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 $S_{n} = X_{1} + \cdots + X_{n}$. Using question VI.A, show that $$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \begin{cases} 0 & \text{if } 0 \leqslant x < \lambda \\ 1 & \text{if } x > \lambda \end{cases}$$
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 $S_{n} = X_{1} + \cdots + X_{n}$.
Using question VI.A, show that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \begin{cases} 0 & \text{if } 0 \leqslant x < \lambda \\ 1 & \text{if } x > \lambda \end{cases}$$