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 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.