Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
For $K\in\mathbb{N}$ and $t\in\left[0,1\left[$, we set $R_K=\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$.
Show that $0\leqslant R_K\leqslant\frac{1}{1-t}\sum_{k=K+1}^\infty P(T=k)$.

Let $X$ and $Y$ be two random variables defined on $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in $\mathbb{N}$.
a) Show that if $A$ and $B$ are events in $\mathcal{A}$, and if $\bar{A}$ and $\bar{B}$ are their respective complementary events, then $$|\mathbb{P}(A) - \mathbb{P}(B)| \leqslant \mathbb{P}(A \cap \bar{B}) + \mathbb{P}(\bar{A} \cap B)$$
b) Deduce that, for all $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.

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 $X$ and $Y$ be two random variables defined on $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in $\mathbb{N}$.
a) Show that if $A$ and $B$ are events in $\mathcal{A}$, and if $\bar{A}$ and $\bar{B}$ are their respective complementary events, then $$|\mathbb{P}(A) - \mathbb{P}(B)| \leqslant \mathbb{P}(A \cap \bar{B}) + \mathbb{P}(\bar{A} \cap B)$$
b) Deduce that, for all $t \in [-1,1], \left|G_{X}(t) - G_{Y}(t)\right| \leqslant 2\mathbb{P}(X \neq Y)$.

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 $[-1,1]$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a sequence $\left( C _ { k } \right) _ { k \in \mathbf { N } }$ of strictly positive reals, independent of $p$, such that
$$\forall k \in \mathbf { N } , \left| \mathbf { E } \left( X ^ { k } \right) - \frac { 1 } { p ^ { k } } \right| \leq \frac { C _ { k } q } { p ^ { k } }$$