In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
a) Show that $\mathbb{E}\left(\mathrm{e}^{X}\right) \leqslant \cosh(c)$.
b) Deduce that $\forall t \in \mathbb{R}^{+*}, \Psi(t) \leqslant \cosh(ct)$.

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. The functions $\Psi$ and $f_{\varepsilon}$ are defined on $\mathbb{R}$, with $f_{\varepsilon}(t) = \mathrm{e}^{-\varepsilon t}\Psi(t)$ (since $m=0$).
Show that $\forall t \in \mathbb{R}^{+*}, f_{\varepsilon}(t) \leqslant \exp\left(-t\varepsilon+\frac{1}{2}c^{2}t^{2}\right)$.

Let $k$ be a strictly positive integer and $U_{1}, \ldots, U_{k}$ a sequence of $k$ random variables taking values in $\{-1,1\}$, independent and uniformly distributed. We also denote $$S_{k} = \sum_{i=1}^{k} U_{i}$$
Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be the function defined by $\varphi(\lambda) = \ln\left(\mathbb{E}\left[e^{\lambda U_{1}}\right]\right)$. Establish that $$\forall \lambda \in \mathbb{R}, \quad \varphi(\lambda) \leqslant \frac{\lambda^{2}}{2}.$$

Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$. We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Deduce that, for every real number $t$, $$\mathbb { E } ( \exp ( t \langle X \mid Y \rangle ) ) \leqslant \exp \left( \frac { t ^ { 2 } } { 2 n } \right)$$

Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that: for all $t \geq 0$, for all $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$, $$\mathbf { E } \left( \exp \left( t \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) \right) \leq \exp \left( \frac { t ^ { 2 } } { 2 } \sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } \right)$$