a) Suppose that $X$ is bounded. Justify that $X$ satisfies $(C_{\tau})$ for all $\tau$ in $\mathbb{R}^{+*}$.
b) Suppose that $X$ follows the geometric distribution with parameter $p \in ]0,1[$ $$\forall k \in \mathbb{N}^{*}, \quad P(X = k) = p(1-p)^{k-1}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.
c) Suppose that $X$ follows the Poisson distribution with parameter $\lambda$: $$\forall k \in \mathbb{N}, \quad P(X = k) = \mathrm{e}^{-\lambda} \frac{\lambda^{k}}{k!} \quad \text{where } \lambda \in \mathbb{R}^{+*}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.

Let $a$ and $b$ be two real numbers such that $a < b$. Suppose $E\left(\mathrm{e}^{aX}\right) < +\infty$ and $E\left(\mathrm{e}^{bX}\right) < +\infty$.
a) Show $\forall t \in [a,b]$, $\mathrm{e}^{tX} \leqslant \mathrm{e}^{aX} + \mathrm{e}^{bX}$. Deduce that $E\left(\mathrm{e}^{tX}\right) < +\infty$. What can we conclude about the set $\left\{t \in \mathbb{R} ; E\left(\mathrm{e}^{tX}\right) < +\infty\right\}$?
b) Let $k$ be in $\mathbb{N}$, $t$ in $]a,b[$. We denote by $\theta_{k,t,a,b}$ the function $y \in \mathbb{R} \mapsto \frac{y^{k} \mathrm{e}^{ty}}{\mathrm{e}^{ay} + \mathrm{e}^{by}}$. Determine the limits of $\theta_{k,t,a,b}$ at $+\infty$ and $-\infty$. Show that this function is bounded on $\mathbb{R}$.
c) Show that $E\left(|X|^{k} \mathrm{e}^{tX}\right) < +\infty$.
d) We return to the notations of question b). Let $k$ be in $\mathbb{N}$, $c$ and $d$ be two real numbers such that $a < c < d < b$. Show that there exists $M_{k,a,b,c,d} \in \mathbb{R}^{+}$ such that for all $t \in [c,d]$ and for all $y \in \mathbb{R}$: $\left|\theta_{k,t,a,b}(y)\right| \leqslant M_{k,a,b,c,d}$.

In this question, $\tau$ is an element of $\mathbb{R}^{+*}$ and $X$ satisfies $(C_{\tau})$.
a) Show that the set of real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$ is an interval $I$ containing $[-\tau, \tau]$. For $t$ in $I$, we denote $\varphi_{X}(t) = E\left(\mathrm{e}^{tX}\right)$.
b) Show that if $X(\Omega)$ is finite, $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
c) Suppose now that $X(\Omega)$ is a countably infinite set. We denote $X(\Omega) = \left\{x_{n} ; n \in \mathbb{N}^{*}\right\}$ where $\left(x_{n}\right)_{n \in \mathbb{N}^{*}}$ is a sequence of pairwise distinct real numbers and we set for all $n \in \mathbb{N}^{*}$, $p_{n} = P\left(X = x_{n}\right)$. Using the results established in question I.A.3 and two theorems relating to series of functions which you will state completely, show that $\varphi_{X}$ is continuous on $I$ and of class $C^{\infty}$ on the interior of $I$.
d) Verify that for $t$ in the interior of $I$ and $k$ in $\mathbb{N}$, $\varphi_{X}^{(k)}(t) = E\left(X^{k} \mathrm{e}^{tX}\right)$.
e) Let $\psi_{X} = \frac{\varphi_{X}^{\prime}}{\varphi_{X}}$. Show that $\psi_{X}$ is increasing on $I$ and that, if $X$ is not almost surely equal to a constant, $\psi_{X}$ is strictly increasing on $I$.

Let $\alpha$ be a strictly positive real and $X$ a discrete random variable admitting an exponential moment of order $\alpha$. Show that the random variable $e^{\alpha X}$ has finite expectation.

For each of the following real random variables, determine the strictly positive reals $\alpha$ such that the random variable admits an exponential moment of order $\alpha$ and calculate $\mathbb{E}\left(\mathrm{e}^{\alpha X}\right)$ in this case.
a) $X$ a random variable following a Poisson distribution with parameter $\lambda$, where $\lambda$ is a strictly positive real.
b) $Y$ a random variable following a geometric distribution with parameter $p$, where $p$ is a real strictly between 0 and 1.
c) $Z$ a random variable following a binomial distribution with parameters $n$ and $p$, where $n$ is a strictly positive integer and $p$ is a real strictly between 0 and 1.

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$.
Show that for every real $t$ belonging to the segment $[-\alpha, \alpha]$ and every $n$ belonging to $\mathbb{N}^{*}$, the real random variable $\mathrm{e}^{t S_{n}}$ has expectation equal to $(\Psi(t))^{n}$.

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$, and $f_{\varepsilon}(t) = \mathrm{e}^{-(m+\varepsilon)t}\Psi(t)$.
a) Let $t$ be a real belonging to the interval $]0, \alpha]$ and let $n$ belong to $\mathbb{N}^{*}$. Show that $\mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right)=\mathbb{P}\left(\mathrm{e}^{t S_{n}} \geqslant\left(\mathrm{e}^{t(m+\varepsilon)}\right)^{n}\right)$, then that $\mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right) \leqslant\left(f_{\varepsilon}(t)\right)^{n}$.
b) Deduce that there exists a real $r$ belonging to the interval $]0,1[$ such that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right) \leqslant r^{n}$.

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real, and $m = \mathbb{E}(X)$.
Show that the sequence defined by: $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}-m\right| \geqslant \varepsilon\right)$ is bounded above by a sequence with limit zero and whose convergence rate is geometric. Compare this result to the upper bound obtained with the weak law of large numbers.

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 $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
Show that the random variable $X$ admits an exponential moment of order $\alpha$ for every strictly positive real number $\alpha$.

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 $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
We consider $Y$ the real random variable defined by $Y=\frac{1}{2}-\frac{X}{2c}$.
a) Verify that $X=-cY+(1-Y)c$.
b) Show that $\mathrm{e}^{X} \leqslant Y \mathrm{e}^{-c}+(1-Y) \mathrm{e}^{c}$.

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 $\forall t \in \mathbb{R}^{+*}, f_{\varepsilon}(t) \leqslant \exp\left(-t\varepsilon+\frac{1}{2}c^{2}t^{2}\right)$.

We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Show conversely that, if there exists a real $R > 0$ such that, for all $t \in ] - R , R [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation, then the domain of definition of the moment generating function of $X$ contains $] - R , R [$ and for all $t \in ] - R , R \left[ , M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right) \right.$.

We assume that $X$ and $Y$ are two independent discrete real-valued random variables with strictly positive values admitting moments of all orders. We denote $R _ { X }$ (respectively $R _ { Y }$) the radius of convergence (assumed strictly positive) associated with the function $M _ { X }$ (respectively $M _ { Y }$).
Show that the random variable $X + Y$ admits moments of all orders and that $$\forall | t | < \min \left( R _ { X } , R _ { Y } \right) , \quad M _ { X + Y } ( t ) = M _ { X } ( t ) \times M _ { Y } ( t )$$

For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.
Calculate the moment generating function of the random variable $Y _ { n }$.

For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.
For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { Y _ { n } } ( t )$.

Let $n$ be a non-zero natural number and $t$ be a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_X(t) = \mathbb{E}(\mathrm{e}^{\mathrm{i}tX})$.
Show $$\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right).$$

Let $n$ be a non-zero natural number and $t$ be a real number. Using the result of Q1, deduce $$\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}.$$

Let $n$ be a non-zero natural number and $t$ be a real number. We have $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$ and $\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}$.
Determine the pointwise limit of the sequence of functions $\left(\Phi_{X_n}\right)_{n \geqslant 1}$.

Study the continuity of $\lim_{n \rightarrow +\infty} \Phi_{X_n}$.

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 $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$

Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$ converge uniformly on $\mathbb{R}$?

Let $Z$ be a discrete real random variable such that $\exp ( \lambda Z )$ has finite expectation for all $\lambda > 0$. Show that for all $\lambda > 0$ and $t \in \mathbb { R }$, $$P [ Z \geqslant t ] \leqslant \exp ( - \lambda t ) E [ \exp ( \lambda Z ) ] .$$

Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ as well as, for all $\lambda \in \mathbb { R }$, $$\psi ( \lambda ) = \log \left( \frac { 1 } { 2 } e ^ { \lambda } + \frac { 1 } { 2 } e ^ { - \lambda } \right)$$ Show that for all $t \in \mathbb { R }$, we have $$\frac { 1 } { n } \log P \left[ S _ { n } \geqslant t \right] \leqslant \inf _ { \lambda \geqslant 0 } ( \psi ( \lambda ) - \lambda t )$$

Let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ Show that the function $m$ is strictly increasing on $\mathbb{R}_{\geqslant 0}$, and that for all $t \in [0,1]$, there exists a unique $\lambda \geqslant 0$ such that $m(\lambda) = t$.