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 $[-\alpha, \alpha]$. We consider the function $f_{\varepsilon}$ defined by $$f_{\varepsilon}:\left\{\begin{array}{l}[-\alpha, \alpha] \rightarrow \mathbb{R}^{+} \\ t \mapsto \mathrm{e}^{-(m+\varepsilon) t} \Psi(t)\end{array}\right.$$ a) Give the values of $f_{\varepsilon}(0)$ and $f_{\varepsilon}^{\prime}(0)$. b) Deduce that there exists a real $t_{0}$ belonging to the interval $]0, \alpha[$ satisfying $0 < f_{\varepsilon}\left(t_{0}\right) < 1$.
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 $[-\alpha, \alpha]$.
We consider the function $f_{\varepsilon}$ defined by
$$f_{\varepsilon}:\left\{\begin{array}{l}[-\alpha, \alpha] \rightarrow \mathbb{R}^{+} \\ t \mapsto \mathrm{e}^{-(m+\varepsilon) t} \Psi(t)\end{array}\right.$$
a) Give the values of $f_{\varepsilon}(0)$ and $f_{\varepsilon}^{\prime}(0)$.
b) Deduce that there exists a real $t_{0}$ belonging to the interval $]0, \alpha[$ satisfying $0 < f_{\varepsilon}\left(t_{0}\right) < 1$.