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 $S(\omega)=S_{T(\omega)}(\omega)$.
Show that, if $X$ and $Y$ are two independent random variables taking values in $\mathbb{N}$, then $G_{X+Y}=G_X G_Y$.

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 $S(\omega)=S_{T(\omega)}(\omega)$.
By admitting that, for all $k\in\mathbb{N}$, $S_k$ is independent of $X_{k+1}$, prove that, for all $k\in\mathbb{N}$, $G_{S_k}=(G_X)^k$.

Let $X$ be a random variable taking values in $\mathbb{N}$ admitting a decomposition $X \sim Y + Z$, where $Y$ and $Z$ are independent random variables taking values in $\mathbb{N}$. What relation links $G_{X}, G_{Y}$ and $G_{Z}$?

Let $X$ and $Y$ be two independent random variables, defined on the space $\Omega$ and taking values in $\mathbb{N}$, and let $H_{X}$ and $H_{Y}$ be their auxiliary power series. Show $H_{X+Y}(t) = H_{X}(t) + H_{Y}(t)$ for all real $t$ satisfying $|t| < \min(\rho(X), \rho(Y))$.

Let $X$ be a random variable taking values in $\mathbb{N}$ admitting a decomposition $X \sim Y + Z$, where $Y$ and $Z$ are independent random variables taking values in $\mathbb{N}$. What relation links $G_{X}, G_{Y}$ and $G_{Z}$?

Let $X$ and $Y$ be two independent random variables, defined on the space $\Omega$ and taking values in $\mathbb{N}$, and let $H_{X}$ and $H_{Y}$ be their auxiliary power series. Show $H_{X+Y}(t) = H_{X}(t) + H_{Y}(t)$ for all real $t$ satisfying $|t| < \min(\rho(X), \rho(Y))$.

Justify that $\forall t \in \mathbb{R}, G_{S_n}(t) = \left(G_{X_1}(t)\right)^n$.