We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set
$$\forall n \in \mathbb{N}^{*}, \quad S_{n} = X_{1} + \cdots + X_{n}$$
VI.A.1) By induction, prove that, for every integer $n \in \mathbb{N}$, $S_{n}$ follows the Poisson distribution with parameter $n\lambda$.
We will admit that, for every integer $n \in \mathbb{N}^{*}$, the variables $S_{n}$ and $X_{n+1}$ are mutually independent.
VI.A.2) Let $\varepsilon \in \mathbb{R}_{+}^{*}$. Prove that
$$\forall n \in \mathbb{N}^{*}, \quad P\left(\left|S_{n} - n\lambda\right| \geqslant n\varepsilon\right) \leqslant \frac{\lambda}{n\varepsilon^{2}}$$
VI.A.3) Let $\varepsilon > 0$. Justify the following two inclusions
$$\begin{aligned} & \left(S_{n} > n(\lambda+\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \\ & \left(S_{n} \leqslant n(\lambda-\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \end{aligned}$$
VI.A.4) In all the following questions, we assume $x \geqslant 0$.
Deduce from VI.A.3 that
$$\begin{cases} \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 0 & \text{if } 0 \leqslant x < \lambda \\ \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 1 & \text{if } x > \lambda \end{cases}$$