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 $n\in\mathbb{N}$, $T$ and $S_n$ are independent, show that $$\forall t\in\left[0,1\left[,\forall K\in\mathbb{N}\quad G_S(t)=\sum_{k=0}^K P(T=k)\left(G_X(t)\right)^k+\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$$
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 $n\in\mathbb{N}$, $T$ and $S_n$ are independent, show that
$$\forall t\in\left[0,1\left[,\forall K\in\mathbb{N}\quad G_S(t)=\sum_{k=0}^K P(T=k)\left(G_X(t)\right)^k+\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$$