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 random variables $(X_{n,i})_{(n,i)\in\mathbb{N}\times\mathbb{N}^*}$ independent and all following distribution $\mu$, with $Y_0=1$ and $$\begin{cases} Y_{n+1}(\omega)=0 & \text{if }Y_n(\omega)=0\\ Y_{n+1}(\omega)=\sum_{i=1}^{Y_n(\omega)}X_{n,i}(\omega) & \text{if }Y_n(\omega)\neq 0\end{cases}$$ We denote by $f$ the generating function of $\mu$ and, for $n\in\mathbb{N}$, $\varphi_n$ the generating function of $Y_n$. We have $\varphi_0(t)=t$ for $t\in[0,1]$.
Show that, for all $n\in\mathbb{N}$, $\varphi_{n+1}=\varphi_n\circ f$.

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 $m$ the expectation of $\mu$.
If $m\leqslant 1$, show that the probability of extinction is equal to 1.

We consider the Galton-Watson process with extinction time $T$ defined by: $$\omega\in\Omega\quad\begin{cases}T(\omega)=\min\{n\in\mathbb{N}\mid Y_n(\omega)=0\} & \text{if there exists }n\in\mathbb{N}\text{ such that }Y_n(\omega)=0\\ T(\omega)=-1 & \text{otherwise}\end{cases}$$
We assume $m<1$. Verify that $T$ has a finite expectation.

We consider the Galton-Watson process with extinction time $T$. We assume $m<1$.
a) Show that, for all integer $n$, $P(Y_n\geqslant 1)\leqslant m^n$.
b) Show that $E(T)=\sum_{n=0}^{+\infty}P(T>n)$.
c) Deduce an upper bound for $E(T)$.

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 admit that $Z$ is a random variable defined on $\bigcup_{k\in\mathbb{N}}\{Y_k=0\}$.
Show that $Z$ is defined on a set of probability 1.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Verify that, for all $t\in\left[0,1\left[$, $\varphi_n(t)\neq 1$.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)\neq 1$ for $t\in[0,1[$, so we can set $a_n(t)=\frac{1}{\varphi_n(t)-1}$.
Show that, for $t\in\left[0,1\left[$, the sequence $(a_n(t))_{n\in\mathbb{N}}$ is arithmetic.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Deduce that, for $t\in\left[0,1\left[$ and $n\in\mathbb{N}$, $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$. The extinction time $T$ is defined as before.
Express, in terms of $n\in\mathbb{N}^*$, the probability of the event $T>n$.
Does the variable $T$ have a finite expectation?

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that $P(W_1>k)>0$.

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ does not take the value $k$ is non-zero; we denote this probability by $u$.
One may study separately the cases $p_0=0$ and $p_0>0$.