Let $\left(U_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of mutually independent random variables taking values in $\mathbb{N}$ such that the series of $\mathbb{P}\left(U_{i} \neq 0\right)$ is convergent. a) Let $Z_{n} = \left\{\omega \in \Omega \mid \exists i \geqslant n, U_{i}(\omega) \neq 0\right\}$. Show that $(Z_{n})$ is a decreasing sequence of events and that $\lim_{n \rightarrow \infty} \mathbb{P}\left(Z_{n}\right) = 0$. b) Deduce that the set $\left\{i \in \mathbb{N}^{*} \mid U_{i} \neq 0\right\}$ is almost surely finite. c) We set $S_{n} = \sum_{i=1}^{n} U_{i}$ and $S = \sum_{i=1}^{\infty} U_{i}$. Justify that $S$ is defined almost surely. Show that $G_{S_{n}}$ converges uniformly to $G_{S}$ on $[-1,1]$.
Let $\left(U_{i}\right)_{i \in \mathbb{N}^{*}}$ be a sequence of mutually independent random variables taking values in $\mathbb{N}$ such that the series of $\mathbb{P}\left(U_{i} \neq 0\right)$ is convergent.
a) Let $Z_{n} = \left\{\omega \in \Omega \mid \exists i \geqslant n, U_{i}(\omega) \neq 0\right\}$. Show that $(Z_{n})$ is a decreasing sequence of events and that $\lim_{n \rightarrow \infty} \mathbb{P}\left(Z_{n}\right) = 0$.
b) Deduce that the set $\left\{i \in \mathbb{N}^{*} \mid U_{i} \neq 0\right\}$ is almost surely finite.
c) We set $S_{n} = \sum_{i=1}^{n} U_{i}$ and $S = \sum_{i=1}^{\infty} U_{i}$. Justify that $S$ is defined almost surely. Show that $G_{S_{n}}$ converges uniformly to $G_{S}$ on $[-1,1]$.