ { 2 } } { 2 } \right| \leq \frac { | \theta | ^ { 3 } } { 3 } \left( \mathbf { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$

Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\delta _ { n } = \operatorname { det } \left( M _ { n } \right)$.
Calculate the expectation of the variable $\delta _ { n }$.

Let $S$ and $T$ be two independent random variables each taking a finite number of real values. Assume that $T$ and $- T$ follow the same distribution.
Show that:
$$E ( \cos ( S + T ) ) = E ( \cos ( S ) ) E ( \cos ( T ) )$$

Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Deduce that for all $n \in \mathbf { N } ^ { * }$, and for all $t \in \mathbf { R }$:
$$E \left( \cos \left( t S _ { n } \right) \right) = ( \cos ( t ) ) ^ { n } .$$