Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
Verify that $f$ is an application from $I$ to $\mathbf{R}$ that is log-convex.

By using the Cauchy-Schwarz inequality, show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) \leq \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} P_{t}\left(\frac{f^{\prime 2}}{f}\right)(x) \varphi(x) \mathrm{d}x$$

By using the Cauchy-Schwarz inequality, show that $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} P_t\!\left(\frac{f'^2}{f}\right)(x)\,\varphi(x)\,\mathrm{d}x.$$

Let $p , q \in ] 1 , + \infty [$ such that $\frac { 1 } { p } + \frac { 1 } { q } = 1$. Let $X , Y \in L ^ { 0 } ( \Omega )$ which we assume are both non-negative. Deduce the following inequality (Hölder's inequality): $$\mathbf { E } ( X Y ) \leq \left( \mathrm { E } \left( X ^ { p } \right) \right) ^ { 1 / p } \left( \mathrm { E } \left( Y ^ { q } \right) \right) ^ { 1 / q } .$$ You may begin by treating the case where $\mathbf { E } \left( X ^ { p } \right) = \mathbf { E } \left( Y ^ { q } \right) = 1$.

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose in this question that $\sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } = 1$. Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$ converges, then that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 4 } \right) \leq 8 \int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Deduce that there exists a real $\beta _ { p } > 0$ such that $$\mathbf { E } \left( \left| \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right| ^ { p } \right) ^ { 1 / p } \leq \beta _ { p } \mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 2 } \right) ^ { 1 / 2 } .$$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Assume $1 \leq p < 2$. Justify that there exists $\theta \in ] 0,1 [$ such that $\frac { 1 } { 2 } = \frac { \theta } { p } + \frac { 1 - \theta } { 4 }$.

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Let the map $\psi : u \in \mathbf { R } ^ { ( \mathbf { N } ) } \mapsto \sum _ { i = 0 } ^ { + \infty } u _ { i } X _ { i }$. We denote $R = \psi \left( \mathbf { R } ^ { ( \mathbf { N } ) } \right)$. Show that for all $p , q \in \left[ 1 , + \infty \right[$, the norms $\| \cdot \| _ { p }$ and $\| \cdot \| _ { q }$ are equivalent on $R$.

In this part, we assume that $n$ is a power of 2: we write $n = 2 ^ { k }$ with $k \in \mathbf { N } ^ { \star }$. Let $\left( a _ { 1 } , \ldots , a _ { k } \right) \in \mathbf { R } ^ { k }$. Show that $$\alpha _ { 1 } n \left\| \left( a _ { 1 } , \ldots , a _ { k } \right) \right\| _ { 2 } ^ { \mathbf { R } ^ { k } } \leq \sum _ { \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { k } \right) \in \{ - 1,1 \} ^ { k } } \left| \sum _ { i = 1 } ^ { k } \varepsilon _ { i } a _ { i } \right| \leq \beta _ { 1 } n \left\| \left( a _ { 1 } , \ldots , a _ { k } \right) \right\| _ { 2 } ^ { \mathbf { R } ^ { k } } .$$ You may use questions 11 and 16.

We assume that the random variables $X_1, \ldots, X_N$ are pairwise uncorrelated, that is: $$\forall 1 \leq m, n \leq N, \quad n \neq m \Rightarrow \mathbb{E}[X_n X_m] = 0.$$ Prove that $$\mathbb{E}\left[|S_N|^2\right] \leq N.$$ Deduce that, for all $t > 0$, $$\mathbb{P}\left(|S_N| > t\sqrt{N}\right) \leq \frac{1}{t^2}$$ where $S_N := X_1 + \cdots + X_N$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$.
Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.

Let $x_1, x_2, \ldots, x_n$ be non-negative real numbers such that $\sum_{i=1}^{n} x_i = 1$. What is the maximum possible value of $\sum_{i=1}^{n} \sqrt{x_i}$?
(A) 1
(B) $\sqrt{n}$
(C) $n^{3/4}$
(D) $n$

Let $x _ { 1 } , \ldots , x _ { 2024 }$ be non-negative real numbers with $\sum _ { i = 1 } ^ { 2024 } x _ { i } = 1$. Find, with proof, the minimum and maximum possible values of the expression
$$\sum _ { i = 1 } ^ { 1012 } x _ { i } + \sum _ { i = 1013 } ^ { 2024 } x _ { i } ^ { 2 }$$