The interval $I$ and the function $\varphi_{X}$ are defined as in question I.A.4. We suppose that $X$ satisfies $(C_{\tau})$ for some $\tau > 0$, is not almost surely constant, and $a > E(X)$. We define the function $\chi : \begin{aligned} & I \rightarrow \mathbb{R} \\ & t \mapsto \ln\left(\varphi_{X}(t)\right) - ta \end{aligned}$ a) Show that the function $\chi$ is bounded below on $I \cap \mathbb{R}^{+}$. We denote by $\eta_{a}$ the infimum of $\chi$ on $I \cap \mathbb{R}^{+}$. b) Give an equivalent of $\chi(t)$ as $t$ tends to 0. Deduce that $\eta_{a} < 0$. c) Show $\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\eta_{a}}$. Deduce that $\gamma_{a} < 0$. d) In each of the following two cases, determine the set of real numbers $a$ satisfying the conditions $P(X \geqslant a) > 0$ and $a > E(X)$; then, for $a$ satisfying these conditions, calculate $\eta_{a}$. i. $X$ follows the Bernoulli distribution $\mathcal{B}(p)$ with $0 < p < 1$. ii. $X$ follows the Poisson distribution $\mathcal{P}(\lambda)$ with $\lambda > 0$.
The interval $I$ and the function $\varphi_{X}$ are defined as in question I.A.4. We suppose that $X$ satisfies $(C_{\tau})$ for some $\tau > 0$, is not almost surely constant, and $a > E(X)$. We define the function $\chi : \begin{aligned} & I \rightarrow \mathbb{R} \\ & t \mapsto \ln\left(\varphi_{X}(t)\right) - ta \end{aligned}$
a) Show that the function $\chi$ is bounded below on $I \cap \mathbb{R}^{+}$. We denote by $\eta_{a}$ the infimum of $\chi$ on $I \cap \mathbb{R}^{+}$.
b) Give an equivalent of $\chi(t)$ as $t$ tends to 0. Deduce that $\eta_{a} < 0$.
c) Show $\forall n \in \mathbb{N}^{*}, \quad P\left(S_{n} \geqslant na\right) \leqslant \mathrm{e}^{n\eta_{a}}$. Deduce that $\gamma_{a} < 0$.
d) In each of the following two cases, determine the set of real numbers $a$ satisfying the conditions $P(X \geqslant a) > 0$ and $a > E(X)$; then, for $a$ satisfying these conditions, calculate $\eta_{a}$.
i. $X$ follows the Bernoulli distribution $\mathcal{B}(p)$ with $0 < p < 1$.
ii. $X$ follows the Poisson distribution $\mathcal{P}(\lambda)$ with $\lambda > 0$.