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 $S(\omega)=S_{T(\omega)}(\omega)$.
Conclude that $G_S=G_T\circ G_X$.

We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We denote by $f$ the generating function of $\mu$ and $m$ the expectation of $\mu$.
a) Express $G_{Z_1}$ in terms of $f$.
b) We admit that, for all natural integer $n$ greater than or equal to 2 and for all $s\in[0,1]$, $G_{Z_n}(s)=sf(G_{Z_{n-1}}(s))$.
Deduce that, for all $s\in\left[0,1\left[$, $G_Z(s)=sf(G_Z(s))$.
c) Show that $Z$ has finite expectation if and only if $m<1$. Calculate the expectation when this is the case.

We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ Show that $$\forall x \in ]-1,1[ , \quad F ( x ) = 1 + F ( x ) G ( x ) .$$ Determine the limit of $F ( x )$ as $x$ tends to $1^{-}$, discussing according to the value of $P ( R \neq + \infty )$.