Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We denote by $f$ the generating function of $\mu$ and $\varphi_n$ the generating function of $Y_n$. a) Verify that the probability of extinction is equal to the limit of the sequence $(\varphi_n(0))_{n\geqslant 0}$. b) Verify that we can apply the results of Part I to the sequence $(\varphi_n(0))_{n\geqslant 0}$.
Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We denote by $f$ the generating function of $\mu$ and $\varphi_n$ the generating function of $Y_n$.
a) Verify that the probability of extinction is equal to the limit of the sequence $(\varphi_n(0))_{n\geqslant 0}$.
b) Verify that we can apply the results of Part I to the sequence $(\varphi_n(0))_{n\geqslant 0}$.